From ff03d3bdecab9ed95c94adaadbe3e8be68b94d97 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Wed, 1 Apr 2026 22:22:47 +0800
Subject: [PATCH 01/69] add two lemmas

---
 Mathlib/RingTheory/Ideal/Height.lean             | 3 +++
 Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean | 3 +++
 2 files changed, 6 insertions(+)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index 28ae520c1cff57..a6cf6647278e54 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -197,6 +197,9 @@ lemma Ideal.primeHeight_eq_zero_iff {I : Ideal R} [I.IsPrime] :
   · rintro ⟨hI, hI'⟩ b hb
     exact hI' (y := b.asIdeal) b.isPrime hb
 
+lemma Ideal.height_eq_zero_iff {I : Ideal R} [I.IsPrime] : height I = 0 ↔ I ∈ minimalPrimes R := by
+  rw [Ideal.height_eq_primeHeight, Ideal.primeHeight_eq_zero_iff]
+
 @[simp]
 lemma Ideal.height_bot [Nontrivial R] : (⊥ : Ideal R).height = 0 := by
   obtain ⟨p, hp⟩ := Ideal.nonempty_minimalPrimes (R := R) (I := ⊥) top_ne_bot.symm
diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
index 6338b28d7d2623..8c16364ef9a978 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
@@ -54,6 +54,9 @@ variable {I J}
 theorem Ideal.minimalPrimes_isPrime {p : Ideal R} (h : p ∈ I.minimalPrimes) : p.IsPrime :=
   h.1.1
 
+theorem minimalPrimes_isPrime {p : Ideal R} (h : p ∈ minimalPrimes R) : p.IsPrime :=
+  h.1.1
+
 theorem Ideal.exists_minimalPrimes_le [J.IsPrime] (e : I ≤ J) : ∃ p ∈ I.minimalPrimes, p ≤ J := by
   set S := { p : (Ideal R)ᵒᵈ | Ideal.IsPrime p ∧ I ≤ OrderDual.ofDual p }
   suffices h : ∃ m, OrderDual.toDual J ≤ m ∧ Maximal (· ∈ S) m by

From 1f742557f1d7946e5462c0858b0ae45267d0252e Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Wed, 1 Apr 2026 23:50:31 +0800
Subject: [PATCH 02/69] reduce .1.1

---
 .../RingTheory/Ideal/AssociatedPrime/Localization.lean    | 2 +-
 Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean         | 8 ++++----
 2 files changed, 5 insertions(+), 5 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.lean b/Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.lean
index 37909f3465e6de..408ebe65b172f3 100644
--- a/Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.lean
+++ b/Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.lean
@@ -131,7 +131,7 @@ variable (R M) in
 lemma minimalPrimes_annihilator_subset_associatedPrimes [IsNoetherianRing R] [Module.Finite R M] :
     (Module.annihilator R M).minimalPrimes ⊆ associatedPrimes R M := by
   intro p hp
-  have prime := hp.1.1
+  have prime := Ideal.minimalPrimes_isPrime hp
   let Rₚ := Localization.AtPrime p
   have : Nontrivial (LocalizedModule p.primeCompl M) := by
     simpa [← Module.mem_support_iff (p := ⟨p, prime⟩), Module.support_eq_zeroLocus] using hp.1.2
diff --git a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
index 0d273c352d3b7b..a8256e59729471 100644
--- a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
+++ b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
@@ -106,7 +106,7 @@ lemma Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes_of_isLocalRing
   has height at most 1. -/
 lemma Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes
     (I : Ideal R) [I.IsPrincipal] (p : Ideal R) (hp : p ∈ I.minimalPrimes) : p.height ≤ 1 := by
-  have := hp.1.1
+  have := Ideal.minimalPrimes_isPrime hp
   let f := algebraMap R (Localization.AtPrime p)
   have := Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes_of_isLocalRing (I.map f) ?_
   · rwa [← IsLocalization.height_comap p.primeCompl,
@@ -117,7 +117,7 @@ lemma Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes
 theorem Ideal.map_height_le_one_of_mem_minimalPrimes {I p : Ideal R} {x : R}
     (hp : p ∈ (I ⊔ span {x}).minimalPrimes) : (p.map (Ideal.Quotient.mk I)).height ≤ 1 :=
   let f := Ideal.Quotient.mk I
-  have : p.IsPrime := hp.1.1
+  have : p.IsPrime := Ideal.minimalPrimes_isPrime hp
   have hfp : RingHom.ker f ≤ p := I.mk_ker.trans_le (le_sup_left.trans hp.1.2)
   height_le_one_of_isPrincipal_of_mem_minimalPrimes ((span {x}).map f) (p.map f)
     ⟨⟨map_isPrime_of_surjective Quotient.mk_surjective hfp, map_mono (le_sup_right.trans hp.1.2)⟩,
@@ -176,7 +176,7 @@ nonrec lemma Ideal.height_le_spanRank_toENat_of_mem_minimal_primes
   induction hn : s.card using Nat.strong_induction_on generalizing R with
   | h n H =>
     replace hn : s.card ≤ n := hn.le
-    have := hp.1.1
+    have := Ideal.minimalPrimes_isPrime hp
     cases n with
     | zero =>
       rw [ENat.coe_zero, nonpos_iff_eq_zero, height_eq_primeHeight p,
@@ -239,7 +239,7 @@ lemma Ideal.height_le_spanRank_toENat (I : Ideal R) (hI : I ≠ ⊤) :
   obtain ⟨J, hJ⟩ := nonempty_minimalPrimes hI
   refine (iInf₂_le J hJ).trans ?_
   convert (I.height_le_spanRank_toENat_of_mem_minimal_primes J hJ)
-  exact Eq.symm (@height_eq_primeHeight _ _ J hJ.1.1)
+  exact Eq.symm (@height_eq_primeHeight _ _ J (Ideal.minimalPrimes_isPrime hJ))
 
 lemma Ideal.height_le_spanFinrank (I : Ideal R) (hI : I ≠ ⊤) :
     I.height ≤ I.spanFinrank := by

From 348b7158aa55be5e6740ec33415be6500e022dee Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Thu, 2 Apr 2026 00:17:15 +0800
Subject: [PATCH 03/69] add more heoght analogus

---
 Mathlib/RingTheory/Ideal/Height.lean | 61 +++++++++++++++-------------
 1 file changed, 33 insertions(+), 28 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index a6cf6647278e54..c8ee1f1bf9f2dd 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -213,10 +213,7 @@ lemma Ideal.height_of_subsingleton [Subsingleton R] : I.height = ⊤ := by
 
 theorem Ideal.isMaximal_of_primeHeight_eq_ringKrullDim {I : Ideal R} [I.IsPrime]
     [FiniteRingKrullDim R] (e : I.primeHeight = ringKrullDim R) : I.IsMaximal := by
-  have h : I ≠ ⊤ := by
-    intro h
-    simp only [h, ← Ideal.height_eq_primeHeight, Ideal.height_top, WithBot.coe_top] at e
-    exact ringKrullDim_ne_top e.symm
+  have h : I ≠ ⊤ := Ideal.IsPrime.ne_top'
   obtain ⟨M, hM, hM'⟩ := Ideal.exists_le_maximal I h
   rcases lt_or_eq_of_le hM' with (hM' | hM')
   · have h1 := Ideal.primeHeight_strict_mono hM'
@@ -224,6 +221,11 @@ theorem Ideal.isMaximal_of_primeHeight_eq_ringKrullDim {I : Ideal R} [I.IsPrime]
     simp [← not_lt, h1] at h2
   · exact hM' ▸ hM
 
+theorem Ideal.isMaximal_of_height_eq_ringKrullDim {I : Ideal R} [I.IsPrime]
+    [FiniteRingKrullDim R] (e : I.height = ringKrullDim R) : I.IsMaximal := by
+  rw [Ideal.height_eq_primeHeight] at e
+  exact Ideal.isMaximal_of_primeHeight_eq_ringKrullDim e
+
 /-- The prime height of the maximal ideal equals the Krull dimension in a local ring -/
 @[simp]
 theorem IsLocalRing.maximalIdeal_primeHeight_eq_ringKrullDim [IsLocalRing R] :
@@ -239,13 +241,17 @@ theorem IsLocalRing.maximalIdeal_height_eq_ringKrullDim [IsLocalRing R] :
 
 /-- For a local ring with finite Krull dimension, a prime ideal has height equal to the Krull
 dimension if and only if it is the maximal ideal. -/
-theorem Ideal.primeHeight_eq_ringKrullDim_iff [FiniteRingKrullDim R] [IsLocalRing R] {I : Ideal R}
-    [I.IsPrime] : Ideal.primeHeight I = ringKrullDim R ↔ I = IsLocalRing.maximalIdeal R := by
+theorem Ideal.height_eq_ringKrullDim_iff [FiniteRingKrullDim R] [IsLocalRing R] {I : Ideal R}
+    [I.IsPrime] : I.height = ringKrullDim R ↔ I = IsLocalRing.maximalIdeal R := by
   constructor
   · intro h
-    exact IsLocalRing.eq_maximalIdeal (Ideal.isMaximal_of_primeHeight_eq_ringKrullDim h)
+    exact IsLocalRing.eq_maximalIdeal (Ideal.isMaximal_of_height_eq_ringKrullDim h)
   · rintro rfl
-    exact IsLocalRing.maximalIdeal_primeHeight_eq_ringKrullDim
+    exact IsLocalRing.maximalIdeal_height_eq_ringKrullDim
+
+theorem Ideal.primeHeight_eq_ringKrullDim_iff [FiniteRingKrullDim R] [IsLocalRing R] {I : Ideal R}
+    [I.IsPrime] : Ideal.primeHeight I = ringKrullDim R ↔ I = IsLocalRing.maximalIdeal R := by
+  rw [← Ideal.height_eq_primeHeight, Ideal.height_eq_ringKrullDim_iff]
 
 lemma Ideal.height_le_iff {p : Ideal R} {n : ℕ} [p.IsPrime] :
     p.height ≤ n ↔ ∀ q : Ideal R, q.IsPrime → q < p → q.height < n := by
@@ -378,7 +384,7 @@ lemma exists_spanRank_le_and_le_height_of_le_height [IsNoetherianRing R] (I : Id
       refine (Ideal.subset_union_prime ⊥ ⊥ ?_).not.mpr ?_
       · rintro K hK - -
         rw [Set.Finite.mem_toFinset] at hK
-        exact hK.1.1.1
+        exact Ideal.minimalPrimes_isPrime hK.1
       · push Not
         intro K hK e
         have := hr.trans (Ideal.height_mono e)
@@ -394,25 +400,18 @@ lemma exists_spanRank_le_and_le_height_of_le_height [IsNoetherianRing R] (I : Id
       push_cast
       exact add_le_add h₂ ((Submodule.spanRank_span_le_card _).trans (by simp))
     · refine le_iInf₂ (fun p hp ↦ ?_)
-      have := hp.1.1
+      have := Ideal.minimalPrimes_isPrime hp
+      rw [← p.height_eq_primeHeight]
       by_cases h : p.height = ⊤
-      · rw [← p.height_eq_primeHeight, h]
-        exact le_top
+      · exact le_of_le_of_eq le_top h.symm
       have : p.FiniteHeight := ⟨Or.inr h⟩
       have := Ideal.height_mono (le_sup_left.trans hp.1.2)
-      suffices h : (r : ℕ∞) ≠ p.primeHeight by
-        push_cast
-        have := h₃.trans this
-        rw [Ideal.height_eq_primeHeight] at this
-        exact Order.add_one_le_of_lt (lt_of_le_of_ne this h)
+      suffices h : (r : ℕ∞) ≠ p.height by
+        exact Order.add_one_le_of_lt (lt_of_le_of_ne (h₃.trans this) h)
       intro e
       apply hx₂ p
-      · have : J.height = p.primeHeight := by
-          apply le_antisymm
-          · rwa [← p.height_eq_primeHeight]
-          · rwa [← e]
-        refine ⟨mem_minimalPrimes_of_primeHeight_eq_height (le_sup_left.trans hp.1.2) this.symm, ?_⟩
-        rwa [p.height_eq_primeHeight, eq_comm]
+      · have hteq : J.height = p.height := this.antisymm (le_of_eq_of_le e.symm h₃)
+        refine ⟨mem_minimalPrimes_of_height_eq (le_sup_left.trans hp.1.2) (ge_of_eq hteq), e.symm⟩
       · exact hp.1.2 <| Ideal.mem_sup_right <| Ideal.subset_span <| Set.mem_singleton x
 
 /-- In a nontrivial commutative ring `R`, the supremum of heights of all ideals is equal to the
@@ -451,16 +450,22 @@ lemma Ideal.sup_primeHeight_eq_ringKrullDim [Nontrivial R] :
 
 /-- In a nontrivial commutative ring `R`, the supremum of prime heights of all maximal ideals is
 equal to the Krull dimension of `R`. -/
-lemma Ideal.sup_primeHeight_of_maximal_eq_ringKrullDim [Nontrivial R] :
-    ↑(⨆ (I : Ideal R) (_ : I.IsMaximal), I.primeHeight) = ringKrullDim R := by
-  rw [← Ideal.sup_primeHeight_eq_ringKrullDim, WithBot.coe_inj]
+lemma Ideal.sup_height_of_maximal_eq_ringKrullDim [Nontrivial R] :
+    ↑(⨆ (I : Ideal R) (_ : I.IsMaximal), I.height) = ringKrullDim R := by
+  rw [← Ideal.sup_height_eq_ringKrullDim, WithBot.coe_inj]
   apply le_antisymm
-  · exact iSup_mono fun I => iSup_mono' fun hI => ⟨IsMaximal.isPrime hI, le_rfl⟩
+  · exact iSup_mono fun I => iSup_mono' fun hI => ⟨hI.isPrime.ne_top , le_rfl⟩
   · refine iSup_mono' fun I => ?_
     obtain rfl | I_top := eq_or_ne I ⊤
     · exact ⟨⊥, by grind [iSup_le_iff, Ideal.IsPrime.ne_top]⟩
     · obtain ⟨M, hM, hIM⟩ := exists_le_maximal I I_top
-      exact ⟨M, iSup_mono' (fun hI ↦ ⟨hM, primeHeight_mono hIM⟩)⟩
+      exact ⟨M, iSup_mono' (fun hI ↦ ⟨hM, height_mono hIM⟩)⟩
+
+/-- In a nontrivial commutative ring `R`, the supremum of prime heights of all maximal ideals is
+equal to the Krull dimension of `R`. -/
+lemma Ideal.sup_primeHeight_of_maximal_eq_ringKrullDim [Nontrivial R] :
+    ↑(⨆ (I : Ideal R) (_ : I.IsMaximal), I.primeHeight) = ringKrullDim R := by
+  simp_rw [← Ideal.height_eq_primeHeight, Ideal.sup_height_of_maximal_eq_ringKrullDim]
 
 section isLocalization
 

From c24b383bc6bebe6acfc7d57e71638332ed5f4372 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Thu, 2 Apr 2026 10:18:05 +0800
Subject: [PATCH 04/69] add le lemma

---
 Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean | 3 +++
 1 file changed, 3 insertions(+)

diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
index 8c16364ef9a978..f2ef9f2c52ac28 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
@@ -54,6 +54,9 @@ variable {I J}
 theorem Ideal.minimalPrimes_isPrime {p : Ideal R} (h : p ∈ I.minimalPrimes) : p.IsPrime :=
   h.1.1
 
+theorem Ideal.le_minimalPrimes {p : Ideal R} (h : p ∈ I.minimalPrimes) : I ≤ p :=
+  h.1.2
+
 theorem minimalPrimes_isPrime {p : Ideal R} (h : p ∈ minimalPrimes R) : p.IsPrime :=
   h.1.1
 

From a12d523d24d56f3e3840ff58daae8fb615796caa Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Thu, 2 Apr 2026 10:59:03 +0800
Subject: [PATCH 05/69] reduce more .1.1 and  .1.2

---
 .../Ideal/AssociatedPrime/Localization.lean   |  3 +-
 Mathlib/RingTheory/Ideal/Height.lean          | 22 ++++++-------
 .../RingTheory/Ideal/MinimalPrime/Basic.lean  | 17 +++++-----
 .../RingTheory/Ideal/MinimalPrime/Colon.lean  | 12 +++----
 .../Ideal/MinimalPrime/Localization.lean      | 33 ++++++++++---------
 5 files changed, 45 insertions(+), 42 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.lean b/Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.lean
index 408ebe65b172f3..2d07b507186198 100644
--- a/Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.lean
+++ b/Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.lean
@@ -134,7 +134,8 @@ lemma minimalPrimes_annihilator_subset_associatedPrimes [IsNoetherianRing R] [Mo
   have prime := Ideal.minimalPrimes_isPrime hp
   let Rₚ := Localization.AtPrime p
   have : Nontrivial (LocalizedModule p.primeCompl M) := by
-    simpa [← Module.mem_support_iff (p := ⟨p, prime⟩), Module.support_eq_zeroLocus] using hp.1.2
+    simpa [← Module.mem_support_iff (p := ⟨p, prime⟩), Module.support_eq_zeroLocus] using
+      Ideal.le_minimalPrimes hp
   rcases associatedPrimes.nonempty Rₚ (LocalizedModule p.primeCompl M) with ⟨q, hq⟩
   have q_prime : q.IsPrime := IsAssociatedPrime.isPrime hq
   simp only [← preimage_comap_associatedPrimes_eq_associatedPrimes_of_isLocalizedModule p.primeCompl
diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index c8ee1f1bf9f2dd..5b458b468e9a25 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -117,7 +117,7 @@ theorem Ideal.height_mono {I J : Ideal R} (h : I ≤ J) : I.height ≤ J.height
   simp only [height]
   refine le_iInf₂ (fun p hp ↦ ?_)
   have := Ideal.minimalPrimes_isPrime hp
-  obtain ⟨q, hq, e⟩ := Ideal.exists_minimalPrimes_le (h.trans hp.1.2)
+  obtain ⟨q, hq, e⟩ := Ideal.exists_minimalPrimes_le (h.trans (Ideal.le_minimalPrimes hp))
   haveI := Ideal.minimalPrimes_isPrime hq
   exact (iInf₂_le q hq).trans (Ideal.primeHeight_mono e)
 
@@ -131,7 +131,7 @@ lemma Ideal.height_strict_mono_of_is_prime {I J : Ideal R} [I.IsPrime]
   · rw [← ENat.add_one_le_iff I.primeHeight_ne_top, Ideal.height]
     refine le_iInf₂ (fun K hK ↦ ?_)
     haveI := Ideal.minimalPrimes_isPrime hK
-    have : I < K := lt_of_lt_of_le h hK.1.2
+    have : I < K := lt_of_lt_of_le h (Ideal.le_minimalPrimes hK)
     exact Ideal.primeHeight_add_one_le_of_lt this
 
 lemma Ideal.primeHeight_le_ringKrullDim {I : Ideal R} [I.IsPrime] :
@@ -179,10 +179,10 @@ lemma Ideal.mem_minimalPrimes_of_height_eq {I J : Ideal R} (e : I ≤ J) [J.IsPr
   obtain ⟨p, h₁, h₂⟩ := Ideal.exists_minimalPrimes_le e
   convert h₁
   refine (eq_of_le_of_not_lt h₂ fun h₃ ↦ ?_).symm
-  have := h₁.1.1
+  have := Ideal.minimalPrimes_isPrime h₁
   have := finiteHeight_of_le h₂ IsPrime.ne_top'
-  exact lt_irrefl _
-     ((height_strict_mono_of_is_prime h₃).trans_le (e'.trans <| height_mono h₁.1.2))
+  exact lt_irrefl _ ((height_strict_mono_of_is_prime h₃).trans_le
+    (e'.trans <| height_mono (Ideal.le_minimalPrimes h₁)))
 
 /-- A prime ideal has height zero if and only if it is minimal -/
 lemma Ideal.primeHeight_eq_zero_iff {I : Ideal R} [I.IsPrime] :
@@ -204,7 +204,7 @@ lemma Ideal.height_eq_zero_iff {I : Ideal R} [I.IsPrime] : height I = 0 ↔ I 
 lemma Ideal.height_bot [Nontrivial R] : (⊥ : Ideal R).height = 0 := by
   obtain ⟨p, hp⟩ := Ideal.nonempty_minimalPrimes (R := R) (I := ⊥) top_ne_bot.symm
   simp only [Ideal.height, ENat.iInf_eq_zero]
-  exact ⟨p, hp, haveI := hp.1.1; primeHeight_eq_zero_iff.mpr hp⟩
+  exact ⟨p, hp, haveI := Ideal.minimalPrimes_isPrime hp; primeHeight_eq_zero_iff.mpr hp⟩
 
 /-- In a trivial commutative ring, the height of any ideal is `∞`. -/
 @[simp, nontriviality]
@@ -290,7 +290,7 @@ lemma RingEquiv.height_comap {S : Type*} [CommRing S] (e : R ≃+* S) (I : Ideal
     rw [← Ideal.comap_coe,
       Ideal.comap_minimalPrimes_eq_of_surjective (f := (↑e : R →+* S)) e.surjective]
     exact e.idealComapOrderIso.injective.mem_set_image.symm
-  · have : J.IsPrime := h.1.1
+  · have : J.IsPrime := Ideal.minimalPrimes_isPrime h
     simp only [EquivLike.coe_coe, RingEquiv.idealComapOrderIso_apply,
       ← Ideal.height_eq_primeHeight, RingEquiv.height_comap_of_isPrime]
 
@@ -405,14 +405,14 @@ lemma exists_spanRank_le_and_le_height_of_le_height [IsNoetherianRing R] (I : Id
       by_cases h : p.height = ⊤
       · exact le_of_le_of_eq le_top h.symm
       have : p.FiniteHeight := ⟨Or.inr h⟩
-      have := Ideal.height_mono (le_sup_left.trans hp.1.2)
+      have := Ideal.height_mono (le_sup_left.trans (Ideal.le_minimalPrimes hp))
       suffices h : (r : ℕ∞) ≠ p.height by
         exact Order.add_one_le_of_lt (lt_of_le_of_ne (h₃.trans this) h)
       intro e
       apply hx₂ p
-      · have hteq : J.height = p.height := this.antisymm (le_of_eq_of_le e.symm h₃)
-        refine ⟨mem_minimalPrimes_of_height_eq (le_sup_left.trans hp.1.2) (ge_of_eq hteq), e.symm⟩
-      · exact hp.1.2 <| Ideal.mem_sup_right <| Ideal.subset_span <| Set.mem_singleton x
+      · refine ⟨mem_minimalPrimes_of_height_eq (le_sup_left.trans (Ideal.le_minimalPrimes hp))
+          (le_of_eq_of_le e.symm h₃), e.symm⟩
+      · exact (Ideal.le_minimalPrimes hp) <| Ideal.mem_sup_right <| mem_span_singleton_self x
 
 /-- In a nontrivial commutative ring `R`, the supremum of heights of all ideals is equal to the
 Krull dimension of `R`. -/
diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
index f2ef9f2c52ac28..689fddcfb9f092 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
@@ -123,17 +123,17 @@ theorem Ideal.minimalPrimes_eq_subsingleton (hI : I.IsPrimary) : I.minimalPrimes
   ext J
   constructor
   · exact fun H =>
-      let e := H.1.1.radical_le_iff.mpr H.1.2
+      let e := (Ideal.minimalPrimes_isPrime H).radical_le_iff.mpr (Ideal.le_minimalPrimes H)
       (H.2 ⟨Ideal.isPrime_radical hI, Ideal.le_radical⟩ e).antisymm e
   · rintro (rfl : J = I.radical)
     exact ⟨⟨Ideal.isPrime_radical hI, Ideal.le_radical⟩, fun _ H _ => H.1.radical_le_iff.mpr H.2⟩
 
 theorem Ideal.minimalPrimes_eq_subsingleton_self [I.IsPrime] : I.minimalPrimes = {I} := by
   ext J
-  constructor
-  · exact fun H => (H.2 ⟨inferInstance, rfl.le⟩ H.1.2).antisymm H.1.2
-  · rintro (rfl : J = I)
-    exact ⟨⟨inferInstance, rfl.le⟩, fun _ h _ => h.2⟩
+  refine ⟨fun H => (H.2 ⟨inferInstance, rfl.le⟩ (Ideal.le_minimalPrimes H)).antisymm
+    (Ideal.le_minimalPrimes H), ?_⟩
+  rintro (rfl : J = I)
+  exact ⟨⟨inferInstance, rfl.le⟩, fun _ h _ => h.2⟩
 
 variable (R) in
 theorem IsDomain.minimalPrimes_eq_singleton_bot [IsDomain R] :
@@ -150,7 +150,7 @@ theorem Ideal.minimalPrimes_top : (⊤ : Ideal R).minimalPrimes = ∅ := by
   ext p
   simp only [Set.notMem_empty, iff_false]
   intro h
-  exact h.1.1.ne_top (top_le_iff.mp h.1.2)
+  exact (Ideal.minimalPrimes_isPrime h).ne_top (top_le_iff.mp (Ideal.le_minimalPrimes h))
 
 theorem Ideal.minimalPrimes_eq_empty_iff (I : Ideal R) :
     I.minimalPrimes = ∅ ↔ I = ⊤ := by
@@ -169,7 +169,8 @@ lemma Ideal.mem_minimalPrimes_sup {R : Type*} [CommRing R] {p I J : Ideal R} [p.
     p ∈ (I ⊔ J).minimalPrimes := by
   refine ⟨⟨‹_›, ?_⟩, fun q ⟨_, hq⟩ hqp ↦ ?_⟩
   · rw [sup_le_iff]
-    exact ⟨hle, by simpa [hle] using Ideal.comap_mono (f := Ideal.Quotient.mk I) h.1.2⟩
+    refine ⟨hle, ?_⟩
+    simpa [hle] using Ideal.comap_mono (f := Ideal.Quotient.mk I) (Ideal.le_minimalPrimes h)
   · rw [sup_le_iff] at hq
     have h2 : p.map (Quotient.mk I) ≤ q.map (Quotient.mk I) :=
       h.2 ⟨isPrime_map_quotientMk_of_isPrime hq.1, map_mono hq.2⟩ (map_mono hqp)
@@ -189,7 +190,7 @@ lemma Ideal.map_sup_mem_minimalPrimes_of_map_quotientMk_mem_minimalPrimes
   refine ⟨⟨inferInstance, sup_le_iff.mpr ?_⟩, fun q ⟨_, hleq⟩ hqle ↦ ?_⟩
   · refine ⟨?_, hJP⟩
     rw [Ideal.map_le_iff_le_comap, ← Ideal.under_def, ← Ideal.over_def P p]
-    exact hI.1.2
+    exact Ideal.le_minimalPrimes hI
   · simp only [sup_le_iff] at hleq
     have h1 : p.map (algebraMap R S) ≤ q := by
       rw [Ideal.map_le_iff_le_comap]
diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Colon.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Colon.lean
index 9ecd208c7666f6..8758fea64c554d 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Colon.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Colon.lean
@@ -48,7 +48,8 @@ theorem exists_eq_colon_of_mem_minimalPrimes [IsNoetherianRing R]
     refine ⟨n, hn0, ((h.toFinset.erase I).inf id) ^ n, hn, ?_⟩
     have (K : Ideal R) (hKI : K ≤ I) (hK : K ∈ ann.minimalPrimes) : K = I :=
       le_antisymm hKI (hI.2 hK.1 hKI)
-    simpa [hI.1.1.pow_le_iff hn0, hI.1.1.inf_le', imp_not_comm, not_imp_not]
+    simpa [(Ideal.minimalPrimes_isPrime hI).pow_le_iff hn0,
+      (Ideal.minimalPrimes_isPrime hI).inf_le', imp_not_comm, not_imp_not]
   obtain ⟨hn0, J, hJ, hJI⟩ := Nat.find_spec key
   -- let `n` be minimal such that there exists an ideal `J` with `I ^ n * J ≤ ann` and `¬ J ≠ I`
   set n := Nat.find key
@@ -74,9 +75,7 @@ theorem exists_eq_colon_of_mem_minimalPrimes [IsNoetherianRing R]
   -- let `z` be the product of these finitely many `f y`'s
   let z := ∏ y ∈ s, f y
   -- then `z ∉ I`
-  have hz : z ∉ I := by
-    simp only [z, hI.1.1.prod_mem_iff, not_exists, not_and_or]
-    exact fun i ↦ Or.inr (h i)
+  have hz : z ∉ I := by simp [z, (Ideal.minimalPrimes_isPrime hI).prod_mem_iff, h]
   -- and `K ≤ colon N {z • x}`
   have hz' : K ≤ colon N {z • x} := by
     rw [← (map_injective_of_injective K.subtype_injective).eq_iff, map_subtype_top] at hs
@@ -92,11 +91,12 @@ theorem exists_eq_colon_of_mem_minimalPrimes [IsNoetherianRing R]
     simpa only [ann, mem_colon_singleton, mul_comm, mul_smul] using hz' hi
   -- but now `K = I ^ (n - 1) * J` contradicts the minimality of `n`
   have hK : I ^ (n - 1) * (J * Ideal.span {z}) ≤ ann ∧ ¬ J * Ideal.span {z} ≤ I := by
-    rw [← mul_assoc, hI.1.1.mul_le, not_or, Ideal.span_singleton_le_iff_mem]
+    rw [← mul_assoc, (Ideal.minimalPrimes_isPrime hI).mul_le, not_or,
+      Ideal.span_singleton_le_iff_mem]
     exact ⟨hz', hJI, hz⟩
   by_cases hn' : n - 1 = 0
   · simp [K, show n = 1 by grind] at hz'
-    exact (hK.2 (hz'.trans hI.1.2)).elim
+    exact (hK.2 (hz'.trans (Ideal.le_minimalPrimes hI))).elim
   · grind [Nat.find_min' key ⟨hn', J * Ideal.span {z}, hK⟩]
 
 end Submodule
diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean
index 7d76dbf783fb02..5b07cd7cc83ae0 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean
@@ -65,8 +65,8 @@ theorem Ideal.iUnion_minimalPrimes :
   · rintro ⟨y, hy, hx⟩
     obtain ⟨p, hp, hyp⟩ : ∃ p ∈ I.minimalPrimes, y ∉ p := by
       simpa [← Ideal.sInf_minimalPrimes] using hy
-    refine ⟨p, hp, (hp.1.1.mem_or_mem ?_).resolve_right hyp⟩
-    exact hp.1.1.radical_le_iff.mpr hp.1.2 hx
+    refine ⟨p, hp, ((Ideal.minimalPrimes_isPrime hp).mem_or_mem ?_).resolve_right hyp⟩
+    exact (Ideal.minimalPrimes_isPrime hp).radical_le_iff.mpr (Ideal.le_minimalPrimes hp) hx
 
 theorem Ideal.exists_mul_mem_of_mem_minimalPrimes
     {p : Ideal R} (hp : p ∈ I.minimalPrimes) {x : R} (hx : x ∈ p) :
@@ -97,8 +97,8 @@ lemma Ideal.disjoint_nonZeroDivisors_of_mem_minimalPrimes {p : Ideal R} (hp : p
 
 theorem Ideal.exists_comap_eq_of_mem_minimalPrimes {I : Ideal S} (f : R →+* S) (p)
     (H : p ∈ (I.comap f).minimalPrimes) : ∃ p' : Ideal S, p'.IsPrime ∧ I ≤ p' ∧ p'.comap f = p :=
-  have := H.1.1
-  have ⟨p', hIp', hp', le⟩ := exists_ideal_comap_le_prime p I H.1.2
+  have := Ideal.minimalPrimes_isPrime H
+  have ⟨p', hIp', hp', le⟩ := exists_ideal_comap_le_prime p I (Ideal.le_minimalPrimes H)
   ⟨p', hp', hIp', le.antisymm (H.2 ⟨inferInstance, comap_mono hIp'⟩ le)⟩
 
 @[stacks 00FK] theorem Ideal.exists_comap_eq_of_mem_minimalPrimes_of_injective {f : R →+* S}
@@ -113,9 +113,9 @@ theorem Ideal.exists_minimalPrimes_comap_eq {I : Ideal S} (f : R →+* S) (p)
   obtain ⟨p', h₁, h₂, h₃⟩ := Ideal.exists_comap_eq_of_mem_minimalPrimes f p H
   obtain ⟨q, hq, hq'⟩ := Ideal.exists_minimalPrimes_le h₂
   refine ⟨q, hq, Eq.symm ?_⟩
-  have := hq.1.1
+  have := Ideal.minimalPrimes_isPrime hq
   have := (Ideal.comap_mono hq').trans_eq h₃
-  exact (H.2 ⟨inferInstance, Ideal.comap_mono hq.1.2⟩ this).antisymm this
+  exact (H.2 ⟨inferInstance, Ideal.comap_mono (Ideal.le_minimalPrimes hq)⟩ this).antisymm this
 
 theorem Ideal.minimalPrimes_comap_subset (f : R →+* S) (J : Ideal S) :
     (J.comap f).minimalPrimes ⊆ Ideal.comap f '' J.minimalPrimes :=
@@ -129,8 +129,8 @@ variable {R S : Type*} [CommRing R] [CommRing S] {I J : Ideal R}
 
 theorem Ideal.minimal_primes_comap_of_surjective {f : R →+* S} (hf : Function.Surjective f)
     {I J : Ideal S} (h : J ∈ I.minimalPrimes) : J.comap f ∈ (I.comap f).minimalPrimes := by
-  have := h.1.1
-  refine ⟨⟨inferInstance, Ideal.comap_mono h.1.2⟩, ?_⟩
+  have := Ideal.minimalPrimes_isPrime h
+  refine ⟨⟨inferInstance, Ideal.comap_mono (Ideal.le_minimalPrimes h)⟩, ?_⟩
   rintro K ⟨hK, e₁⟩ e₂
   have : RingHom.ker f ≤ K := (Ideal.comap_mono bot_le).trans e₁
   rw [← sup_eq_left.mpr this, RingHom.ker_eq_comap_bot, ← Ideal.comap_map_of_surjective f hf]
@@ -156,7 +156,8 @@ lemma Ideal.minimalPrimes_map_of_surjective {S : Type*} [CommRing S] {f : R →+
   rw [← Ideal.comap_minimalPrimes_eq_of_surjective hf, ← Set.image_comp,
     Ideal.comap_map_of_surjective f hf, Set.image_congr, Set.image_id, RingHom.ker]
   intro x hx
-  exact (Ideal.comap_map_of_surjective f hf _).trans (sup_eq_left.mpr <| le_sup_right.trans hx.1.2)
+  exact (Ideal.comap_map_of_surjective f hf _).trans
+    (sup_eq_left.mpr <| le_sup_right.trans (Ideal.le_minimalPrimes hx))
 
 theorem Ideal.minimalPrimes_eq_comap :
     I.minimalPrimes = Ideal.comap (Ideal.Quotient.mk I) '' minimalPrimes (R ⧸ I) := by
@@ -174,21 +175,21 @@ theorem IsLocalization.minimalPrimes_map [IsLocalization S A] (J : Ideal R) :
   ext p
   constructor
   · intro hp
-    haveI := hp.1.1
-    refine ⟨⟨Ideal.IsPrime.comap _, Ideal.map_le_iff_le_comap.mp hp.1.2⟩, ?_⟩
+    haveI := Ideal.minimalPrimes_isPrime hp
+    refine ⟨⟨Ideal.IsPrime.comap _, Ideal.map_le_iff_le_comap.mp (Ideal.le_minimalPrimes hp)⟩, ?_⟩
     rintro I hI e
     have hI' : Disjoint (S : Set R) I := Set.disjoint_of_subset_right e
-      ((IsLocalization.isPrime_iff_isPrime_disjoint S A _).mp hp.1.1).2
+      ((IsLocalization.isPrime_iff_isPrime_disjoint S A _).mp (Ideal.minimalPrimes_isPrime hp)).2
     refine (Ideal.comap_mono <|
       hp.2 ⟨?_, Ideal.map_mono hI.2⟩ (Ideal.map_le_iff_le_comap.mpr e)).trans_eq ?_
     · exact IsLocalization.isPrime_of_isPrime_disjoint S A I hI.1 hI'
     · exact IsLocalization.comap_map_of_isPrime_disjoint S A hI.1 hI'
   · intro hp
-    refine ⟨⟨?_, Ideal.map_le_iff_le_comap.mpr hp.1.2⟩, ?_⟩
+    refine ⟨⟨?_, Ideal.map_le_iff_le_comap.mpr (Ideal.le_minimalPrimes hp)⟩, ?_⟩
     · rw [IsLocalization.isPrime_iff_isPrime_disjoint S A, IsLocalization.disjoint_comap_iff S]
-      refine ⟨hp.1.1, ?_⟩
+      refine ⟨Ideal.minimalPrimes_isPrime hp, ?_⟩
       rintro rfl
-      exact hp.1.1.ne_top rfl
+      exact (Ideal.minimalPrimes_isPrime hp).ne_top rfl
     · intro I hI e
       rw [← IsLocalization.map_comap S A I, ← IsLocalization.map_comap S A p]
       haveI := hI.1
@@ -212,7 +213,7 @@ theorem IsLocalization.AtPrime.radical_map_of_mem_minimalPrimes
   · rw [← IsLocalization.comap_le_comap_iff q.primeCompl A,
       AtPrime.map_eq_maximalIdeal q A, AtPrime.comap_maximalIdeal A q]
     apply hIq.2 hJ.1
-    have := hJ.1.1.ne_top
+    have := (Ideal.minimalPrimes_isPrime hJ).ne_top
     rw [ne_eq, Ideal.comap_eq_top_iff, ← ne_eq, ← disjoint_comap_iff q.primeCompl A J] at this
     exact Set.disjoint_compl_left_iff_subset.mp this
 

From e7fd6eeb910a31c9d04f331bc3c57e462d141be8 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Thu, 2 Apr 2026 11:09:54 +0800
Subject: [PATCH 06/69] add private and deprecation

---
 Mathlib/RingTheory/Ideal/Height.lean | 22 +++++++++++-----------
 1 file changed, 11 insertions(+), 11 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index 5b458b468e9a25..ef2315e404be85 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -134,7 +134,7 @@ lemma Ideal.height_strict_mono_of_is_prime {I J : Ideal R} [I.IsPrime]
     have : I < K := lt_of_lt_of_le h (Ideal.le_minimalPrimes hK)
     exact Ideal.primeHeight_add_one_le_of_lt this
 
-lemma Ideal.primeHeight_le_ringKrullDim {I : Ideal R} [I.IsPrime] :
+private lemma Ideal.primeHeight_le_ringKrullDim {I : Ideal R} [I.IsPrime] :
     I.primeHeight ≤ ringKrullDim R := Order.height_le_krullDim _
 
 lemma Ideal.height_le_ringKrullDim_of_ne_top {I : Ideal R} (h : I ≠ ⊤) :
@@ -185,7 +185,7 @@ lemma Ideal.mem_minimalPrimes_of_height_eq {I J : Ideal R} (e : I ≤ J) [J.IsPr
     (e'.trans <| height_mono (Ideal.le_minimalPrimes h₁)))
 
 /-- A prime ideal has height zero if and only if it is minimal -/
-lemma Ideal.primeHeight_eq_zero_iff {I : Ideal R} [I.IsPrime] :
+private lemma Ideal.primeHeight_eq_zero_iff {I : Ideal R} [I.IsPrime] :
     primeHeight I = 0 ↔ I ∈ minimalPrimes R := by
   rw [Ideal.primeHeight, Order.height_eq_zero, minimalPrimes, Ideal.minimalPrimes]
   simp only [bot_le, and_true, Set.mem_setOf_eq, Minimal, IsMin]
@@ -211,7 +211,7 @@ lemma Ideal.height_bot [Nontrivial R] : (⊥ : Ideal R).height = 0 := by
 lemma Ideal.height_of_subsingleton [Subsingleton R] : I.height = ⊤ := by
   rw [Subsingleton.elim I ⊤, Ideal.height_top]
 
-theorem Ideal.isMaximal_of_primeHeight_eq_ringKrullDim {I : Ideal R} [I.IsPrime]
+private theorem Ideal.isMaximal_of_primeHeight_eq_ringKrullDim {I : Ideal R} [I.IsPrime]
     [FiniteRingKrullDim R] (e : I.primeHeight = ringKrullDim R) : I.IsMaximal := by
   have h : I ≠ ⊤ := Ideal.IsPrime.ne_top'
   obtain ⟨M, hM, hM'⟩ := Ideal.exists_le_maximal I h
@@ -227,8 +227,7 @@ theorem Ideal.isMaximal_of_height_eq_ringKrullDim {I : Ideal R} [I.IsPrime]
   exact Ideal.isMaximal_of_primeHeight_eq_ringKrullDim e
 
 /-- The prime height of the maximal ideal equals the Krull dimension in a local ring -/
-@[simp]
-theorem IsLocalRing.maximalIdeal_primeHeight_eq_ringKrullDim [IsLocalRing R] :
+private theorem IsLocalRing.maximalIdeal_primeHeight_eq_ringKrullDim [IsLocalRing R] :
     (IsLocalRing.maximalIdeal R).primeHeight = ringKrullDim R := by
   rw [ringKrullDim, Ideal.primeHeight, ← Order.height_top_eq_krullDim]
   rfl
@@ -249,6 +248,7 @@ theorem Ideal.height_eq_ringKrullDim_iff [FiniteRingKrullDim R] [IsLocalRing R]
   · rintro rfl
     exact IsLocalRing.maximalIdeal_height_eq_ringKrullDim
 
+@[deprecated "Use `Ideal.height_eq_ringKrullDim_iff` instead." (since := "2026-04-02")]
 theorem Ideal.primeHeight_eq_ringKrullDim_iff [FiniteRingKrullDim R] [IsLocalRing R] {I : Ideal R}
     [I.IsPrime] : Ideal.primeHeight I = ringKrullDim R ↔ I = IsLocalRing.maximalIdeal R := by
   rw [← Ideal.height_eq_primeHeight, Ideal.height_eq_ringKrullDim_iff]
@@ -319,8 +319,8 @@ lemma ringKrullDim_le_iff_isMaximal_height_le {R : Type*} [CommRing R] (n : With
   norm_cast
   exact Ideal.height_mono hle
 
-theorem IsLocalization.primeHeight_comap (S : Submonoid R) {A : Type*} [CommRing A] [Algebra R A]
-    [IsLocalization S A] (J : Ideal A) [J.IsPrime] :
+private theorem IsLocalization.primeHeight_comap (S : Submonoid R) {A : Type*} [CommRing A]
+    [Algebra R A] [IsLocalization S A] (J : Ideal A) [J.IsPrime] :
     (J.comap (algebraMap R A)).primeHeight = J.primeHeight := by
   rw [eq_comm, Ideal.primeHeight, Ideal.primeHeight, ← WithBot.coe_inj,
     Order.height_eq_krullDim_Iic, Order.height_eq_krullDim_Iic]
@@ -346,10 +346,9 @@ theorem IsLocalization.AtPrime.ringKrullDim_eq_height (I : Ideal R) [I.IsPrime]
     [CommRing A] [Algebra R A] [IsLocalization.AtPrime A I] :
     ringKrullDim A = I.height := by
   have := IsLocalization.AtPrime.isLocalRing A I
-  rw [← IsLocalRing.maximalIdeal_primeHeight_eq_ringKrullDim,
-      ← IsLocalization.primeHeight_comap I.primeCompl,
-      ← IsLocalization.AtPrime.comap_maximalIdeal A I,
-      Ideal.height_eq_primeHeight]
+  rw [← IsLocalRing.maximalIdeal_height_eq_ringKrullDim,
+      ← IsLocalization.height_comap I.primeCompl,
+      ← IsLocalization.AtPrime.comap_maximalIdeal A I]
 
 lemma IsLocalization.height_map_of_disjoint {S : Type*} [CommRing S] [Algebra R S] (M : Submonoid R)
     [IsLocalization M S] (p : Ideal R) [p.IsPrime] (h : Disjoint (M : Set R) (p : Set R)) :
@@ -363,6 +362,7 @@ lemma IsLocalization.height_map_of_disjoint {S : Type*} [CommRing S] [Algebra R
   rw [AtPrime.ringKrullDim_eq_height P, AtPrime.ringKrullDim_eq_height p] at this
   exact WithBot.coe_eq_coe.mp this
 
+@[deprecated "Use `mem_minimalPrimes_of_height_eq` instead." (since := "2026-04-02")]
 lemma mem_minimalPrimes_of_primeHeight_eq_height {I J : Ideal R} [J.IsPrime] (e : I ≤ J)
     (e' : J.primeHeight = I.height) [J.FiniteHeight] : J ∈ I.minimalPrimes := by
   rw [← J.height_eq_primeHeight] at e'

From 43fc664d699b6a0e68fc59258bef8bb9a886b8f1 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Thu, 2 Apr 2026 11:17:04 +0800
Subject: [PATCH 07/69] golf Ideal.sup_primeHeight_eq_ringKrullDim

---
 Mathlib/RingTheory/Ideal/Height.lean | 19 ++++++++++++-------
 1 file changed, 12 insertions(+), 7 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index ef2315e404be85..0c0c0c0f92533d 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -433,20 +433,25 @@ lemma Ideal.sup_height_eq_ringKrullDim [Nontrivial R] :
       · exact le_iSup_of_le p.last.isPrime.ne_top' le_rfl
       · exact le_iSup (fun I => ⨆ _, I.height) p.last.asIdeal
 
-/-- In a nontrivial commutative ring `R`, the supremum of prime heights of all prime ideals is
+/-- In a nontrivial commutative ring `R`, the supremum of heights of all prime ideals is
 equal to the Krull dimension of `R`. -/
-lemma Ideal.sup_primeHeight_eq_ringKrullDim [Nontrivial R] :
-    ↑(⨆ (I : Ideal R) (_ : I.IsPrime), I.primeHeight) = ringKrullDim R := by
+lemma Ideal.sup_height_isPrime_eq_ringKrullDim [Nontrivial R] :
+    ↑(⨆ (I : Ideal R) (_ : I.IsPrime), I.height) = ringKrullDim R := by
   rw [← sup_height_eq_ringKrullDim, WithBot.coe_inj]
   apply le_antisymm
-  · exact iSup_mono fun I => iSup_mono' fun hI => ⟨hI.ne_top, by rw [← height_eq_primeHeight]⟩
+  · exact iSup_mono fun I => iSup_mono' fun hI => ⟨hI.ne_top, le_refl _⟩
   · refine iSup_mono' fun I => ?_
     by_cases I_top : I = ⊤
     · exact ⟨⊥, by simp [I_top]⟩
-    · obtain ⟨P, hP⟩ : I.minimalPrimes.Nonempty :=
-        Set.nonempty_coe_sort.mp (nonempty_minimalPrimes I_top)
+    · obtain ⟨P, hP⟩ := Set.nonempty_coe_sort.mp (nonempty_minimalPrimes I_top)
       refine ⟨P, iSup_pos (α := ℕ∞) I_top ▸ le_iSup_of_le (hP.left.left) ?_⟩
-      exact iInf_le_of_le P (iInf_le_of_le hP le_rfl)
+      have := Ideal.minimalPrimes_isPrime hP
+      exact iInf_le_of_le P (iInf_le_of_le hP (ge_of_eq (Ideal.height_eq_primeHeight P)))
+
+@[deprecated "Use `Ideal.sup_height_isPrime_eq_ringKrullDim` instead." (since := "2026-04-02")]
+lemma Ideal.sup_primeHeight_eq_ringKrullDim [Nontrivial R] :
+    ↑(⨆ (I : Ideal R) (_ : I.IsPrime), I.primeHeight) = ringKrullDim R := by
+  simp [← Ideal.height_eq_primeHeight, Ideal.sup_height_isPrime_eq_ringKrullDim]
 
 /-- In a nontrivial commutative ring `R`, the supremum of prime heights of all maximal ideals is
 equal to the Krull dimension of `R`. -/

From a580c9ead4e4f4ee19329c12df1a6194402ed41d Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Thu, 2 Apr 2026 11:19:51 +0800
Subject: [PATCH 08/69] fix

---
 .../RingTheory/Ideal/KrullsHeightTheorem.lean | 30 ++++++++++---------
 1 file changed, 16 insertions(+), 14 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
index a8256e59729471..786dd1087da2e9 100644
--- a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
+++ b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
@@ -83,7 +83,7 @@ lemma Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes_of_isLocalRing
     ← (IsLocalization.orderEmbedding q.primeCompl (Localization.AtPrime q)).map_rel_iff]
   refine Submodule.le_of_le_smul_of_le_jacobson_bot (I := I) (IsNoetherian.noetherian _) ?_ ?_
   · rw [IsLocalRing.jacobson_eq_maximalIdeal]
-    exacts [hp.1.2, bot_ne_top]
+    exacts [Ideal.le_minimalPrimes hp, bot_ne_top]
   · replace hn := congr(Ideal.comap (Ideal.Quotient.mk I) $(hn _ n.le_succ))
     simp only [qs, OrderHom.coe_mk, ← RingHom.ker_eq_comap_bot, Ideal.mk_ker,
       Ideal.comap_map_of_surjective _ Ideal.Quotient.mk_surjective] at hn
@@ -118,13 +118,14 @@ theorem Ideal.map_height_le_one_of_mem_minimalPrimes {I p : Ideal R} {x : R}
     (hp : p ∈ (I ⊔ span {x}).minimalPrimes) : (p.map (Ideal.Quotient.mk I)).height ≤ 1 :=
   let f := Ideal.Quotient.mk I
   have : p.IsPrime := Ideal.minimalPrimes_isPrime hp
-  have hfp : RingHom.ker f ≤ p := I.mk_ker.trans_le (le_sup_left.trans hp.1.2)
+  have hfp : RingHom.ker f ≤ p := I.mk_ker.trans_le (le_sup_left.trans (Ideal.le_minimalPrimes hp))
   height_le_one_of_isPrincipal_of_mem_minimalPrimes ((span {x}).map f) (p.map f)
-    ⟨⟨map_isPrime_of_surjective Quotient.mk_surjective hfp, map_mono (le_sup_right.trans hp.1.2)⟩,
-      fun _ ⟨hr, hxr⟩ hrp ↦ map_le_iff_le_comap.mpr <| hp.2 ⟨hr.comap f, sup_le_iff.mpr
-        ⟨I.mk_ker.symm.trans_le <| ker_le_comap (Ideal.Quotient.mk I), le_comap_of_map_le hxr⟩⟩ <|
-          (comap_mono hrp).trans <| Eq.le <|
-            (p.comap_map_of_surjective _ Quotient.mk_surjective).trans <| sup_eq_left.mpr hfp⟩
+    ⟨⟨map_isPrime_of_surjective Quotient.mk_surjective hfp,
+      map_mono (le_sup_right.trans (Ideal.le_minimalPrimes hp))⟩,
+        fun _ ⟨hr, hxr⟩ hrp ↦ map_le_iff_le_comap.mpr <| hp.2 ⟨hr.comap f, sup_le_iff.mpr
+          ⟨I.mk_ker.symm.trans_le <| ker_le_comap (Ideal.Quotient.mk I), le_comap_of_map_le hxr⟩⟩ <|
+            (comap_mono hrp).trans <| Eq.le <|
+              (p.comap_map_of_surjective _ Quotient.mk_surjective).trans <| sup_eq_left.mpr hfp⟩
 
 /-- If `q < p` are prime ideals such that `p` is minimal over `span (s ∪ {x})` and
 `t` is a set contained in `q` such that `s ⊆ √span (t ∪ {x})`, then `q` is minimal over `span t`.
@@ -147,12 +148,13 @@ theorem Ideal.mem_minimalPrimes_span_of_mem_minimalPrimes_span_insert {q p : Ide
     have : (p.map f).FiniteHeight := ⟨Or.inr (h.trans_lt (WithTop.coe_lt_top 1)).ne⟩
     rw [height_eq_primeHeight] at h
     have := (primeHeight_strict_mono h_lt).trans_le h
-    rw [ENat.lt_one_iff_eq_zero, primeHeight_eq_zero_iff] at this
+    rw [ENat.lt_one_iff_eq_zero, ← Ideal.height_eq_primeHeight, height_eq_zero_iff] at this
     have := minimal_primes_comap_of_surjective hf this
     rwa [comap_map_of_surjective f hf, ← RingHom.ker_eq_comap_bot,
       mk_ker, sup_eq_left.mpr hI'q] at this
   refine height_le_one_of_isPrincipal_of_mem_minimalPrimes ((span {x}).map f) (p.map f) ⟨⟨this,
-    map_mono <| span_le.mpr <| Set.singleton_subset_iff.mpr <| hp.1.2 <| subset_span <| .inl rfl⟩,
+    map_mono <| span_le.mpr <| Set.singleton_subset_iff.mpr <| (Ideal.le_minimalPrimes hp) <|
+      subset_span <| .inl rfl⟩,
     fun r ⟨hr, hxr⟩ hrp ↦ map_le_iff_le_comap.mpr (hp.2 ⟨hr.comap f, ?_⟩ ?_)⟩
   · rw [span_le, Set.insert_subset_iff]
     have := map_le_iff_le_comap.mp hxr (subset_span rfl)
@@ -179,8 +181,7 @@ nonrec lemma Ideal.height_le_spanRank_toENat_of_mem_minimal_primes
     have := Ideal.minimalPrimes_isPrime hp
     cases n with
     | zero =>
-      rw [ENat.coe_zero, nonpos_iff_eq_zero, height_eq_primeHeight p,
-        primeHeight_eq_zero_iff, minimalPrimes]
+      rw [ENat.coe_zero, nonpos_iff_eq_zero, height_eq_zero_iff, minimalPrimes]
       simp_all
     | succ n =>
       wlog hR : ∃ (_ : IsLocalRing R), p = maximalIdeal R
@@ -204,7 +205,7 @@ nonrec lemma Ideal.height_le_spanRank_toENat_of_mem_minimal_primes
           ⟨le_sup_left, x, mem_sup_right (mem_span_singleton_self _), hxq⟩).trans_le hJ)
           ((le_maximalIdeal hJ'.ne_top).lt_of_not_ge h)
       have h : (s' : Set R) ⊆ (q ⊔ span {x}).radical := by
-        have := hp.1.2.trans this
+        have := (Ideal.le_minimalPrimes hp).trans this
         rw [span_le, Finset.coe_insert, Set.insert_subset_iff] at this
         exact this.2
       obtain ⟨t, ht, hspan⟩ := exists_subset_radical_span_sup_of_subset_radical_sup _ _ _ h
@@ -318,7 +319,8 @@ lemma Ideal.height_le_height_add_spanFinrank_of_le {I p : Ideal R} [p.IsPrime] (
   let p' := p.map (algebraMap R (R ⧸ I))
   have : p'.IsPrime := isPrime_map_quotientMk_of_isPrime hrp
   obtain ⟨s, hps, hs⟩ := exists_finset_card_eq_height_of_isNoetherianRing p'
-  have hsp' : (s : Set (R ⧸ I)) ⊆ (p' : Set _) := fun _ hx ↦ hps.1.2 (subset_span hx)
+  have hsp' : (s : Set (R ⧸ I)) ⊆ (p' : Set _) :=
+    fun _ hx ↦ (Ideal.le_minimalPrimes hps) (subset_span hx)
   have : Set.SurjOn (Ideal.Quotient.mk I) p s := by
     refine Set.SurjOn.mono subset_rfl hsp' fun x hx ↦ ?_
     obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x
@@ -415,7 +417,7 @@ lemma Ideal.height_le_height_add_of_liesOver [IsNoetherianRing S] (p : Ideal R)
   let P' := P.map (Ideal.Quotient.mk <| p.map (algebraMap R S))
   obtain ⟨s', hP', heq'⟩ := P'.exists_finset_card_eq_height_of_isNoetherianRing
   have hsP'sub : (s' : Set <| S ⧸ (Ideal.map (algebraMap R S) p)) ⊆ (P' : Set <| S ⧸ _) :=
-    fun x hx ↦ hP'.1.2 (Ideal.subset_span hx)
+    fun x hx ↦ (Ideal.le_minimalPrimes hP') (Ideal.subset_span hx)
   have : Set.SurjOn (Ideal.Quotient.mk (p.map (algebraMap R S))) P s' := by
     refine Set.SurjOn.mono subset_rfl hsP'sub fun x hx ↦ ?_
     obtain ⟨y, rfl⟩ := Ideal.Quotient.mk_surjective x

From b5387efe52ac544b18593bb673925a58d3298d35 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Fri, 3 Apr 2026 18:47:56 +0800
Subject: [PATCH 09/69] make strict mono private

---
 Mathlib/RingTheory/Ideal/Height.lean          | 22 +++++++++++--------
 .../RingTheory/Ideal/KrullsHeightTheorem.lean |  5 ++---
 2 files changed, 15 insertions(+), 12 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index 0c0c0c0f92533d..f9b73a2cd944b9 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -92,26 +92,30 @@ lemma Ideal.primeHeight_mono {I J : Ideal R} [I.IsPrime] [J.IsPrime] (h : I ≤
   gcongr
   exact h
 
-lemma Ideal.primeHeight_add_one_le_of_lt {I J : Ideal R} [I.IsPrime] [J.IsPrime] (h : I < J) :
-    I.primeHeight + 1 ≤ J.primeHeight := by
-  unfold primeHeight
-  exact Order.height_add_one_le h
+private lemma Ideal.primeHeight_add_one_le_of_lt {I J : Ideal R} [I.IsPrime] [J.IsPrime]
+    (h : I < J) : I.primeHeight + 1 ≤ J.primeHeight :=
+  Order.height_add_one_le h
+
+lemma Ideal.height_add_one_le_of_lt_of_isPrime {I J : Ideal R} [I.IsPrime] [J.IsPrime] (h : I < J) :
+    I.height + 1 ≤ J.height := by
+  simpa [Ideal.height_eq_primeHeight] using Ideal.primeHeight_add_one_le_of_lt h
 
 @[simp]
 theorem Ideal.height_top : (⊤ : Ideal R).height = ⊤ := by
   simp [height, minimalPrimes_top]
 
-@[gcongr]
-lemma Ideal.primeHeight_strict_mono {I J : Ideal R} [I.IsPrime] [J.IsPrime]
-    (h : I < J) [J.FiniteHeight] :
-    I.primeHeight < J.primeHeight := by
-  rw [primeHeight]
+private lemma Ideal.primeHeight_strict_mono {I J : Ideal R} [I.IsPrime] [J.IsPrime] (h : I < J)
+    [J.FiniteHeight] : I.primeHeight < J.primeHeight := by
   have : I.FiniteHeight := by
     rw [Ideal.finiteHeight_iff, ← lt_top_iff_ne_top, Ideal.height_eq_primeHeight]
     right
     exact lt_of_le_of_lt (Ideal.primeHeight_mono h.le) (Ideal.primeHeight_lt_top J)
   exact Order.height_strictMono h (Ideal.primeHeight_lt_top _)
 
+lemma Ideal.height_strict_mono_of_isPrime {I J : Ideal R} [I.IsPrime] [J.IsPrime] (h : I < J)
+    [J.FiniteHeight] : I.height < J.height := by
+  simpa [Ideal.height_eq_primeHeight] using Ideal.primeHeight_strict_mono h
+
 @[gcongr]
 theorem Ideal.height_mono {I J : Ideal R} (h : I ≤ J) : I.height ≤ J.height := by
   simp only [height]
diff --git a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
index 786dd1087da2e9..eff8c67f42326a 100644
--- a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
+++ b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
@@ -146,9 +146,8 @@ theorem Ideal.mem_minimalPrimes_span_of_mem_minimalPrimes_span_insert {q p : Ide
         sup_eq_left.mpr hI'q, sup_eq_left.mpr hI'p] using comap_mono (f := f) e
     have : (q.map f).IsPrime := map_isPrime_of_surjective hf (by rwa [mk_ker])
     have : (p.map f).FiniteHeight := ⟨Or.inr (h.trans_lt (WithTop.coe_lt_top 1)).ne⟩
-    rw [height_eq_primeHeight] at h
-    have := (primeHeight_strict_mono h_lt).trans_le h
-    rw [ENat.lt_one_iff_eq_zero, ← Ideal.height_eq_primeHeight, height_eq_zero_iff] at this
+    have := (height_strict_mono_of_isPrime h_lt).trans_le h
+    rw [ENat.lt_one_iff_eq_zero, height_eq_zero_iff] at this
     have := minimal_primes_comap_of_surjective hf this
     rwa [comap_map_of_surjective f hf, ← RingHom.ker_eq_comap_bot,
       mk_ker, sup_eq_left.mpr hI'q] at this

From 732ccc0ff775da9e891bae1449b6fe36c9e1f73a Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Fri, 3 Apr 2026 19:56:16 +0800
Subject: [PATCH 10/69] fix

---
 Mathlib/RingTheory/Ideal/Height.lean | 40 +++++++++-------------------
 1 file changed, 13 insertions(+), 27 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index f9b73a2cd944b9..f161d0b916d261 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -108,14 +108,9 @@ private lemma Ideal.primeHeight_strict_mono {I J : Ideal R} [I.IsPrime] [J.IsPri
     [J.FiniteHeight] : I.primeHeight < J.primeHeight := by
   have : I.FiniteHeight := by
     rw [Ideal.finiteHeight_iff, ← lt_top_iff_ne_top, Ideal.height_eq_primeHeight]
-    right
-    exact lt_of_le_of_lt (Ideal.primeHeight_mono h.le) (Ideal.primeHeight_lt_top J)
+    exact Or.inr (lt_of_le_of_lt (Ideal.primeHeight_mono h.le) (Ideal.primeHeight_lt_top J))
   exact Order.height_strictMono h (Ideal.primeHeight_lt_top _)
 
-lemma Ideal.height_strict_mono_of_isPrime {I J : Ideal R} [I.IsPrime] [J.IsPrime] (h : I < J)
-    [J.FiniteHeight] : I.height < J.height := by
-  simpa [Ideal.height_eq_primeHeight] using Ideal.primeHeight_strict_mono h
-
 @[gcongr]
 theorem Ideal.height_mono {I J : Ideal R} (h : I ≤ J) : I.height ≤ J.height := by
   simp only [height]
@@ -126,15 +121,14 @@ theorem Ideal.height_mono {I J : Ideal R} (h : I ≤ J) : I.height ≤ J.height
   exact (iInf₂_le q hq).trans (Ideal.primeHeight_mono e)
 
 @[gcongr]
-lemma Ideal.height_strict_mono_of_is_prime {I J : Ideal R} [I.IsPrime]
+lemma Ideal.height_strict_mono_of_isPrime {I J : Ideal R} [I.IsPrime]
     (h : I < J) [I.FiniteHeight] : I.height < J.height := by
-  rw [Ideal.height_eq_primeHeight I]
   by_cases hJ : J = ⊤
-  · rw [hJ, height_top]
-    exact I.primeHeight_lt_top
-  · rw [← ENat.add_one_le_iff I.primeHeight_ne_top, Ideal.height]
+  · grw [hJ, height_top]
+    exact I.height_lt_top IsPrime.ne_top'
+  · rw [← ENat.add_one_le_iff (I.height_ne_top IsPrime.ne_top'), I.height_eq_primeHeight]
     refine le_iInf₂ (fun K hK ↦ ?_)
-    haveI := Ideal.minimalPrimes_isPrime hK
+    have := Ideal.minimalPrimes_isPrime hK
     have : I < K := lt_of_lt_of_le h (Ideal.le_minimalPrimes hK)
     exact Ideal.primeHeight_add_one_le_of_lt this
 
@@ -164,17 +158,13 @@ lemma Ideal.exists_isMaximal_height [FiniteRingKrullDim R] :
 instance (priority := 900) Ideal.finiteHeight_of_finiteRingKrullDim {I : Ideal R}
     [FiniteRingKrullDim R] : I.FiniteHeight := by
   rw [finiteHeight_iff, or_iff_not_imp_left, ← lt_top_iff_ne_top, ← WithBot.coe_lt_coe]
-  intro h
-  have h1 := ringKrullDim_lt_top (R := R)
-  have h2 := Ideal.height_le_ringKrullDim_of_ne_top h
-  exact lt_of_le_of_lt h2 h1
+  exact fun h ↦ lt_of_le_of_lt (Ideal.height_le_ringKrullDim_of_ne_top h) ringKrullDim_lt_top
 
 /-- If J has finite height and I ≤ J, then I has finite height -/
 lemma Ideal.finiteHeight_of_le {I J : Ideal R} (e : I ≤ J) (hJ : J ≠ ⊤) [FiniteHeight J] :
     FiniteHeight I where
-  eq_top_or_height_ne_top := Or.inr <| by
-    rw [← lt_top_iff_ne_top]
-    exact (height_mono e).trans_lt (height_lt_top hJ)
+  eq_top_or_height_ne_top := Or.inr <|
+    lt_top_iff_ne_top.mp ((height_mono e).trans_lt (height_lt_top hJ))
 
 /-- If J is a prime ideal containing I, and its height is less than or equal to the height of I,
 then J is a minimal prime over I -/
@@ -185,7 +175,7 @@ lemma Ideal.mem_minimalPrimes_of_height_eq {I J : Ideal R} (e : I ≤ J) [J.IsPr
   refine (eq_of_le_of_not_lt h₂ fun h₃ ↦ ?_).symm
   have := Ideal.minimalPrimes_isPrime h₁
   have := finiteHeight_of_le h₂ IsPrime.ne_top'
-  exact lt_irrefl _ ((height_strict_mono_of_is_prime h₃).trans_le
+  exact lt_irrefl _ ((height_strict_mono_of_isPrime h₃).trans_le
     (e'.trans <| height_mono (Ideal.le_minimalPrimes h₁)))
 
 /-- A prime ideal has height zero if and only if it is minimal -/
@@ -193,13 +183,9 @@ private lemma Ideal.primeHeight_eq_zero_iff {I : Ideal R} [I.IsPrime] :
     primeHeight I = 0 ↔ I ∈ minimalPrimes R := by
   rw [Ideal.primeHeight, Order.height_eq_zero, minimalPrimes, Ideal.minimalPrimes]
   simp only [bot_le, and_true, Set.mem_setOf_eq, Minimal, IsMin]
-  constructor
-  · intro h
-    refine ⟨inferInstance, ?_⟩
-    by_contra! ⟨P, ⟨hP₁, ⟨hP₂, hP₃⟩⟩⟩
-    exact hP₃ (h (b := ⟨P, hP₁⟩) hP₂)
-  · rintro ⟨hI, hI'⟩ b hb
-    exact hI' (y := b.asIdeal) b.isPrime hb
+  refine ⟨fun h ↦ ⟨‹_›, ?_⟩, fun ⟨hI, hI'⟩ b hb ↦ hI' b.isPrime hb⟩
+  by_contra! ⟨P, ⟨hP₁, ⟨hP₂, hP₃⟩⟩⟩
+  exact hP₃ (h (b := ⟨P, hP₁⟩) hP₂)
 
 lemma Ideal.height_eq_zero_iff {I : Ideal R} [I.IsPrime] : height I = 0 ↔ I ∈ minimalPrimes R := by
   rw [Ideal.height_eq_primeHeight, Ideal.primeHeight_eq_zero_iff]

From ae7cff95c92f5cf506a13a1d79ce83b33566ebde Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Fri, 3 Apr 2026 20:21:44 +0800
Subject: [PATCH 11/69] deprecate strict mono

---
 Mathlib/RingTheory/Ideal/Height.lean | 36 ++++++++++++++++------------
 1 file changed, 21 insertions(+), 15 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index f161d0b916d261..f0af50ba765314 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -104,13 +104,6 @@ lemma Ideal.height_add_one_le_of_lt_of_isPrime {I J : Ideal R} [I.IsPrime] [J.Is
 theorem Ideal.height_top : (⊤ : Ideal R).height = ⊤ := by
   simp [height, minimalPrimes_top]
 
-private lemma Ideal.primeHeight_strict_mono {I J : Ideal R} [I.IsPrime] [J.IsPrime] (h : I < J)
-    [J.FiniteHeight] : I.primeHeight < J.primeHeight := by
-  have : I.FiniteHeight := by
-    rw [Ideal.finiteHeight_iff, ← lt_top_iff_ne_top, Ideal.height_eq_primeHeight]
-    exact Or.inr (lt_of_le_of_lt (Ideal.primeHeight_mono h.le) (Ideal.primeHeight_lt_top J))
-  exact Order.height_strictMono h (Ideal.primeHeight_lt_top _)
-
 @[gcongr]
 theorem Ideal.height_mono {I J : Ideal R} (h : I ≤ J) : I.height ≤ J.height := by
   simp only [height]
@@ -132,6 +125,19 @@ lemma Ideal.height_strict_mono_of_isPrime {I J : Ideal R} [I.IsPrime]
     have : I < K := lt_of_lt_of_le h (Ideal.le_minimalPrimes hK)
     exact Ideal.primeHeight_add_one_le_of_lt this
 
+lemma Ideal.height_strict_mono_of_isPrime_of_isPrime {I J : Ideal R} [I.IsPrime] [J.IsPrime]
+    (h : I < J) [J.FiniteHeight] : I.height < J.height := by
+  have : I.FiniteHeight := by
+    simpa [Ideal.finiteHeight_iff, ‹I.IsPrime›.ne_top] using
+      (lt_of_le_of_lt (Ideal.height_mono h.le) (J.height_lt_top IsPrime.ne_top')).ne
+  exact Ideal.height_strict_mono_of_isPrime h
+
+@[deprecated "Use `Ideal.height_strict_mono_of_isPrime_of_isPrime` instead."
+  (since := "2026-04-02")]
+lemma Ideal.primeHeight_strict_mono {I J : Ideal R} [I.IsPrime] [J.IsPrime] (h : I < J)
+    [J.FiniteHeight] : I.primeHeight < J.primeHeight := by
+  simpa [← Ideal.height_eq_primeHeight] using Ideal.height_strict_mono_of_isPrime_of_isPrime h
+
 private lemma Ideal.primeHeight_le_ringKrullDim {I : Ideal R} [I.IsPrime] :
     I.primeHeight ≤ ringKrullDim R := Order.height_le_krullDim _
 
@@ -201,20 +207,20 @@ lemma Ideal.height_bot [Nontrivial R] : (⊥ : Ideal R).height = 0 := by
 lemma Ideal.height_of_subsingleton [Subsingleton R] : I.height = ⊤ := by
   rw [Subsingleton.elim I ⊤, Ideal.height_top]
 
-private theorem Ideal.isMaximal_of_primeHeight_eq_ringKrullDim {I : Ideal R} [I.IsPrime]
-    [FiniteRingKrullDim R] (e : I.primeHeight = ringKrullDim R) : I.IsMaximal := by
+theorem Ideal.isMaximal_of_height_eq_ringKrullDim {I : Ideal R} [I.IsPrime]
+    [FiniteRingKrullDim R] (e : I.height = ringKrullDim R) : I.IsMaximal := by
   have h : I ≠ ⊤ := Ideal.IsPrime.ne_top'
   obtain ⟨M, hM, hM'⟩ := Ideal.exists_le_maximal I h
   rcases lt_or_eq_of_le hM' with (hM' | hM')
-  · have h1 := Ideal.primeHeight_strict_mono hM'
-    have h2 := e ▸ M.primeHeight_le_ringKrullDim
+  · have h1 := Ideal.height_strict_mono_of_isPrime hM'
+    have h2 := e ▸ M.height_le_ringKrullDim_of_ne_top hM.ne_top
     simp [← not_lt, h1] at h2
   · exact hM' ▸ hM
 
-theorem Ideal.isMaximal_of_height_eq_ringKrullDim {I : Ideal R} [I.IsPrime]
-    [FiniteRingKrullDim R] (e : I.height = ringKrullDim R) : I.IsMaximal := by
-  rw [Ideal.height_eq_primeHeight] at e
-  exact Ideal.isMaximal_of_primeHeight_eq_ringKrullDim e
+@[deprecated "Use `Ideal.isMaximal_of_height_eq_ringKrullDim` instead." (since := "2026-04-02")]
+theorem Ideal.isMaximal_of_primeHeight_eq_ringKrullDim {I : Ideal R} [I.IsPrime]
+    [FiniteRingKrullDim R] (e : I.primeHeight = ringKrullDim R) : I.IsMaximal :=
+  Ideal.isMaximal_of_height_eq_ringKrullDim (by simpa [Ideal.height_eq_primeHeight])
 
 /-- The prime height of the maximal ideal equals the Krull dimension in a local ring -/
 private theorem IsLocalRing.maximalIdeal_primeHeight_eq_ringKrullDim [IsLocalRing R] :

From facecaee3c25ee7101375e07fd8058e93fd92920 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Fri, 3 Apr 2026 20:32:45 +0800
Subject: [PATCH 12/69] golf

---
 Mathlib/RingTheory/Ideal/Height.lean | 9 +++------
 1 file changed, 3 insertions(+), 6 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index f0af50ba765314..d3bcfb4374b962 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -127,9 +127,8 @@ lemma Ideal.height_strict_mono_of_isPrime {I J : Ideal R} [I.IsPrime]
 
 lemma Ideal.height_strict_mono_of_isPrime_of_isPrime {I J : Ideal R} [I.IsPrime] [J.IsPrime]
     (h : I < J) [J.FiniteHeight] : I.height < J.height := by
-  have : I.FiniteHeight := by
-    simpa [Ideal.finiteHeight_iff, ‹I.IsPrime›.ne_top] using
-      (lt_of_le_of_lt (Ideal.height_mono h.le) (J.height_lt_top IsPrime.ne_top')).ne
+  have : I.FiniteHeight := I.finiteHeight_iff.mpr
+    (Or.inr (lt_of_le_of_lt (Ideal.height_mono h.le) (J.height_lt_top IsPrime.ne_top')).ne)
   exact Ideal.height_strict_mono_of_isPrime h
 
 @[deprecated "Use `Ideal.height_strict_mono_of_isPrime_of_isPrime` instead."
@@ -143,11 +142,9 @@ private lemma Ideal.primeHeight_le_ringKrullDim {I : Ideal R} [I.IsPrime] :
 
 lemma Ideal.height_le_ringKrullDim_of_ne_top {I : Ideal R} (h : I ≠ ⊤) :
     I.height ≤ ringKrullDim R := by
-  rw [Ideal.height]
   obtain ⟨P, hP⟩ : Nonempty (I.minimalPrimes) := Ideal.nonempty_minimalPrimes h
   have := Ideal.minimalPrimes_isPrime hP
-  refine le_trans ?_ (Ideal.primeHeight_le_ringKrullDim (I := P))
-  simpa using iInf₂_le _ hP
+  exact (WithBot.coe_le_coe.mpr (iInf₂_le _ hP)).trans P.primeHeight_le_ringKrullDim
 
 /-- If `R` has finite Krull dimension, there exists a maximal ideal `m` with `ht m = dim R`. -/
 lemma Ideal.exists_isMaximal_height [FiniteRingKrullDim R] :

From bc6aba4914b853772e36973ae595c0455f02b20d Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Fri, 3 Apr 2026 20:33:02 +0800
Subject: [PATCH 13/69] deprecate primeHeight_eq_zero

---
 Mathlib/RingTheory/Ideal/Height.lean | 17 ++++++++++-------
 1 file changed, 10 insertions(+), 7 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index d3bcfb4374b962..d9b438d25a8514 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -182,22 +182,25 @@ lemma Ideal.mem_minimalPrimes_of_height_eq {I J : Ideal R} (e : I ≤ J) [J.IsPr
     (e'.trans <| height_mono (Ideal.le_minimalPrimes h₁)))
 
 /-- A prime ideal has height zero if and only if it is minimal -/
-private lemma Ideal.primeHeight_eq_zero_iff {I : Ideal R} [I.IsPrime] :
-    primeHeight I = 0 ↔ I ∈ minimalPrimes R := by
-  rw [Ideal.primeHeight, Order.height_eq_zero, minimalPrimes, Ideal.minimalPrimes]
-  simp only [bot_le, and_true, Set.mem_setOf_eq, Minimal, IsMin]
+lemma Ideal.height_eq_zero_iff {I : Ideal R} [I.IsPrime] : height I = 0 ↔ I ∈ minimalPrimes R := by
+  rw [Ideal.height_eq_primeHeight, Ideal.primeHeight, Order.height_eq_zero, minimalPrimes]
+  simp only [Ideal.minimalPrimes, bot_le, and_true, Set.mem_setOf_eq, Minimal, IsMin]
   refine ⟨fun h ↦ ⟨‹_›, ?_⟩, fun ⟨hI, hI'⟩ b hb ↦ hI' b.isPrime hb⟩
   by_contra! ⟨P, ⟨hP₁, ⟨hP₂, hP₃⟩⟩⟩
   exact hP₃ (h (b := ⟨P, hP₁⟩) hP₂)
 
-lemma Ideal.height_eq_zero_iff {I : Ideal R} [I.IsPrime] : height I = 0 ↔ I ∈ minimalPrimes R := by
-  rw [Ideal.height_eq_primeHeight, Ideal.primeHeight_eq_zero_iff]
+@[deprecated "Use `Ideal.height_eq_zero_iff` instead." (since := "2026-04-02")]
+private lemma Ideal.primeHeight_eq_zero_iff {I : Ideal R} [I.IsPrime] :
+    primeHeight I = 0 ↔ I ∈ minimalPrimes R := by
+  rw [← Ideal.height_eq_primeHeight, Ideal.height_eq_zero_iff]
 
 @[simp]
 lemma Ideal.height_bot [Nontrivial R] : (⊥ : Ideal R).height = 0 := by
   obtain ⟨p, hp⟩ := Ideal.nonempty_minimalPrimes (R := R) (I := ⊥) top_ne_bot.symm
   simp only [Ideal.height, ENat.iInf_eq_zero]
-  exact ⟨p, hp, haveI := Ideal.minimalPrimes_isPrime hp; primeHeight_eq_zero_iff.mpr hp⟩
+  refine ⟨p, hp, ?_⟩
+  have := Ideal.minimalPrimes_isPrime hp
+  rw [← Ideal.height_eq_primeHeight, height_eq_zero_iff.mpr hp]
 
 /-- In a trivial commutative ring, the height of any ideal is `∞`. -/
 @[simp, nontriviality]

From c043789e59ad21d1edcd2a07f0a14e5e6b5dab67 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Fri, 3 Apr 2026 20:37:45 +0800
Subject: [PATCH 14/69] fix

---
 Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
index eff8c67f42326a..bf90d4a888ced7 100644
--- a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
+++ b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
@@ -146,7 +146,7 @@ theorem Ideal.mem_minimalPrimes_span_of_mem_minimalPrimes_span_insert {q p : Ide
         sup_eq_left.mpr hI'q, sup_eq_left.mpr hI'p] using comap_mono (f := f) e
     have : (q.map f).IsPrime := map_isPrime_of_surjective hf (by rwa [mk_ker])
     have : (p.map f).FiniteHeight := ⟨Or.inr (h.trans_lt (WithTop.coe_lt_top 1)).ne⟩
-    have := (height_strict_mono_of_isPrime h_lt).trans_le h
+    have := (height_strict_mono_of_isPrime_of_isPrime h_lt).trans_le h
     rw [ENat.lt_one_iff_eq_zero, height_eq_zero_iff] at this
     have := minimal_primes_comap_of_surjective hf this
     rwa [comap_map_of_surjective f hf, ← RingHom.ker_eq_comap_bot,

From a35ed7f1f833053975aea424b7ed239e25155289 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Sat, 4 Apr 2026 14:28:02 +0800
Subject: [PATCH 15/69] add height version of definition

---
 Mathlib/RingTheory/Ideal/Height.lean | 13 ++++++++++---
 1 file changed, 10 insertions(+), 3 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index d9b438d25a8514..ac3ee6fa917874 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -43,9 +43,16 @@ noncomputable def Ideal.height : ℕ∞ :=
 
 /-- For a prime ideal, its height equals its prime height. -/
 lemma Ideal.height_eq_primeHeight [I.IsPrime] : I.height = I.primeHeight := by
-  unfold height primeHeight
-  simp_rw [Ideal.minimalPrimes_eq_subsingleton_self]
-  simp
+  simp [height, primeHeight, Ideal.minimalPrimes_eq_subsingleton_self]
+
+lemma Ideal.height_eq_order_height_of_isPrime [I.IsPrime] : I.height =
+    Order.height (⟨I, ‹I.IsPrime›⟩ : PrimeSpectrum R) :=
+  I.height_eq_primeHeight
+
+lemma Ideal.height_eq_inf_minimalPrimes : I.height = ⨅ J ∈ I.minimalPrimes, J.height := by
+  apply iInf_congr (fun p ↦ iInf_congr fun hp ↦ ?_)
+  have := I.minimalPrimes_isPrime hp
+  exact (Ideal.height_eq_primeHeight _).symm
 
 /-- An ideal has finite height if it is either the unit ideal or its height is finite.
 We include the unit ideal in order to have the instance `IsNoetherianRing R → FiniteHeight I`. -/

From 2eef1814e73e3a2436575228a079a766ecdbd81a Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Sat, 4 Apr 2026 14:32:21 +0800
Subject: [PATCH 16/69] deprecate more lemmas

---
 Mathlib/RingTheory/Ideal/Height.lean | 11 ++++++++++-
 1 file changed, 10 insertions(+), 1 deletion(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index ac3ee6fa917874..e72d0fddc1575c 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -72,11 +72,21 @@ lemma Ideal.height_lt_top {I : Ideal R} (hI : I ≠ ⊤) [I.FiniteHeight] :
     I.height < ⊤ :=
   (Ideal.height_ne_top hI).lt_top
 
+lemma Ideal.height_ne_top_of_isPrime {I : Ideal R} [I.FiniteHeight] [I.IsPrime] :
+    I.height ≠ ⊤ :=
+  Ideal.height_ne_top ‹I.IsPrime›.ne_top
+
+@[deprecated "Use `Ideal.height_ne_top_of_isPrime` instead." (since := "2026-04-04")]
 lemma Ideal.primeHeight_ne_top (I : Ideal R) [I.FiniteHeight] [I.IsPrime] :
     I.primeHeight ≠ ⊤ := by
   rw [← I.height_eq_primeHeight]
   exact Ideal.height_ne_top ‹I.IsPrime›.ne_top
 
+lemma Ideal.height_lt_top_of_isPrime {I : Ideal R} [I.FiniteHeight] [I.IsPrime] :
+    I.height < ⊤ :=
+  Ideal.height_lt_top ‹I.IsPrime›.ne_top
+
+@[deprecated "Use `Ideal.height_lt_top_of_isPrime` instead." (since := "2026-04-04")]
 lemma Ideal.primeHeight_lt_top (I : Ideal R) [I.FiniteHeight] [I.IsPrime] :
     I.primeHeight < ⊤ := by
   rw [← I.height_eq_primeHeight]
@@ -92,7 +102,6 @@ lemma Ideal.exists_ltSeries_length_eq_height (p : Ideal R) [p.IsPrime] [p.Finite
   rw [hn]
   exact ⟨l, last, by rw [len, WithTop.some_eq_coe, ENat.some_eq_coe]⟩
 
-@[gcongr]
 lemma Ideal.primeHeight_mono {I J : Ideal R} [I.IsPrime] [J.IsPrime] (h : I ≤ J) :
     I.primeHeight ≤ J.primeHeight := by
   unfold primeHeight

From e09911d9a6e7ff1e3ea120e6a35011494d5333a1 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Sat, 4 Apr 2026 14:40:54 +0800
Subject: [PATCH 17/69] deprecate more

---
 Mathlib/RingTheory/Ideal/Height.lean | 31 +++++++++++++++++-----------
 1 file changed, 19 insertions(+), 12 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index e72d0fddc1575c..2d6c8196ceea16 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -102,19 +102,26 @@ lemma Ideal.exists_ltSeries_length_eq_height (p : Ideal R) [p.IsPrime] [p.Finite
   rw [hn]
   exact ⟨l, last, by rw [len, WithTop.some_eq_coe, ENat.some_eq_coe]⟩
 
-lemma Ideal.primeHeight_mono {I J : Ideal R} [I.IsPrime] [J.IsPrime] (h : I ≤ J) :
-    I.primeHeight ≤ J.primeHeight := by
-  unfold primeHeight
+private lemma Ideal.height_mono_of_isPrime {I J : Ideal R} [I.IsPrime] [J.IsPrime] (h : I ≤ J) :
+    I.height ≤ J.height := by
+  simp only [Ideal.height_eq_order_height_of_isPrime]
   gcongr
   exact h
 
-private lemma Ideal.primeHeight_add_one_le_of_lt {I J : Ideal R} [I.IsPrime] [J.IsPrime]
-    (h : I < J) : I.primeHeight + 1 ≤ J.primeHeight :=
-  Order.height_add_one_le h
+@[deprecated "Use `Ideal.height_mono_of_isPrime` instead." (since := "2026-04-04")]
+lemma Ideal.primeHeight_mono {I J : Ideal R} [I.IsPrime] [J.IsPrime] (h : I ≤ J) :
+    I.primeHeight ≤ J.primeHeight := by
+  simpa [Ideal.height_eq_primeHeight] using Ideal.height_mono_of_isPrime h
 
 lemma Ideal.height_add_one_le_of_lt_of_isPrime {I J : Ideal R} [I.IsPrime] [J.IsPrime] (h : I < J) :
     I.height + 1 ≤ J.height := by
-  simpa [Ideal.height_eq_primeHeight] using Ideal.primeHeight_add_one_le_of_lt h
+  simp only [Ideal.height_eq_order_height_of_isPrime]
+  exact Order.height_add_one_le h
+
+@[deprecated "Use `Ideal.height_add_one_le_of_lt_of_isPrime` instead." (since := "2026-04-04")]
+lemma Ideal.primeHeight_add_one_le_of_lt {I J : Ideal R} [I.IsPrime] [J.IsPrime]
+    (h : I < J) : I.primeHeight + 1 ≤ J.primeHeight := by
+  simpa [Ideal.height_eq_primeHeight] using Ideal.height_add_one_le_of_lt_of_isPrime h
 
 @[simp]
 theorem Ideal.height_top : (⊤ : Ideal R).height = ⊤ := by
@@ -122,12 +129,12 @@ theorem Ideal.height_top : (⊤ : Ideal R).height = ⊤ := by
 
 @[gcongr]
 theorem Ideal.height_mono {I J : Ideal R} (h : I ≤ J) : I.height ≤ J.height := by
-  simp only [height]
+  simp only [I.height_eq_inf_minimalPrimes, J.height_eq_inf_minimalPrimes]
   refine le_iInf₂ (fun p hp ↦ ?_)
   have := Ideal.minimalPrimes_isPrime hp
   obtain ⟨q, hq, e⟩ := Ideal.exists_minimalPrimes_le (h.trans (Ideal.le_minimalPrimes hp))
-  haveI := Ideal.minimalPrimes_isPrime hq
-  exact (iInf₂_le q hq).trans (Ideal.primeHeight_mono e)
+  have := Ideal.minimalPrimes_isPrime hq
+  exact (iInf₂_le q hq).trans (Ideal.height_mono_of_isPrime e)
 
 @[gcongr]
 lemma Ideal.height_strict_mono_of_isPrime {I J : Ideal R} [I.IsPrime]
@@ -135,11 +142,11 @@ lemma Ideal.height_strict_mono_of_isPrime {I J : Ideal R} [I.IsPrime]
   by_cases hJ : J = ⊤
   · grw [hJ, height_top]
     exact I.height_lt_top IsPrime.ne_top'
-  · rw [← ENat.add_one_le_iff (I.height_ne_top IsPrime.ne_top'), I.height_eq_primeHeight]
+  · rw [← ENat.add_one_le_iff (I.height_ne_top IsPrime.ne_top'), J.height_eq_inf_minimalPrimes]
     refine le_iInf₂ (fun K hK ↦ ?_)
     have := Ideal.minimalPrimes_isPrime hK
     have : I < K := lt_of_lt_of_le h (Ideal.le_minimalPrimes hK)
-    exact Ideal.primeHeight_add_one_le_of_lt this
+    exact Ideal.height_add_one_le_of_lt_of_isPrime this
 
 lemma Ideal.height_strict_mono_of_isPrime_of_isPrime {I J : Ideal R} [I.IsPrime] [J.IsPrime]
     (h : I < J) [J.FiniteHeight] : I.height < J.height := by

From 55ca9bfedb316516ccb8c89e6d762de800cb9cd8 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Sat, 4 Apr 2026 14:48:04 +0800
Subject: [PATCH 18/69] deprecate even more

---
 Mathlib/RingTheory/Ideal/Height.lean | 11 +++++++++--
 1 file changed, 9 insertions(+), 2 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index 2d6c8196ceea16..65bb5520ff13cc 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -160,14 +160,21 @@ lemma Ideal.primeHeight_strict_mono {I J : Ideal R} [I.IsPrime] [J.IsPrime] (h :
     [J.FiniteHeight] : I.primeHeight < J.primeHeight := by
   simpa [← Ideal.height_eq_primeHeight] using Ideal.height_strict_mono_of_isPrime_of_isPrime h
 
-private lemma Ideal.primeHeight_le_ringKrullDim {I : Ideal R} [I.IsPrime] :
+lemma Ideal.height_le_ringKrullDim_of_isPrime {I : Ideal R} [I.IsPrime] :
+    I.height ≤ ringKrullDim R := by
+  rw [I.height_eq_order_height_of_isPrime]
+  exact Order.height_le_krullDim _
+
+@[deprecated "Use `Ideal.height_add_one_le_of_lt_of_isPrime` instead." (since := "2026-04-04")]
+lemma Ideal.primeHeight_le_ringKrullDim {I : Ideal R} [I.IsPrime] :
     I.primeHeight ≤ ringKrullDim R := Order.height_le_krullDim _
 
 lemma Ideal.height_le_ringKrullDim_of_ne_top {I : Ideal R} (h : I ≠ ⊤) :
     I.height ≤ ringKrullDim R := by
   obtain ⟨P, hP⟩ : Nonempty (I.minimalPrimes) := Ideal.nonempty_minimalPrimes h
+  rw [I.height_eq_inf_minimalPrimes]
   have := Ideal.minimalPrimes_isPrime hP
-  exact (WithBot.coe_le_coe.mpr (iInf₂_le _ hP)).trans P.primeHeight_le_ringKrullDim
+  refine (WithBot.coe_le_coe.mpr (iInf₂_le _ hP)).trans P.height_le_ringKrullDim_of_isPrime
 
 /-- If `R` has finite Krull dimension, there exists a maximal ideal `m` with `ht m = dim R`. -/
 lemma Ideal.exists_isMaximal_height [FiniteRingKrullDim R] :

From 618cfa23224cb857db11cd124b0a4550667b5ff8 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Sat, 4 Apr 2026 14:50:08 +0800
Subject: [PATCH 19/69] deprecate even more

---
 Mathlib/RingTheory/Ideal/Height.lean | 15 ++++++++-------
 1 file changed, 8 insertions(+), 7 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index 65bb5520ff13cc..54a4f100209f74 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -252,17 +252,18 @@ theorem Ideal.isMaximal_of_primeHeight_eq_ringKrullDim {I : Ideal R} [I.IsPrime]
     [FiniteRingKrullDim R] (e : I.primeHeight = ringKrullDim R) : I.IsMaximal :=
   Ideal.isMaximal_of_height_eq_ringKrullDim (by simpa [Ideal.height_eq_primeHeight])
 
-/-- The prime height of the maximal ideal equals the Krull dimension in a local ring -/
-private theorem IsLocalRing.maximalIdeal_primeHeight_eq_ringKrullDim [IsLocalRing R] :
-    (IsLocalRing.maximalIdeal R).primeHeight = ringKrullDim R := by
-  rw [ringKrullDim, Ideal.primeHeight, ← Order.height_top_eq_krullDim]
-  rfl
-
 /-- The height of the maximal ideal equals the Krull dimension in a local ring. -/
 @[simp]
 theorem IsLocalRing.maximalIdeal_height_eq_ringKrullDim [IsLocalRing R] :
     (IsLocalRing.maximalIdeal R).height = ringKrullDim R := by
-  rw [Ideal.height_eq_primeHeight, IsLocalRing.maximalIdeal_primeHeight_eq_ringKrullDim]
+  rw [Ideal.height_eq_order_height_of_isPrime]
+  exact Order.height_top_eq_krullDim
+
+@[deprecated "Use `IsLocalRing.maximalIdeal_height_eq_ringKrullDim` instead."
+  (since := "2026-04-04")]
+private theorem IsLocalRing.maximalIdeal_primeHeight_eq_ringKrullDim [IsLocalRing R] :
+    (IsLocalRing.maximalIdeal R).primeHeight = ringKrullDim R := by
+  simp [← Ideal.height_eq_primeHeight]
 
 /-- For a local ring with finite Krull dimension, a prime ideal has height equal to the Krull
 dimension if and only if it is the maximal ideal. -/

From ab901dba051e5bb622cec0732f71963243eb7cfb Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Sat, 4 Apr 2026 15:00:48 +0800
Subject: [PATCH 20/69] deprecate a prime height usage

---
 Mathlib/RingTheory/Ideal/Height.lean | 10 ++++------
 1 file changed, 4 insertions(+), 6 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index d9b438d25a8514..aa35455cd75b9e 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -449,7 +449,7 @@ lemma Ideal.sup_primeHeight_eq_ringKrullDim [Nontrivial R] :
     ↑(⨆ (I : Ideal R) (_ : I.IsPrime), I.primeHeight) = ringKrullDim R := by
   simp [← Ideal.height_eq_primeHeight, Ideal.sup_height_isPrime_eq_ringKrullDim]
 
-/-- In a nontrivial commutative ring `R`, the supremum of prime heights of all maximal ideals is
+/-- In a nontrivial commutative ring `R`, the supremum of heights of all maximal ideals is
 equal to the Krull dimension of `R`. -/
 lemma Ideal.sup_height_of_maximal_eq_ringKrullDim [Nontrivial R] :
     ↑(⨆ (I : Ideal R) (_ : I.IsMaximal), I.height) = ringKrullDim R := by
@@ -462,8 +462,7 @@ lemma Ideal.sup_height_of_maximal_eq_ringKrullDim [Nontrivial R] :
     · obtain ⟨M, hM, hIM⟩ := exists_le_maximal I I_top
       exact ⟨M, iSup_mono' (fun hI ↦ ⟨hM, height_mono hIM⟩)⟩
 
-/-- In a nontrivial commutative ring `R`, the supremum of prime heights of all maximal ideals is
-equal to the Krull dimension of `R`. -/
+@[deprecated "Use `Ideal.sup_height_of_maximal_eq_ringKrullDim` instead." (since := "2026-04-02")]
 lemma Ideal.sup_primeHeight_of_maximal_eq_ringKrullDim [Nontrivial R] :
     ↑(⨆ (I : Ideal R) (_ : I.IsMaximal), I.primeHeight) = ringKrullDim R := by
   simp_rw [← Ideal.height_eq_primeHeight, Ideal.sup_height_of_maximal_eq_ringKrullDim]
@@ -481,10 +480,9 @@ lemma Ring.krullDimLE_of_isLocalization_maximal {n : ℕ}
     Ring.KrullDimLE n R := by
   simp_rw [Ring.krullDimLE_iff] at h ⊢
   nontriviality R
-  rw [← Ideal.sup_primeHeight_of_maximal_eq_ringKrullDim]
+  rw [← Ideal.sup_height_of_maximal_eq_ringKrullDim]
   refine (WithBot.coe_le_coe).mpr (iSup₂_le_iff.mpr fun P hP ↦ ?_)
-  rw [← Ideal.height_eq_primeHeight, ← WithBot.coe_le_coe]
-  rw [← IsLocalization.AtPrime.ringKrullDim_eq_height P (Rₚ P)]
+  rw [← WithBot.coe_le_coe, ← IsLocalization.AtPrime.ringKrullDim_eq_height P (Rₚ P)]
   exact h P
 
 end isLocalization

From cd6b68c9b806c66b9320f92e8546b8c50ec520da Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Sat, 4 Apr 2026 15:06:07 +0800
Subject: [PATCH 21/69] deprecate further more

---
 Mathlib/RingTheory/Ideal/Height.lean | 22 +++++++++++++++-------
 1 file changed, 15 insertions(+), 7 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index 9d619647c6c5a3..9232a3e5a3c758 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -346,11 +346,11 @@ lemma ringKrullDim_le_iff_isMaximal_height_le {R : Type*} [CommRing R] (n : With
   norm_cast
   exact Ideal.height_mono hle
 
-private theorem IsLocalization.primeHeight_comap (S : Submonoid R) {A : Type*} [CommRing A]
+private theorem IsLocalization.height_comap_eq_of_isPrime (S : Submonoid R) {A : Type*} [CommRing A]
     [Algebra R A] [IsLocalization S A] (J : Ideal A) [J.IsPrime] :
-    (J.comap (algebraMap R A)).primeHeight = J.primeHeight := by
-  rw [eq_comm, Ideal.primeHeight, Ideal.primeHeight, ← WithBot.coe_inj,
-    Order.height_eq_krullDim_Iic, Order.height_eq_krullDim_Iic]
+    (J.comap (algebraMap R A)).height = J.height := by
+  rw [eq_comm, Ideal.height_eq_order_height_of_isPrime, Ideal.height_eq_order_height_of_isPrime,
+    ← WithBot.coe_inj, Order.height_eq_krullDim_Iic, Order.height_eq_krullDim_Iic]
   let e := IsLocalization.orderIsoOfPrime S A
   have H (p : Ideal R) (hp : p ≤ J.comap (algebraMap R A)) : Disjoint (S : Set R) p :=
     Set.disjoint_of_subset_right hp (e ⟨_, ‹J.IsPrime›⟩).2.2
@@ -363,11 +363,19 @@ private theorem IsLocalization.primeHeight_comap (S : Submonoid R) {A : Type*} [
         congrArg (fun I ↦ I.1) (e.right_inv ⟨_, I.1.2, H _ I.2⟩)
       map_rel_iff' {I₁ I₂} := @RelIso.map_rel_iff _ _ _ _ e ⟨_, I₁.1.2⟩ ⟨_, I₂.1.2⟩ }
 
+@[deprecated "Use `Ideal.height_ne_top_of_isPrime` instead." (since := "2026-04-04")]
+private theorem IsLocalization.primeHeight_comap (S : Submonoid R) {A : Type*} [CommRing A]
+    [Algebra R A] [IsLocalization S A] (J : Ideal A) [J.IsPrime] :
+    (J.comap (algebraMap R A)).primeHeight = J.primeHeight := by
+  simpa [Ideal.height_eq_primeHeight] using IsLocalization.height_comap_eq_of_isPrime S J
+
 theorem IsLocalization.height_comap (S : Submonoid R) {A : Type*} [CommRing A] [Algebra R A]
     [IsLocalization S A] (J : Ideal A) : (J.comap (algebraMap R A)).height = J.height := by
-  rw [Ideal.height, Ideal.height]
-  simp_rw [← IsLocalization.primeHeight_comap S, IsLocalization.minimalPrimes_comap S A,
-    ← Ideal.height_eq_primeHeight, iInf_image]
+  rw [(J.comap _).height_eq_inf_minimalPrimes, J.height_eq_inf_minimalPrimes]
+  simp only [IsLocalization.minimalPrimes_comap S A, iInf_image]
+  apply iInf_congr (fun p ↦ iInf_congr fun hp ↦ ?_)
+  have := Ideal.minimalPrimes_isPrime hp
+  exact IsLocalization.height_comap_eq_of_isPrime S _
 
 theorem IsLocalization.AtPrime.ringKrullDim_eq_height (I : Ideal R) [I.IsPrime] (A : Type*)
     [CommRing A] [Algebra R A] [IsLocalization.AtPrime A I] :

From 375cdb828900d5dbaf9e6bdac5139f1a24aa6d42 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Sat, 4 Apr 2026 15:14:39 +0800
Subject: [PATCH 22/69] golf

---
 Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean | 7 +++----
 1 file changed, 3 insertions(+), 4 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
index bf90d4a888ced7..aed2a13495c071 100644
--- a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
+++ b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
@@ -237,9 +237,8 @@ than or equal to the minimum number of generators for this ideal. -/
 lemma Ideal.height_le_spanRank_toENat (I : Ideal R) (hI : I ≠ ⊤) :
     I.height ≤ I.spanRank.toENat := by
   obtain ⟨J, hJ⟩ := nonempty_minimalPrimes hI
-  refine (iInf₂_le J hJ).trans ?_
-  convert (I.height_le_spanRank_toENat_of_mem_minimal_primes J hJ)
-  exact Eq.symm (@height_eq_primeHeight _ _ J (Ideal.minimalPrimes_isPrime hJ))
+  rw [I.height_eq_inf_minimalPrimes]
+  exact (iInf₂_le J hJ).trans (I.height_le_spanRank_toENat_of_mem_minimal_primes J hJ)
 
 lemma Ideal.height_le_spanFinrank (I : Ideal R) (hI : I ≠ ⊤) :
     I.height ≤ I.spanFinrank := by
@@ -466,7 +465,7 @@ lemma Ideal.height_eq_height_add_of_liesOver_of_hasGoingDown [IsNoetherianRing S
       simp [hlq, map_le_iff_le_comap, LiesOver.over (p := p) (P := P)]
   obtain ⟨lp', hlp'len, hlp', _⟩ := exists_ltSeries_of_hasGoingDown lp l'.head.asIdeal
   have : (lp'.smash l' hlp').length = lp.length + lq.length := by simp [hlp'len, l']
-  rw [← hlenp, ← hlenq, ← Nat.cast_add, ← this, height_eq_primeHeight]
+  rw [← hlenp, ← hlenq, ← Nat.cast_add, ← this, Ideal.height_eq_order_height_of_isPrime]
   apply Order.length_le_height
   simp [hlq, l', ← PrimeSpectrum.asIdeal_le_asIdeal, map_le_iff_le_comap,
     LiesOver.over (p := p) (P := P)]

From 9790c4fb12712acd3eb242c1b76827e0d9b5fc7e Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Tue, 14 Apr 2026 15:08:27 +0800
Subject: [PATCH 23/69] add lemma to avoid def eq

---
 .../RingTheory/Ideal/MinimalPrime/Basic.lean  | 23 ++++++++-----
 .../RingTheory/Ideal/MinimalPrime/Colon.lean  | 12 +++----
 .../Ideal/MinimalPrime/Localization.lean      | 33 ++++++++++---------
 3 files changed, 38 insertions(+), 30 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
index 6338b28d7d2623..689fddcfb9f092 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
@@ -54,6 +54,12 @@ variable {I J}
 theorem Ideal.minimalPrimes_isPrime {p : Ideal R} (h : p ∈ I.minimalPrimes) : p.IsPrime :=
   h.1.1
 
+theorem Ideal.le_minimalPrimes {p : Ideal R} (h : p ∈ I.minimalPrimes) : I ≤ p :=
+  h.1.2
+
+theorem minimalPrimes_isPrime {p : Ideal R} (h : p ∈ minimalPrimes R) : p.IsPrime :=
+  h.1.1
+
 theorem Ideal.exists_minimalPrimes_le [J.IsPrime] (e : I ≤ J) : ∃ p ∈ I.minimalPrimes, p ≤ J := by
   set S := { p : (Ideal R)ᵒᵈ | Ideal.IsPrime p ∧ I ≤ OrderDual.ofDual p }
   suffices h : ∃ m, OrderDual.toDual J ≤ m ∧ Maximal (· ∈ S) m by
@@ -117,17 +123,17 @@ theorem Ideal.minimalPrimes_eq_subsingleton (hI : I.IsPrimary) : I.minimalPrimes
   ext J
   constructor
   · exact fun H =>
-      let e := H.1.1.radical_le_iff.mpr H.1.2
+      let e := (Ideal.minimalPrimes_isPrime H).radical_le_iff.mpr (Ideal.le_minimalPrimes H)
       (H.2 ⟨Ideal.isPrime_radical hI, Ideal.le_radical⟩ e).antisymm e
   · rintro (rfl : J = I.radical)
     exact ⟨⟨Ideal.isPrime_radical hI, Ideal.le_radical⟩, fun _ H _ => H.1.radical_le_iff.mpr H.2⟩
 
 theorem Ideal.minimalPrimes_eq_subsingleton_self [I.IsPrime] : I.minimalPrimes = {I} := by
   ext J
-  constructor
-  · exact fun H => (H.2 ⟨inferInstance, rfl.le⟩ H.1.2).antisymm H.1.2
-  · rintro (rfl : J = I)
-    exact ⟨⟨inferInstance, rfl.le⟩, fun _ h _ => h.2⟩
+  refine ⟨fun H => (H.2 ⟨inferInstance, rfl.le⟩ (Ideal.le_minimalPrimes H)).antisymm
+    (Ideal.le_minimalPrimes H), ?_⟩
+  rintro (rfl : J = I)
+  exact ⟨⟨inferInstance, rfl.le⟩, fun _ h _ => h.2⟩
 
 variable (R) in
 theorem IsDomain.minimalPrimes_eq_singleton_bot [IsDomain R] :
@@ -144,7 +150,7 @@ theorem Ideal.minimalPrimes_top : (⊤ : Ideal R).minimalPrimes = ∅ := by
   ext p
   simp only [Set.notMem_empty, iff_false]
   intro h
-  exact h.1.1.ne_top (top_le_iff.mp h.1.2)
+  exact (Ideal.minimalPrimes_isPrime h).ne_top (top_le_iff.mp (Ideal.le_minimalPrimes h))
 
 theorem Ideal.minimalPrimes_eq_empty_iff (I : Ideal R) :
     I.minimalPrimes = ∅ ↔ I = ⊤ := by
@@ -163,7 +169,8 @@ lemma Ideal.mem_minimalPrimes_sup {R : Type*} [CommRing R] {p I J : Ideal R} [p.
     p ∈ (I ⊔ J).minimalPrimes := by
   refine ⟨⟨‹_›, ?_⟩, fun q ⟨_, hq⟩ hqp ↦ ?_⟩
   · rw [sup_le_iff]
-    exact ⟨hle, by simpa [hle] using Ideal.comap_mono (f := Ideal.Quotient.mk I) h.1.2⟩
+    refine ⟨hle, ?_⟩
+    simpa [hle] using Ideal.comap_mono (f := Ideal.Quotient.mk I) (Ideal.le_minimalPrimes h)
   · rw [sup_le_iff] at hq
     have h2 : p.map (Quotient.mk I) ≤ q.map (Quotient.mk I) :=
       h.2 ⟨isPrime_map_quotientMk_of_isPrime hq.1, map_mono hq.2⟩ (map_mono hqp)
@@ -183,7 +190,7 @@ lemma Ideal.map_sup_mem_minimalPrimes_of_map_quotientMk_mem_minimalPrimes
   refine ⟨⟨inferInstance, sup_le_iff.mpr ?_⟩, fun q ⟨_, hleq⟩ hqle ↦ ?_⟩
   · refine ⟨?_, hJP⟩
     rw [Ideal.map_le_iff_le_comap, ← Ideal.under_def, ← Ideal.over_def P p]
-    exact hI.1.2
+    exact Ideal.le_minimalPrimes hI
   · simp only [sup_le_iff] at hleq
     have h1 : p.map (algebraMap R S) ≤ q := by
       rw [Ideal.map_le_iff_le_comap]
diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Colon.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Colon.lean
index 9ecd208c7666f6..8758fea64c554d 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Colon.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Colon.lean
@@ -48,7 +48,8 @@ theorem exists_eq_colon_of_mem_minimalPrimes [IsNoetherianRing R]
     refine ⟨n, hn0, ((h.toFinset.erase I).inf id) ^ n, hn, ?_⟩
     have (K : Ideal R) (hKI : K ≤ I) (hK : K ∈ ann.minimalPrimes) : K = I :=
       le_antisymm hKI (hI.2 hK.1 hKI)
-    simpa [hI.1.1.pow_le_iff hn0, hI.1.1.inf_le', imp_not_comm, not_imp_not]
+    simpa [(Ideal.minimalPrimes_isPrime hI).pow_le_iff hn0,
+      (Ideal.minimalPrimes_isPrime hI).inf_le', imp_not_comm, not_imp_not]
   obtain ⟨hn0, J, hJ, hJI⟩ := Nat.find_spec key
   -- let `n` be minimal such that there exists an ideal `J` with `I ^ n * J ≤ ann` and `¬ J ≠ I`
   set n := Nat.find key
@@ -74,9 +75,7 @@ theorem exists_eq_colon_of_mem_minimalPrimes [IsNoetherianRing R]
   -- let `z` be the product of these finitely many `f y`'s
   let z := ∏ y ∈ s, f y
   -- then `z ∉ I`
-  have hz : z ∉ I := by
-    simp only [z, hI.1.1.prod_mem_iff, not_exists, not_and_or]
-    exact fun i ↦ Or.inr (h i)
+  have hz : z ∉ I := by simp [z, (Ideal.minimalPrimes_isPrime hI).prod_mem_iff, h]
   -- and `K ≤ colon N {z • x}`
   have hz' : K ≤ colon N {z • x} := by
     rw [← (map_injective_of_injective K.subtype_injective).eq_iff, map_subtype_top] at hs
@@ -92,11 +91,12 @@ theorem exists_eq_colon_of_mem_minimalPrimes [IsNoetherianRing R]
     simpa only [ann, mem_colon_singleton, mul_comm, mul_smul] using hz' hi
   -- but now `K = I ^ (n - 1) * J` contradicts the minimality of `n`
   have hK : I ^ (n - 1) * (J * Ideal.span {z}) ≤ ann ∧ ¬ J * Ideal.span {z} ≤ I := by
-    rw [← mul_assoc, hI.1.1.mul_le, not_or, Ideal.span_singleton_le_iff_mem]
+    rw [← mul_assoc, (Ideal.minimalPrimes_isPrime hI).mul_le, not_or,
+      Ideal.span_singleton_le_iff_mem]
     exact ⟨hz', hJI, hz⟩
   by_cases hn' : n - 1 = 0
   · simp [K, show n = 1 by grind] at hz'
-    exact (hK.2 (hz'.trans hI.1.2)).elim
+    exact (hK.2 (hz'.trans (Ideal.le_minimalPrimes hI))).elim
   · grind [Nat.find_min' key ⟨hn', J * Ideal.span {z}, hK⟩]
 
 end Submodule
diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean
index 7d76dbf783fb02..5b07cd7cc83ae0 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean
@@ -65,8 +65,8 @@ theorem Ideal.iUnion_minimalPrimes :
   · rintro ⟨y, hy, hx⟩
     obtain ⟨p, hp, hyp⟩ : ∃ p ∈ I.minimalPrimes, y ∉ p := by
       simpa [← Ideal.sInf_minimalPrimes] using hy
-    refine ⟨p, hp, (hp.1.1.mem_or_mem ?_).resolve_right hyp⟩
-    exact hp.1.1.radical_le_iff.mpr hp.1.2 hx
+    refine ⟨p, hp, ((Ideal.minimalPrimes_isPrime hp).mem_or_mem ?_).resolve_right hyp⟩
+    exact (Ideal.minimalPrimes_isPrime hp).radical_le_iff.mpr (Ideal.le_minimalPrimes hp) hx
 
 theorem Ideal.exists_mul_mem_of_mem_minimalPrimes
     {p : Ideal R} (hp : p ∈ I.minimalPrimes) {x : R} (hx : x ∈ p) :
@@ -97,8 +97,8 @@ lemma Ideal.disjoint_nonZeroDivisors_of_mem_minimalPrimes {p : Ideal R} (hp : p
 
 theorem Ideal.exists_comap_eq_of_mem_minimalPrimes {I : Ideal S} (f : R →+* S) (p)
     (H : p ∈ (I.comap f).minimalPrimes) : ∃ p' : Ideal S, p'.IsPrime ∧ I ≤ p' ∧ p'.comap f = p :=
-  have := H.1.1
-  have ⟨p', hIp', hp', le⟩ := exists_ideal_comap_le_prime p I H.1.2
+  have := Ideal.minimalPrimes_isPrime H
+  have ⟨p', hIp', hp', le⟩ := exists_ideal_comap_le_prime p I (Ideal.le_minimalPrimes H)
   ⟨p', hp', hIp', le.antisymm (H.2 ⟨inferInstance, comap_mono hIp'⟩ le)⟩
 
 @[stacks 00FK] theorem Ideal.exists_comap_eq_of_mem_minimalPrimes_of_injective {f : R →+* S}
@@ -113,9 +113,9 @@ theorem Ideal.exists_minimalPrimes_comap_eq {I : Ideal S} (f : R →+* S) (p)
   obtain ⟨p', h₁, h₂, h₃⟩ := Ideal.exists_comap_eq_of_mem_minimalPrimes f p H
   obtain ⟨q, hq, hq'⟩ := Ideal.exists_minimalPrimes_le h₂
   refine ⟨q, hq, Eq.symm ?_⟩
-  have := hq.1.1
+  have := Ideal.minimalPrimes_isPrime hq
   have := (Ideal.comap_mono hq').trans_eq h₃
-  exact (H.2 ⟨inferInstance, Ideal.comap_mono hq.1.2⟩ this).antisymm this
+  exact (H.2 ⟨inferInstance, Ideal.comap_mono (Ideal.le_minimalPrimes hq)⟩ this).antisymm this
 
 theorem Ideal.minimalPrimes_comap_subset (f : R →+* S) (J : Ideal S) :
     (J.comap f).minimalPrimes ⊆ Ideal.comap f '' J.minimalPrimes :=
@@ -129,8 +129,8 @@ variable {R S : Type*} [CommRing R] [CommRing S] {I J : Ideal R}
 
 theorem Ideal.minimal_primes_comap_of_surjective {f : R →+* S} (hf : Function.Surjective f)
     {I J : Ideal S} (h : J ∈ I.minimalPrimes) : J.comap f ∈ (I.comap f).minimalPrimes := by
-  have := h.1.1
-  refine ⟨⟨inferInstance, Ideal.comap_mono h.1.2⟩, ?_⟩
+  have := Ideal.minimalPrimes_isPrime h
+  refine ⟨⟨inferInstance, Ideal.comap_mono (Ideal.le_minimalPrimes h)⟩, ?_⟩
   rintro K ⟨hK, e₁⟩ e₂
   have : RingHom.ker f ≤ K := (Ideal.comap_mono bot_le).trans e₁
   rw [← sup_eq_left.mpr this, RingHom.ker_eq_comap_bot, ← Ideal.comap_map_of_surjective f hf]
@@ -156,7 +156,8 @@ lemma Ideal.minimalPrimes_map_of_surjective {S : Type*} [CommRing S] {f : R →+
   rw [← Ideal.comap_minimalPrimes_eq_of_surjective hf, ← Set.image_comp,
     Ideal.comap_map_of_surjective f hf, Set.image_congr, Set.image_id, RingHom.ker]
   intro x hx
-  exact (Ideal.comap_map_of_surjective f hf _).trans (sup_eq_left.mpr <| le_sup_right.trans hx.1.2)
+  exact (Ideal.comap_map_of_surjective f hf _).trans
+    (sup_eq_left.mpr <| le_sup_right.trans (Ideal.le_minimalPrimes hx))
 
 theorem Ideal.minimalPrimes_eq_comap :
     I.minimalPrimes = Ideal.comap (Ideal.Quotient.mk I) '' minimalPrimes (R ⧸ I) := by
@@ -174,21 +175,21 @@ theorem IsLocalization.minimalPrimes_map [IsLocalization S A] (J : Ideal R) :
   ext p
   constructor
   · intro hp
-    haveI := hp.1.1
-    refine ⟨⟨Ideal.IsPrime.comap _, Ideal.map_le_iff_le_comap.mp hp.1.2⟩, ?_⟩
+    haveI := Ideal.minimalPrimes_isPrime hp
+    refine ⟨⟨Ideal.IsPrime.comap _, Ideal.map_le_iff_le_comap.mp (Ideal.le_minimalPrimes hp)⟩, ?_⟩
     rintro I hI e
     have hI' : Disjoint (S : Set R) I := Set.disjoint_of_subset_right e
-      ((IsLocalization.isPrime_iff_isPrime_disjoint S A _).mp hp.1.1).2
+      ((IsLocalization.isPrime_iff_isPrime_disjoint S A _).mp (Ideal.minimalPrimes_isPrime hp)).2
     refine (Ideal.comap_mono <|
       hp.2 ⟨?_, Ideal.map_mono hI.2⟩ (Ideal.map_le_iff_le_comap.mpr e)).trans_eq ?_
     · exact IsLocalization.isPrime_of_isPrime_disjoint S A I hI.1 hI'
     · exact IsLocalization.comap_map_of_isPrime_disjoint S A hI.1 hI'
   · intro hp
-    refine ⟨⟨?_, Ideal.map_le_iff_le_comap.mpr hp.1.2⟩, ?_⟩
+    refine ⟨⟨?_, Ideal.map_le_iff_le_comap.mpr (Ideal.le_minimalPrimes hp)⟩, ?_⟩
     · rw [IsLocalization.isPrime_iff_isPrime_disjoint S A, IsLocalization.disjoint_comap_iff S]
-      refine ⟨hp.1.1, ?_⟩
+      refine ⟨Ideal.minimalPrimes_isPrime hp, ?_⟩
       rintro rfl
-      exact hp.1.1.ne_top rfl
+      exact (Ideal.minimalPrimes_isPrime hp).ne_top rfl
     · intro I hI e
       rw [← IsLocalization.map_comap S A I, ← IsLocalization.map_comap S A p]
       haveI := hI.1
@@ -212,7 +213,7 @@ theorem IsLocalization.AtPrime.radical_map_of_mem_minimalPrimes
   · rw [← IsLocalization.comap_le_comap_iff q.primeCompl A,
       AtPrime.map_eq_maximalIdeal q A, AtPrime.comap_maximalIdeal A q]
     apply hIq.2 hJ.1
-    have := hJ.1.1.ne_top
+    have := (Ideal.minimalPrimes_isPrime hJ).ne_top
     rw [ne_eq, Ideal.comap_eq_top_iff, ← ne_eq, ← disjoint_comap_iff q.primeCompl A J] at this
     exact Set.disjoint_compl_left_iff_subset.mp this
 

From 903a565b18e95a20baa90f755ce74d295a272d07 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Tue, 14 Apr 2026 15:22:13 +0800
Subject: [PATCH 24/69] replace more .1.1 and .1.2

---
 .../Ideal/AssociatedPrime/Localization.lean   |  5 +--
 Mathlib/RingTheory/Ideal/Height.lean          | 25 +++++++-------
 .../RingTheory/Ideal/KrullsHeightTheorem.lean | 33 ++++++++++---------
 3 files changed, 35 insertions(+), 28 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.lean b/Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.lean
index 37909f3465e6de..2d07b507186198 100644
--- a/Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.lean
+++ b/Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.lean
@@ -131,10 +131,11 @@ variable (R M) in
 lemma minimalPrimes_annihilator_subset_associatedPrimes [IsNoetherianRing R] [Module.Finite R M] :
     (Module.annihilator R M).minimalPrimes ⊆ associatedPrimes R M := by
   intro p hp
-  have prime := hp.1.1
+  have prime := Ideal.minimalPrimes_isPrime hp
   let Rₚ := Localization.AtPrime p
   have : Nontrivial (LocalizedModule p.primeCompl M) := by
-    simpa [← Module.mem_support_iff (p := ⟨p, prime⟩), Module.support_eq_zeroLocus] using hp.1.2
+    simpa [← Module.mem_support_iff (p := ⟨p, prime⟩), Module.support_eq_zeroLocus] using
+      Ideal.le_minimalPrimes hp
   rcases associatedPrimes.nonempty Rₚ (LocalizedModule p.primeCompl M) with ⟨q, hq⟩
   have q_prime : q.IsPrime := IsAssociatedPrime.isPrime hq
   simp only [← preimage_comap_associatedPrimes_eq_associatedPrimes_of_isLocalizedModule p.primeCompl
diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index 28ae520c1cff57..4e84e280aea7c9 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -117,7 +117,7 @@ theorem Ideal.height_mono {I J : Ideal R} (h : I ≤ J) : I.height ≤ J.height
   simp only [height]
   refine le_iInf₂ (fun p hp ↦ ?_)
   have := Ideal.minimalPrimes_isPrime hp
-  obtain ⟨q, hq, e⟩ := Ideal.exists_minimalPrimes_le (h.trans hp.1.2)
+  obtain ⟨q, hq, e⟩ := Ideal.exists_minimalPrimes_le (h.trans (Ideal.le_minimalPrimes hp))
   haveI := Ideal.minimalPrimes_isPrime hq
   exact (iInf₂_le q hq).trans (Ideal.primeHeight_mono e)
 
@@ -131,7 +131,7 @@ lemma Ideal.height_strict_mono_of_is_prime {I J : Ideal R} [I.IsPrime]
   · rw [← ENat.add_one_le_iff I.primeHeight_ne_top, Ideal.height]
     refine le_iInf₂ (fun K hK ↦ ?_)
     haveI := Ideal.minimalPrimes_isPrime hK
-    have : I < K := lt_of_lt_of_le h hK.1.2
+    have : I < K := lt_of_lt_of_le h (Ideal.le_minimalPrimes hK)
     exact Ideal.primeHeight_add_one_le_of_lt this
 
 lemma Ideal.primeHeight_le_ringKrullDim {I : Ideal R} [I.IsPrime] :
@@ -179,10 +179,10 @@ lemma Ideal.mem_minimalPrimes_of_height_eq {I J : Ideal R} (e : I ≤ J) [J.IsPr
   obtain ⟨p, h₁, h₂⟩ := Ideal.exists_minimalPrimes_le e
   convert h₁
   refine (eq_of_le_of_not_lt h₂ fun h₃ ↦ ?_).symm
-  have := h₁.1.1
+  have := Ideal.minimalPrimes_isPrime h₁
   have := finiteHeight_of_le h₂ IsPrime.ne_top'
-  exact lt_irrefl _
-     ((height_strict_mono_of_is_prime h₃).trans_le (e'.trans <| height_mono h₁.1.2))
+  exact lt_irrefl _ ((height_strict_mono_of_is_prime h₃).trans_le
+    (e'.trans <| height_mono (Ideal.le_minimalPrimes h₁)))
 
 /-- A prime ideal has height zero if and only if it is minimal -/
 lemma Ideal.primeHeight_eq_zero_iff {I : Ideal R} [I.IsPrime] :
@@ -201,7 +201,9 @@ lemma Ideal.primeHeight_eq_zero_iff {I : Ideal R} [I.IsPrime] :
 lemma Ideal.height_bot [Nontrivial R] : (⊥ : Ideal R).height = 0 := by
   obtain ⟨p, hp⟩ := Ideal.nonempty_minimalPrimes (R := R) (I := ⊥) top_ne_bot.symm
   simp only [Ideal.height, ENat.iInf_eq_zero]
-  exact ⟨p, hp, haveI := hp.1.1; primeHeight_eq_zero_iff.mpr hp⟩
+  refine ⟨p, hp, ?_⟩
+  have := Ideal.minimalPrimes_isPrime hp
+  rw [primeHeight_eq_zero_iff.mpr hp]
 
 /-- In a trivial commutative ring, the height of any ideal is `∞`. -/
 @[simp, nontriviality]
@@ -281,7 +283,7 @@ lemma RingEquiv.height_comap {S : Type*} [CommRing S] (e : R ≃+* S) (I : Ideal
     rw [← Ideal.comap_coe,
       Ideal.comap_minimalPrimes_eq_of_surjective (f := (↑e : R →+* S)) e.surjective]
     exact e.idealComapOrderIso.injective.mem_set_image.symm
-  · have : J.IsPrime := h.1.1
+  · have : J.IsPrime := Ideal.minimalPrimes_isPrime h
     simp only [EquivLike.coe_coe, RingEquiv.idealComapOrderIso_apply,
       ← Ideal.height_eq_primeHeight, RingEquiv.height_comap_of_isPrime]
 
@@ -375,7 +377,7 @@ lemma exists_spanRank_le_and_le_height_of_le_height [IsNoetherianRing R] (I : Id
       refine (Ideal.subset_union_prime ⊥ ⊥ ?_).not.mpr ?_
       · rintro K hK - -
         rw [Set.Finite.mem_toFinset] at hK
-        exact hK.1.1.1
+        exact Ideal.minimalPrimes_isPrime hK.1
       · push Not
         intro K hK e
         have := hr.trans (Ideal.height_mono e)
@@ -391,7 +393,7 @@ lemma exists_spanRank_le_and_le_height_of_le_height [IsNoetherianRing R] (I : Id
       push_cast
       exact add_le_add h₂ ((Submodule.spanRank_span_le_card _).trans (by simp))
     · refine le_iInf₂ (fun p hp ↦ ?_)
-      have := hp.1.1
+      have := Ideal.minimalPrimes_isPrime hp
       by_cases h : p.height = ⊤
       · rw [← p.height_eq_primeHeight, h]
         exact le_top
@@ -408,9 +410,10 @@ lemma exists_spanRank_le_and_le_height_of_le_height [IsNoetherianRing R] (I : Id
           apply le_antisymm
           · rwa [← p.height_eq_primeHeight]
           · rwa [← e]
-        refine ⟨mem_minimalPrimes_of_primeHeight_eq_height (le_sup_left.trans hp.1.2) this.symm, ?_⟩
+        refine ⟨mem_minimalPrimes_of_primeHeight_eq_height
+          (le_sup_left.trans (Ideal.le_minimalPrimes hp)) this.symm, ?_⟩
         rwa [p.height_eq_primeHeight, eq_comm]
-      · exact hp.1.2 <| Ideal.mem_sup_right <| Ideal.subset_span <| Set.mem_singleton x
+      · exact (Ideal.le_minimalPrimes hp) <| Ideal.mem_sup_right <| mem_span_singleton_self x
 
 /-- In a nontrivial commutative ring `R`, the supremum of heights of all ideals is equal to the
 Krull dimension of `R`. -/
diff --git a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
index 0d273c352d3b7b..f34d9276bde91f 100644
--- a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
+++ b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
@@ -83,7 +83,7 @@ lemma Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes_of_isLocalRing
     ← (IsLocalization.orderEmbedding q.primeCompl (Localization.AtPrime q)).map_rel_iff]
   refine Submodule.le_of_le_smul_of_le_jacobson_bot (I := I) (IsNoetherian.noetherian _) ?_ ?_
   · rw [IsLocalRing.jacobson_eq_maximalIdeal]
-    exacts [hp.1.2, bot_ne_top]
+    exacts [Ideal.le_minimalPrimes hp, bot_ne_top]
   · replace hn := congr(Ideal.comap (Ideal.Quotient.mk I) $(hn _ n.le_succ))
     simp only [qs, OrderHom.coe_mk, ← RingHom.ker_eq_comap_bot, Ideal.mk_ker,
       Ideal.comap_map_of_surjective _ Ideal.Quotient.mk_surjective] at hn
@@ -106,7 +106,7 @@ lemma Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes_of_isLocalRing
   has height at most 1. -/
 lemma Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes
     (I : Ideal R) [I.IsPrincipal] (p : Ideal R) (hp : p ∈ I.minimalPrimes) : p.height ≤ 1 := by
-  have := hp.1.1
+  have := Ideal.minimalPrimes_isPrime hp
   let f := algebraMap R (Localization.AtPrime p)
   have := Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes_of_isLocalRing (I.map f) ?_
   · rwa [← IsLocalization.height_comap p.primeCompl,
@@ -117,14 +117,15 @@ lemma Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes
 theorem Ideal.map_height_le_one_of_mem_minimalPrimes {I p : Ideal R} {x : R}
     (hp : p ∈ (I ⊔ span {x}).minimalPrimes) : (p.map (Ideal.Quotient.mk I)).height ≤ 1 :=
   let f := Ideal.Quotient.mk I
-  have : p.IsPrime := hp.1.1
-  have hfp : RingHom.ker f ≤ p := I.mk_ker.trans_le (le_sup_left.trans hp.1.2)
+  have : p.IsPrime := Ideal.minimalPrimes_isPrime hp
+  have hfp : RingHom.ker f ≤ p := I.mk_ker.trans_le (le_sup_left.trans (Ideal.le_minimalPrimes hp))
   height_le_one_of_isPrincipal_of_mem_minimalPrimes ((span {x}).map f) (p.map f)
-    ⟨⟨map_isPrime_of_surjective Quotient.mk_surjective hfp, map_mono (le_sup_right.trans hp.1.2)⟩,
-      fun _ ⟨hr, hxr⟩ hrp ↦ map_le_iff_le_comap.mpr <| hp.2 ⟨hr.comap f, sup_le_iff.mpr
-        ⟨I.mk_ker.symm.trans_le <| ker_le_comap (Ideal.Quotient.mk I), le_comap_of_map_le hxr⟩⟩ <|
-          (comap_mono hrp).trans <| Eq.le <|
-            (p.comap_map_of_surjective _ Quotient.mk_surjective).trans <| sup_eq_left.mpr hfp⟩
+    ⟨⟨map_isPrime_of_surjective Quotient.mk_surjective hfp,
+      map_mono (le_sup_right.trans (Ideal.le_minimalPrimes hp))⟩,
+        fun _ ⟨hr, hxr⟩ hrp ↦ map_le_iff_le_comap.mpr <| hp.2 ⟨hr.comap f, sup_le_iff.mpr
+          ⟨I.mk_ker.symm.trans_le <| ker_le_comap (Ideal.Quotient.mk I), le_comap_of_map_le hxr⟩⟩ <|
+            (comap_mono hrp).trans <| Eq.le <|
+              (p.comap_map_of_surjective _ Quotient.mk_surjective).trans <| sup_eq_left.mpr hfp⟩
 
 /-- If `q < p` are prime ideals such that `p` is minimal over `span (s ∪ {x})` and
 `t` is a set contained in `q` such that `s ⊆ √span (t ∪ {x})`, then `q` is minimal over `span t`.
@@ -152,7 +153,8 @@ theorem Ideal.mem_minimalPrimes_span_of_mem_minimalPrimes_span_insert {q p : Ide
     rwa [comap_map_of_surjective f hf, ← RingHom.ker_eq_comap_bot,
       mk_ker, sup_eq_left.mpr hI'q] at this
   refine height_le_one_of_isPrincipal_of_mem_minimalPrimes ((span {x}).map f) (p.map f) ⟨⟨this,
-    map_mono <| span_le.mpr <| Set.singleton_subset_iff.mpr <| hp.1.2 <| subset_span <| .inl rfl⟩,
+    map_mono <| span_le.mpr <| Set.singleton_subset_iff.mpr <| (Ideal.le_minimalPrimes hp) <|
+      subset_span <| .inl rfl⟩,
     fun r ⟨hr, hxr⟩ hrp ↦ map_le_iff_le_comap.mpr (hp.2 ⟨hr.comap f, ?_⟩ ?_)⟩
   · rw [span_le, Set.insert_subset_iff]
     have := map_le_iff_le_comap.mp hxr (subset_span rfl)
@@ -176,7 +178,7 @@ nonrec lemma Ideal.height_le_spanRank_toENat_of_mem_minimal_primes
   induction hn : s.card using Nat.strong_induction_on generalizing R with
   | h n H =>
     replace hn : s.card ≤ n := hn.le
-    have := hp.1.1
+    have := Ideal.minimalPrimes_isPrime hp
     cases n with
     | zero =>
       rw [ENat.coe_zero, nonpos_iff_eq_zero, height_eq_primeHeight p,
@@ -204,7 +206,7 @@ nonrec lemma Ideal.height_le_spanRank_toENat_of_mem_minimal_primes
           ⟨le_sup_left, x, mem_sup_right (mem_span_singleton_self _), hxq⟩).trans_le hJ)
           ((le_maximalIdeal hJ'.ne_top).lt_of_not_ge h)
       have h : (s' : Set R) ⊆ (q ⊔ span {x}).radical := by
-        have := hp.1.2.trans this
+        have := (Ideal.le_minimalPrimes hp).trans this
         rw [span_le, Finset.coe_insert, Set.insert_subset_iff] at this
         exact this.2
       obtain ⟨t, ht, hspan⟩ := exists_subset_radical_span_sup_of_subset_radical_sup _ _ _ h
@@ -239,7 +241,7 @@ lemma Ideal.height_le_spanRank_toENat (I : Ideal R) (hI : I ≠ ⊤) :
   obtain ⟨J, hJ⟩ := nonempty_minimalPrimes hI
   refine (iInf₂_le J hJ).trans ?_
   convert (I.height_le_spanRank_toENat_of_mem_minimal_primes J hJ)
-  exact Eq.symm (@height_eq_primeHeight _ _ J hJ.1.1)
+  exact (@height_eq_primeHeight _ _ J (Ideal.minimalPrimes_isPrime hJ)).symm
 
 lemma Ideal.height_le_spanFinrank (I : Ideal R) (hI : I ≠ ⊤) :
     I.height ≤ I.spanFinrank := by
@@ -318,7 +320,8 @@ lemma Ideal.height_le_height_add_spanFinrank_of_le {I p : Ideal R} [p.IsPrime] (
   let p' := p.map (algebraMap R (R ⧸ I))
   have : p'.IsPrime := isPrime_map_quotientMk_of_isPrime hrp
   obtain ⟨s, hps, hs⟩ := exists_finset_card_eq_height_of_isNoetherianRing p'
-  have hsp' : (s : Set (R ⧸ I)) ⊆ (p' : Set _) := fun _ hx ↦ hps.1.2 (subset_span hx)
+  have hsp' : (s : Set (R ⧸ I)) ⊆ (p' : Set _) :=
+    fun _ hx ↦ (Ideal.le_minimalPrimes hps) (subset_span hx)
   have : Set.SurjOn (Ideal.Quotient.mk I) p s := by
     refine Set.SurjOn.mono subset_rfl hsp' fun x hx ↦ ?_
     obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x
@@ -415,7 +418,7 @@ lemma Ideal.height_le_height_add_of_liesOver [IsNoetherianRing S] (p : Ideal R)
   let P' := P.map (Ideal.Quotient.mk <| p.map (algebraMap R S))
   obtain ⟨s', hP', heq'⟩ := P'.exists_finset_card_eq_height_of_isNoetherianRing
   have hsP'sub : (s' : Set <| S ⧸ (Ideal.map (algebraMap R S) p)) ⊆ (P' : Set <| S ⧸ _) :=
-    fun x hx ↦ hP'.1.2 (Ideal.subset_span hx)
+    fun x hx ↦ (Ideal.le_minimalPrimes hP') (Ideal.subset_span hx)
   have : Set.SurjOn (Ideal.Quotient.mk (p.map (algebraMap R S))) P s' := by
     refine Set.SurjOn.mono subset_rfl hsP'sub fun x hx ↦ ?_
     obtain ⟨y, rfl⟩ := Ideal.Quotient.mk_surjective x

From 663dafa3d5db14b6119a56fb99fda4e93f91d80d Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Tue, 14 Apr 2026 15:23:18 +0800
Subject: [PATCH 25/69] fix

---
 Mathlib/RingTheory/Ideal/Height.lean | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index 4e84e280aea7c9..c63c53f6d27026 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -130,7 +130,7 @@ lemma Ideal.height_strict_mono_of_is_prime {I J : Ideal R} [I.IsPrime]
     exact I.primeHeight_lt_top
   · rw [← ENat.add_one_le_iff I.primeHeight_ne_top, Ideal.height]
     refine le_iInf₂ (fun K hK ↦ ?_)
-    haveI := Ideal.minimalPrimes_isPrime hK
+    have := Ideal.minimalPrimes_isPrime hK
     have : I < K := lt_of_lt_of_le h (Ideal.le_minimalPrimes hK)
     exact Ideal.primeHeight_add_one_le_of_lt this
 

From 99d74db08b6dfd61ef6fa6cf6024066849014b06 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Wed, 15 Apr 2026 13:16:01 +0800
Subject: [PATCH 26/69] fix doc

---
 Mathlib/RingTheory/Ideal/Height.lean | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index 90b2ea119c741a..075d158ba45537 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -170,7 +170,7 @@ lemma Ideal.eq_of_le_of_height_le [I.IsPrime] [I.FiniteHeight]
     {J : Ideal R} (h : I ≤ J) (h_height : J.height ≤ I.height) : I = J :=
   eq_of_le_of_not_lt h fun hlt => not_le.mpr (Ideal.height_strict_mono_of_is_prime hlt) h_height
 
-@[deprecated "Use `Ideal.height_add_one_le_of_lt_of_isPrime` instead." (since := "2026-04-04")]
+@[deprecated "Use `Ideal.height_le_ringKrullDim_of_isPrime` instead." (since := "2026-04-04")]
 private lemma Ideal.primeHeight_le_ringKrullDim {I : Ideal R} [I.IsPrime] :
     I.primeHeight ≤ ringKrullDim R := Order.height_le_krullDim _
 

From 6d956b634f6e2e730d36863cd6acf3f39c832673 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Wed, 15 Apr 2026 13:30:05 +0800
Subject: [PATCH 27/69] fix and add deprecation

---
 Mathlib/RingTheory/Ideal/Height.lean | 6 +++++-
 1 file changed, 5 insertions(+), 1 deletion(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index 2c9fb5c2d44815..b2681c76294800 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -131,6 +131,10 @@ lemma Ideal.height_strict_mono_of_isPrime_of_isPrime {I J : Ideal R} [I.IsPrime]
     (Or.inr (lt_of_le_of_lt (Ideal.height_mono h.le) (J.height_lt_top IsPrime.ne_top')).ne)
   exact Ideal.height_strict_mono_of_isPrime h
 
+@[deprecated (since := "2026-04-02")]
+alias Ideal.height_strict_mono_of_isPrime_of_is_prime :=
+  Ideal.height_strict_mono_of_isPrime_of_isPrime
+
 @[deprecated "Use `Ideal.height_strict_mono_of_isPrime_of_isPrime` instead."
   (since := "2026-04-02")]
 lemma Ideal.primeHeight_strict_mono {I J : Ideal R} [I.IsPrime] [J.IsPrime] (h : I < J)
@@ -140,7 +144,7 @@ lemma Ideal.primeHeight_strict_mono {I J : Ideal R} [I.IsPrime] [J.IsPrime] (h :
 /-- A prime ideal of finite height is equal to any ideal that contains it with no greater height. -/
 lemma Ideal.eq_of_le_of_height_le [I.IsPrime] [I.FiniteHeight]
     {J : Ideal R} (h : I ≤ J) (h_height : J.height ≤ I.height) : I = J :=
-  eq_of_le_of_not_lt h fun hlt => not_le.mpr (Ideal.height_strict_mono_of_is_prime hlt) h_height
+  eq_of_le_of_not_lt h fun hlt => not_le.mpr (Ideal.height_strict_mono_of_isPrime hlt) h_height
 
 private lemma Ideal.primeHeight_le_ringKrullDim {I : Ideal R} [I.IsPrime] :
     I.primeHeight ≤ ringKrullDim R := Order.height_le_krullDim _

From 5b55fdc247193c18c5318309029d33a919320620 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Wed, 15 Apr 2026 22:16:16 +0800
Subject: [PATCH 28/69] Update KrullsHeightTheorem.lean

---
 Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
index f10627b26cf219..c08bb61cff9381 100644
--- a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
+++ b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
@@ -148,7 +148,7 @@ theorem Ideal.mem_minimalPrimes_span_of_mem_minimalPrimes_span_insert {q p : Ide
     have : (p.map f).FiniteHeight := ⟨Or.inr (h.trans_lt (WithTop.coe_lt_top 1)).ne⟩
     have := (height_strict_mono_of_isPrime_of_isPrime h_lt).trans_le h
     rw [ENat.lt_one_iff_eq_zero, height_eq_zero_iff] at this
-    have := minimal_primes_comap_of_surjective hf this
+    have := minimalPrimes_comap_of_surjective hf this
     rwa [comap_map_of_surjective f hf, ← RingHom.ker_eq_comap_bot,
       mk_ker, sup_eq_left.mpr hI'q] at this
   refine height_le_one_of_isPrincipal_of_mem_minimalPrimes ((span {x}).map f) (p.map f) ⟨⟨this,

From c03a1f7948a55fb2cebdf1fbc6bb547260f3743c Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Wed, 15 Apr 2026 22:34:53 +0800
Subject: [PATCH 29/69] fix

---
 Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
index 0e412542aaedb1..9a9adf388f6ff5 100644
--- a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
+++ b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
@@ -241,7 +241,7 @@ lemma Ideal.height_le_spanRank_toENat (I : Ideal R) (hI : I ≠ ⊤) :
     I.height ≤ I.spanRank.toENat := by
   obtain ⟨J, hJ⟩ := nonempty_minimalPrimes hI
   rw [I.height_eq_inf_minimalPrimes]
-  exact (iInf₂_le J hJ).trans (I.height_le_spanRank_toENat_of_mem_minimal_primes J hJ)
+  exact (iInf₂_le J hJ).trans (I.height_le_spanRank_toENat_of_mem_minimalPrimes J hJ)
 
 lemma Ideal.height_le_spanFinrank (I : Ideal R) (hI : I ≠ ⊤) :
     I.height ≤ I.spanFinrank := by

From 4ffe3c92d1ebf035c56390c202d78b9881a8b9cf Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Fri, 24 Apr 2026 22:39:30 +0800
Subject: [PATCH 30/69] fix naming

---
 .../Ideal/AssociatedPrime/Localization.lean      |  2 +-
 Mathlib/RingTheory/Ideal/Height.lean             | 10 +++++-----
 .../RingTheory/Ideal/KrullsHeightTheorem.lean    | 14 +++++++-------
 Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean | 16 ++++++++--------
 Mathlib/RingTheory/Ideal/MinimalPrime/Colon.lean |  2 +-
 .../Ideal/MinimalPrime/Localization.lean         | 14 +++++++-------
 6 files changed, 29 insertions(+), 29 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.lean b/Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.lean
index 2d07b507186198..e3dc5170f20330 100644
--- a/Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.lean
+++ b/Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.lean
@@ -135,7 +135,7 @@ lemma minimalPrimes_annihilator_subset_associatedPrimes [IsNoetherianRing R] [Mo
   let Rₚ := Localization.AtPrime p
   have : Nontrivial (LocalizedModule p.primeCompl M) := by
     simpa [← Module.mem_support_iff (p := ⟨p, prime⟩), Module.support_eq_zeroLocus] using
-      Ideal.le_minimalPrimes hp
+      Ideal.le_of_mem_minimalPrimes hp
   rcases associatedPrimes.nonempty Rₚ (LocalizedModule p.primeCompl M) with ⟨q, hq⟩
   have q_prime : q.IsPrime := IsAssociatedPrime.isPrime hq
   simp only [← preimage_comap_associatedPrimes_eq_associatedPrimes_of_isLocalizedModule p.primeCompl
diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index dd5ca9b542be75..ff558449ad202e 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -117,7 +117,7 @@ theorem Ideal.height_mono {I J : Ideal R} (h : I ≤ J) : I.height ≤ J.height
   simp only [height]
   refine le_iInf₂ (fun p hp ↦ ?_)
   have := Ideal.minimalPrimes_isPrime hp
-  obtain ⟨q, hq, e⟩ := Ideal.exists_minimalPrimes_le (h.trans (Ideal.le_minimalPrimes hp))
+  obtain ⟨q, hq, e⟩ := Ideal.exists_minimalPrimes_le (h.trans (Ideal.le_of_mem_minimalPrimes hp))
   haveI := Ideal.minimalPrimes_isPrime hq
   exact (iInf₂_le q hq).trans (Ideal.primeHeight_mono e)
 
@@ -131,7 +131,7 @@ lemma Ideal.height_strict_mono_of_is_prime {I J : Ideal R} [I.IsPrime]
   · rw [← ENat.add_one_le_iff I.primeHeight_ne_top, Ideal.height]
     refine le_iInf₂ (fun K hK ↦ ?_)
     have := Ideal.minimalPrimes_isPrime hK
-    have : I < K := lt_of_lt_of_le h (Ideal.le_minimalPrimes hK)
+    have : I < K := lt_of_lt_of_le h (Ideal.le_of_mem_minimalPrimes hK)
     exact Ideal.primeHeight_add_one_le_of_lt this
 
 /-- A prime ideal of finite height is equal to any ideal that contains it with no greater height. -/
@@ -187,7 +187,7 @@ lemma Ideal.mem_minimalPrimes_of_height_eq {I J : Ideal R} (e : I ≤ J) [J.IsPr
   have := Ideal.minimalPrimes_isPrime h₁
   have := finiteHeight_of_le h₂ IsPrime.ne_top'
   exact lt_irrefl _ ((height_strict_mono_of_is_prime h₃).trans_le
-    (e'.trans <| height_mono (Ideal.le_minimalPrimes h₁)))
+    (e'.trans <| height_mono (Ideal.le_of_mem_minimalPrimes h₁)))
 
 /-- A prime ideal has height zero if and only if it is minimal -/
 lemma Ideal.primeHeight_eq_zero_iff {I : Ideal R} [I.IsPrime] :
@@ -416,9 +416,9 @@ lemma exists_spanRank_le_and_le_height_of_le_height [IsNoetherianRing R] (I : Id
           · rwa [← p.height_eq_primeHeight]
           · rwa [← e]
         refine ⟨mem_minimalPrimes_of_primeHeight_eq_height
-          (le_sup_left.trans (Ideal.le_minimalPrimes hp)) this.symm, ?_⟩
+          (le_sup_left.trans (Ideal.le_of_mem_minimalPrimes hp)) this.symm, ?_⟩
         rwa [p.height_eq_primeHeight, eq_comm]
-      · exact (Ideal.le_minimalPrimes hp) <| Ideal.mem_sup_right <| mem_span_singleton_self x
+      · exact (Ideal.le_of_mem_minimalPrimes hp) <| Ideal.mem_sup_right <| mem_span_singleton_self x
 
 /-- In a nontrivial commutative ring `R`, the supremum of heights of all ideals is equal to the
 Krull dimension of `R`. -/
diff --git a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
index d5a4597fe3c9b6..91de20c5386a1b 100644
--- a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
+++ b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
@@ -83,7 +83,7 @@ lemma Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes_of_isLocalRing
     ← (IsLocalization.orderEmbedding q.primeCompl (Localization.AtPrime q)).map_rel_iff]
   refine Submodule.le_of_le_smul_of_le_jacobson_bot (I := I) (IsNoetherian.noetherian _) ?_ ?_
   · rw [IsLocalRing.jacobson_eq_maximalIdeal]
-    exacts [Ideal.le_minimalPrimes hp, bot_ne_top]
+    exacts [Ideal.le_of_mem_minimalPrimes hp, bot_ne_top]
   · replace hn := congr(Ideal.comap (Ideal.Quotient.mk I) $(hn _ n.le_succ))
     simp only [qs, OrderHom.coe_mk, ← RingHom.ker_eq_comap_bot, Ideal.mk_ker,
       Ideal.comap_map_of_surjective _ Ideal.Quotient.mk_surjective] at hn
@@ -118,10 +118,10 @@ theorem Ideal.map_height_le_one_of_mem_minimalPrimes {I p : Ideal R} {x : R}
     (hp : p ∈ (I ⊔ span {x}).minimalPrimes) : (p.map (Ideal.Quotient.mk I)).height ≤ 1 :=
   let f := Ideal.Quotient.mk I
   have : p.IsPrime := Ideal.minimalPrimes_isPrime hp
-  have hfp : RingHom.ker f ≤ p := I.mk_ker.trans_le (le_sup_left.trans (Ideal.le_minimalPrimes hp))
+  have hfp : RingHom.ker f ≤ p := I.mk_ker.trans_le (le_sup_left.trans (Ideal.le_of_mem_minimalPrimes hp))
   height_le_one_of_isPrincipal_of_mem_minimalPrimes ((span {x}).map f) (p.map f)
     ⟨⟨map_isPrime_of_surjective Quotient.mk_surjective hfp,
-      map_mono (le_sup_right.trans (Ideal.le_minimalPrimes hp))⟩,
+      map_mono (le_sup_right.trans (Ideal.le_of_mem_minimalPrimes hp))⟩,
         fun _ ⟨hr, hxr⟩ hrp ↦ map_le_iff_le_comap.mpr <| hp.2 ⟨hr.comap f, sup_le_iff.mpr
           ⟨I.mk_ker.symm.trans_le <| ker_le_comap (Ideal.Quotient.mk I), le_comap_of_map_le hxr⟩⟩ <|
             (comap_mono hrp).trans <| Eq.le <|
@@ -153,7 +153,7 @@ theorem Ideal.mem_minimalPrimes_span_of_mem_minimalPrimes_span_insert {q p : Ide
     rwa [comap_map_of_surjective f hf, ← RingHom.ker_eq_comap_bot,
       mk_ker, sup_eq_left.mpr hI'q] at this
   refine height_le_one_of_isPrincipal_of_mem_minimalPrimes ((span {x}).map f) (p.map f) ⟨⟨this,
-    map_mono <| span_le.mpr <| Set.singleton_subset_iff.mpr <| (Ideal.le_minimalPrimes hp) <|
+    map_mono <| span_le.mpr <| Set.singleton_subset_iff.mpr <| (Ideal.le_of_mem_minimalPrimes hp) <|
       subset_span <| .inl rfl⟩,
     fun r ⟨hr, hxr⟩ hrp ↦ map_le_iff_le_comap.mpr (hp.2 ⟨hr.comap f, ?_⟩ ?_)⟩
   · rw [span_le, Set.insert_subset_iff]
@@ -206,7 +206,7 @@ nonrec lemma Ideal.height_le_spanRank_toENat_of_mem_minimalPrimes
           ⟨le_sup_left, x, mem_sup_right (mem_span_singleton_self _), hxq⟩).trans_le hJ)
           ((le_maximalIdeal hJ'.ne_top).lt_of_not_ge h)
       have h : (s' : Set R) ⊆ (q ⊔ span {x}).radical := by
-        have := (Ideal.le_minimalPrimes hp).trans this
+        have := (Ideal.le_of_mem_minimalPrimes hp).trans this
         rw [span_le, Finset.coe_insert, Set.insert_subset_iff] at this
         exact this.2
       obtain ⟨t, ht, hspan⟩ := exists_subset_radical_span_sup_of_subset_radical_sup _ _ _ h
@@ -324,7 +324,7 @@ lemma Ideal.height_le_height_add_spanFinrank_of_le {I p : Ideal R} [p.IsPrime] (
   have : p'.IsPrime := isPrime_map_quotientMk_of_isPrime hrp
   obtain ⟨s, hps, hs⟩ := exists_finset_card_eq_height_of_isNoetherianRing p'
   have hsp' : (s : Set (R ⧸ I)) ⊆ (p' : Set _) :=
-    fun _ hx ↦ (Ideal.le_minimalPrimes hps) (subset_span hx)
+    fun _ hx ↦ (Ideal.le_of_mem_minimalPrimes hps) (subset_span hx)
   have : Set.SurjOn (Ideal.Quotient.mk I) p s := by
     refine Set.SurjOn.mono subset_rfl hsp' fun x hx ↦ ?_
     obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x
@@ -421,7 +421,7 @@ lemma Ideal.height_le_height_add_of_liesOver [IsNoetherianRing S] (p : Ideal R)
   let P' := P.map (Ideal.Quotient.mk <| p.map (algebraMap R S))
   obtain ⟨s', hP', heq'⟩ := P'.exists_finset_card_eq_height_of_isNoetherianRing
   have hsP'sub : (s' : Set <| S ⧸ (Ideal.map (algebraMap R S) p)) ⊆ (P' : Set <| S ⧸ _) :=
-    fun x hx ↦ (Ideal.le_minimalPrimes hP') (Ideal.subset_span hx)
+    fun x hx ↦ (Ideal.le_of_mem_minimalPrimes hP') (Ideal.subset_span hx)
   have : Set.SurjOn (Ideal.Quotient.mk (p.map (algebraMap R S))) P s' := by
     refine Set.SurjOn.mono subset_rfl hsP'sub fun x hx ↦ ?_
     obtain ⟨y, rfl⟩ := Ideal.Quotient.mk_surjective x
diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
index 689fddcfb9f092..17483edc6cccfd 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
@@ -54,10 +54,10 @@ variable {I J}
 theorem Ideal.minimalPrimes_isPrime {p : Ideal R} (h : p ∈ I.minimalPrimes) : p.IsPrime :=
   h.1.1
 
-theorem Ideal.le_minimalPrimes {p : Ideal R} (h : p ∈ I.minimalPrimes) : I ≤ p :=
+theorem Ideal.le_of_mem_minimalPrimes {p : Ideal R} (h : p ∈ I.minimalPrimes) : I ≤ p :=
   h.1.2
 
-theorem minimalPrimes_isPrime {p : Ideal R} (h : p ∈ minimalPrimes R) : p.IsPrime :=
+theorem Ideal.IsPrime.of_mem_minimalPrimes {p : Ideal R} (h : p ∈ minimalPrimes R) : p.IsPrime :=
   h.1.1
 
 theorem Ideal.exists_minimalPrimes_le [J.IsPrime] (e : I ≤ J) : ∃ p ∈ I.minimalPrimes, p ≤ J := by
@@ -123,15 +123,15 @@ theorem Ideal.minimalPrimes_eq_subsingleton (hI : I.IsPrimary) : I.minimalPrimes
   ext J
   constructor
   · exact fun H =>
-      let e := (Ideal.minimalPrimes_isPrime H).radical_le_iff.mpr (Ideal.le_minimalPrimes H)
+      let e := (Ideal.minimalPrimes_isPrime H).radical_le_iff.mpr (Ideal.le_of_mem_minimalPrimes H)
       (H.2 ⟨Ideal.isPrime_radical hI, Ideal.le_radical⟩ e).antisymm e
   · rintro (rfl : J = I.radical)
     exact ⟨⟨Ideal.isPrime_radical hI, Ideal.le_radical⟩, fun _ H _ => H.1.radical_le_iff.mpr H.2⟩
 
 theorem Ideal.minimalPrimes_eq_subsingleton_self [I.IsPrime] : I.minimalPrimes = {I} := by
   ext J
-  refine ⟨fun H => (H.2 ⟨inferInstance, rfl.le⟩ (Ideal.le_minimalPrimes H)).antisymm
-    (Ideal.le_minimalPrimes H), ?_⟩
+  refine ⟨fun H => (H.2 ⟨inferInstance, rfl.le⟩ (Ideal.le_of_mem_minimalPrimes H)).antisymm
+    (Ideal.le_of_mem_minimalPrimes H), ?_⟩
   rintro (rfl : J = I)
   exact ⟨⟨inferInstance, rfl.le⟩, fun _ h _ => h.2⟩
 
@@ -150,7 +150,7 @@ theorem Ideal.minimalPrimes_top : (⊤ : Ideal R).minimalPrimes = ∅ := by
   ext p
   simp only [Set.notMem_empty, iff_false]
   intro h
-  exact (Ideal.minimalPrimes_isPrime h).ne_top (top_le_iff.mp (Ideal.le_minimalPrimes h))
+  exact (Ideal.minimalPrimes_isPrime h).ne_top (top_le_iff.mp (Ideal.le_of_mem_minimalPrimes h))
 
 theorem Ideal.minimalPrimes_eq_empty_iff (I : Ideal R) :
     I.minimalPrimes = ∅ ↔ I = ⊤ := by
@@ -170,7 +170,7 @@ lemma Ideal.mem_minimalPrimes_sup {R : Type*} [CommRing R] {p I J : Ideal R} [p.
   refine ⟨⟨‹_›, ?_⟩, fun q ⟨_, hq⟩ hqp ↦ ?_⟩
   · rw [sup_le_iff]
     refine ⟨hle, ?_⟩
-    simpa [hle] using Ideal.comap_mono (f := Ideal.Quotient.mk I) (Ideal.le_minimalPrimes h)
+    simpa [hle] using Ideal.comap_mono (f := Ideal.Quotient.mk I) (Ideal.le_of_mem_minimalPrimes h)
   · rw [sup_le_iff] at hq
     have h2 : p.map (Quotient.mk I) ≤ q.map (Quotient.mk I) :=
       h.2 ⟨isPrime_map_quotientMk_of_isPrime hq.1, map_mono hq.2⟩ (map_mono hqp)
@@ -190,7 +190,7 @@ lemma Ideal.map_sup_mem_minimalPrimes_of_map_quotientMk_mem_minimalPrimes
   refine ⟨⟨inferInstance, sup_le_iff.mpr ?_⟩, fun q ⟨_, hleq⟩ hqle ↦ ?_⟩
   · refine ⟨?_, hJP⟩
     rw [Ideal.map_le_iff_le_comap, ← Ideal.under_def, ← Ideal.over_def P p]
-    exact Ideal.le_minimalPrimes hI
+    exact Ideal.le_of_mem_minimalPrimes hI
   · simp only [sup_le_iff] at hleq
     have h1 : p.map (algebraMap R S) ≤ q := by
       rw [Ideal.map_le_iff_le_comap]
diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Colon.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Colon.lean
index 4b76a27a9235e3..27693479951ca1 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Colon.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Colon.lean
@@ -96,7 +96,7 @@ theorem exists_eq_colon_of_mem_minimalPrimes [IsNoetherianRing R]
     exact ⟨hz', hJI, hz⟩
   by_cases hn' : n - 1 = 0
   · simp [K, show n = 1 by grind] at hz'
-    exact (hK.2 (hz'.trans (Ideal.le_minimalPrimes hI))).elim
+    exact (hK.2 (hz'.trans (Ideal.le_of_mem_minimalPrimes hI))).elim
   · grind [Nat.find_min' key ⟨hn', J * Ideal.span {z}, hK⟩]
 
 end Submodule
diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean
index 0afbe2e5f9e851..cdb9c17bc7a400 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean
@@ -66,7 +66,7 @@ theorem Ideal.iUnion_minimalPrimes :
     obtain ⟨p, hp, hyp⟩ : ∃ p ∈ I.minimalPrimes, y ∉ p := by
       simpa [← Ideal.sInf_minimalPrimes] using hy
     refine ⟨p, hp, ((Ideal.minimalPrimes_isPrime hp).mem_or_mem ?_).resolve_right hyp⟩
-    exact (Ideal.minimalPrimes_isPrime hp).radical_le_iff.mpr (Ideal.le_minimalPrimes hp) hx
+    exact (Ideal.minimalPrimes_isPrime hp).radical_le_iff.mpr (Ideal.le_of_mem_minimalPrimes hp) hx
 
 theorem Ideal.exists_mul_mem_of_mem_minimalPrimes
     {p : Ideal R} (hp : p ∈ I.minimalPrimes) {x : R} (hx : x ∈ p) :
@@ -98,7 +98,7 @@ lemma Ideal.disjoint_nonZeroDivisors_of_mem_minimalPrimes {p : Ideal R} (hp : p
 theorem Ideal.exists_comap_eq_of_mem_minimalPrimes {I : Ideal S} (f : R →+* S) (p)
     (H : p ∈ (I.comap f).minimalPrimes) : ∃ p' : Ideal S, p'.IsPrime ∧ I ≤ p' ∧ p'.comap f = p :=
   have := Ideal.minimalPrimes_isPrime H
-  have ⟨p', hIp', hp', le⟩ := exists_ideal_comap_le_prime p I (Ideal.le_minimalPrimes H)
+  have ⟨p', hIp', hp', le⟩ := exists_ideal_comap_le_prime p I (Ideal.le_of_mem_minimalPrimes H)
   ⟨p', hp', hIp', le.antisymm (H.2 ⟨inferInstance, comap_mono hIp'⟩ le)⟩
 
 @[stacks 00FK] theorem Ideal.exists_comap_eq_of_mem_minimalPrimes_of_injective {f : R →+* S}
@@ -115,7 +115,7 @@ theorem Ideal.exists_minimalPrimes_comap_eq {I : Ideal S} (f : R →+* S) (p)
   refine ⟨q, hq, Eq.symm ?_⟩
   have := Ideal.minimalPrimes_isPrime hq
   have := (Ideal.comap_mono hq').trans_eq h₃
-  exact (H.2 ⟨inferInstance, Ideal.comap_mono (Ideal.le_minimalPrimes hq)⟩ this).antisymm this
+  exact (H.2 ⟨inferInstance, Ideal.comap_mono (Ideal.le_of_mem_minimalPrimes hq)⟩ this).antisymm this
 
 theorem Ideal.minimalPrimes_comap_subset (f : R →+* S) (J : Ideal S) :
     (J.comap f).minimalPrimes ⊆ Ideal.comap f '' J.minimalPrimes :=
@@ -130,7 +130,7 @@ variable {R S : Type*} [CommRing R] [CommRing S] {I J : Ideal R}
 theorem Ideal.minimalPrimes_comap_of_surjective {f : R →+* S} (hf : Function.Surjective f)
     {I J : Ideal S} (h : J ∈ I.minimalPrimes) : J.comap f ∈ (I.comap f).minimalPrimes := by
   have := Ideal.minimalPrimes_isPrime h
-  refine ⟨⟨inferInstance, Ideal.comap_mono (Ideal.le_minimalPrimes h)⟩, ?_⟩
+  refine ⟨⟨inferInstance, Ideal.comap_mono (Ideal.le_of_mem_minimalPrimes h)⟩, ?_⟩
   rintro K ⟨hK, e₁⟩ e₂
   have : RingHom.ker f ≤ K := (Ideal.comap_mono bot_le).trans e₁
   rw [← sup_eq_left.mpr this, RingHom.ker_eq_comap_bot, ← Ideal.comap_map_of_surjective f hf]
@@ -160,7 +160,7 @@ lemma Ideal.minimalPrimes_map_of_surjective {S : Type*} [CommRing S] {f : R →+
     Ideal.comap_map_of_surjective f hf, Set.image_congr, Set.image_id, RingHom.ker]
   intro x hx
   exact (Ideal.comap_map_of_surjective f hf _).trans
-    (sup_eq_left.mpr <| le_sup_right.trans (Ideal.le_minimalPrimes hx))
+    (sup_eq_left.mpr <| le_sup_right.trans (Ideal.le_of_mem_minimalPrimes hx))
 
 theorem Ideal.minimalPrimes_eq_comap :
     I.minimalPrimes = Ideal.comap (Ideal.Quotient.mk I) '' minimalPrimes (R ⧸ I) := by
@@ -179,7 +179,7 @@ theorem IsLocalization.minimalPrimes_map [IsLocalization S A] (J : Ideal R) :
   constructor
   · intro hp
     haveI := Ideal.minimalPrimes_isPrime hp
-    refine ⟨⟨Ideal.IsPrime.comap _, Ideal.map_le_iff_le_comap.mp (Ideal.le_minimalPrimes hp)⟩, ?_⟩
+    refine ⟨⟨Ideal.IsPrime.comap _, Ideal.map_le_iff_le_comap.mp (Ideal.le_of_mem_minimalPrimes hp)⟩, ?_⟩
     rintro I hI e
     have hI' : Disjoint (S : Set R) I := Set.disjoint_of_subset_right e
       ((IsLocalization.isPrime_iff_isPrime_disjoint S A _).mp (Ideal.minimalPrimes_isPrime hp)).2
@@ -188,7 +188,7 @@ theorem IsLocalization.minimalPrimes_map [IsLocalization S A] (J : Ideal R) :
     · exact IsLocalization.isPrime_of_isPrime_disjoint S A I hI.1 hI'
     · exact IsLocalization.comap_map_of_isPrime_disjoint S A hI.1 hI'
   · intro hp
-    refine ⟨⟨?_, Ideal.map_le_iff_le_comap.mpr (Ideal.le_minimalPrimes hp)⟩, ?_⟩
+    refine ⟨⟨?_, Ideal.map_le_iff_le_comap.mpr (Ideal.le_of_mem_minimalPrimes hp)⟩, ?_⟩
     · rw [IsLocalization.isPrime_iff_isPrime_disjoint S A, IsLocalization.disjoint_comap_iff S]
       refine ⟨Ideal.minimalPrimes_isPrime hp, ?_⟩
       rintro rfl

From 90805b47929166aee9aa9aff2d032f16f544d051 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Fri, 24 Apr 2026 23:28:28 +0800
Subject: [PATCH 31/69] fix

---
 Mathlib/RingTheory/Ideal/Height.lean                    | 2 +-
 Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean       | 3 ++-
 Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean | 6 ++++--
 3 files changed, 7 insertions(+), 4 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index ff558449ad202e..2e2e034fe0d23d 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -186,7 +186,7 @@ lemma Ideal.mem_minimalPrimes_of_height_eq {I J : Ideal R} (e : I ≤ J) [J.IsPr
   refine (eq_of_le_of_not_lt h₂ fun h₃ ↦ ?_).symm
   have := Ideal.minimalPrimes_isPrime h₁
   have := finiteHeight_of_le h₂ IsPrime.ne_top'
-  exact lt_irrefl _ ((height_strict_mono_of_is_prime h₃).trans_le
+  exact lt_irrefl _ ((height_strict_mono_of_isPrime h₃).trans_le
     (e'.trans <| height_mono (Ideal.le_of_mem_minimalPrimes h₁)))
 
 /-- A prime ideal has height zero if and only if it is minimal -/
diff --git a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
index 91de20c5386a1b..dd0087b6d0add1 100644
--- a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
+++ b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
@@ -118,7 +118,8 @@ theorem Ideal.map_height_le_one_of_mem_minimalPrimes {I p : Ideal R} {x : R}
     (hp : p ∈ (I ⊔ span {x}).minimalPrimes) : (p.map (Ideal.Quotient.mk I)).height ≤ 1 :=
   let f := Ideal.Quotient.mk I
   have : p.IsPrime := Ideal.minimalPrimes_isPrime hp
-  have hfp : RingHom.ker f ≤ p := I.mk_ker.trans_le (le_sup_left.trans (Ideal.le_of_mem_minimalPrimes hp))
+  have hfp : RingHom.ker f ≤ p :=
+    I.mk_ker.trans_le (le_sup_left.trans (Ideal.le_of_mem_minimalPrimes hp))
   height_le_one_of_isPrincipal_of_mem_minimalPrimes ((span {x}).map f) (p.map f)
     ⟨⟨map_isPrime_of_surjective Quotient.mk_surjective hfp,
       map_mono (le_sup_right.trans (Ideal.le_of_mem_minimalPrimes hp))⟩,
diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean
index cdb9c17bc7a400..a5c4cfb5f300f5 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean
@@ -115,7 +115,8 @@ theorem Ideal.exists_minimalPrimes_comap_eq {I : Ideal S} (f : R →+* S) (p)
   refine ⟨q, hq, Eq.symm ?_⟩
   have := Ideal.minimalPrimes_isPrime hq
   have := (Ideal.comap_mono hq').trans_eq h₃
-  exact (H.2 ⟨inferInstance, Ideal.comap_mono (Ideal.le_of_mem_minimalPrimes hq)⟩ this).antisymm this
+  exact (H.2 ⟨inferInstance, Ideal.comap_mono (Ideal.le_of_mem_minimalPrimes hq)⟩ this).antisymm
+    this
 
 theorem Ideal.minimalPrimes_comap_subset (f : R →+* S) (J : Ideal S) :
     (J.comap f).minimalPrimes ⊆ Ideal.comap f '' J.minimalPrimes :=
@@ -179,7 +180,8 @@ theorem IsLocalization.minimalPrimes_map [IsLocalization S A] (J : Ideal R) :
   constructor
   · intro hp
     haveI := Ideal.minimalPrimes_isPrime hp
-    refine ⟨⟨Ideal.IsPrime.comap _, Ideal.map_le_iff_le_comap.mp (Ideal.le_of_mem_minimalPrimes hp)⟩, ?_⟩
+    refine ⟨⟨Ideal.IsPrime.comap _,
+      Ideal.map_le_iff_le_comap.mp (Ideal.le_of_mem_minimalPrimes hp)⟩, ?_⟩
     rintro I hI e
     have hI' : Disjoint (S : Set R) I := Set.disjoint_of_subset_right e
       ((IsLocalization.isPrime_iff_isPrime_disjoint S A _).mp (Ideal.minimalPrimes_isPrime hp)).2

From 10af9921cde5469f493b9be7a7c02f363f1b74a8 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Fri, 24 Apr 2026 23:37:07 +0800
Subject: [PATCH 32/69] fix

---
 Mathlib/RingTheory/Ideal/Height.lean | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index 2e2e034fe0d23d..ff558449ad202e 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -186,7 +186,7 @@ lemma Ideal.mem_minimalPrimes_of_height_eq {I J : Ideal R} (e : I ≤ J) [J.IsPr
   refine (eq_of_le_of_not_lt h₂ fun h₃ ↦ ?_).symm
   have := Ideal.minimalPrimes_isPrime h₁
   have := finiteHeight_of_le h₂ IsPrime.ne_top'
-  exact lt_irrefl _ ((height_strict_mono_of_isPrime h₃).trans_le
+  exact lt_irrefl _ ((height_strict_mono_of_is_prime h₃).trans_le
     (e'.trans <| height_mono (Ideal.le_of_mem_minimalPrimes h₁)))
 
 /-- A prime ideal has height zero if and only if it is minimal -/

From 08f157e20c4d617ff613cd968fe028532d3a01f2 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Fri, 24 Apr 2026 23:37:31 +0800
Subject: [PATCH 33/69] fix

---
 Mathlib/RingTheory/Ideal/Height.lean | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index 494e24046c9de3..7d1693f6ebf1a4 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -187,7 +187,7 @@ lemma Ideal.mem_minimalPrimes_of_height_eq {I J : Ideal R} (e : I ≤ J) [J.IsPr
   refine (eq_of_le_of_not_lt h₂ fun h₃ ↦ ?_).symm
   have := Ideal.minimalPrimes_isPrime h₁
   have := finiteHeight_of_le h₂ IsPrime.ne_top'
-  exact lt_irrefl _ ((height_strict_mono_of_is_prime h₃).trans_le
+  exact lt_irrefl _ ((height_strict_mono_of_isPrime h₃).trans_le
     (e'.trans <| height_mono (Ideal.le_of_mem_minimalPrimes h₁)))
 
 /-- A prime ideal has height zero if and only if it is minimal -/

From 7648ee642c39f7a70133d1e67ffb495522f9f0c5 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Tue, 5 May 2026 20:44:31 +0800
Subject: [PATCH 34/69] fix naming

---
 .../Ideal/AssociatedPrime/Localization.lean   |  2 +-
 Mathlib/RingTheory/Ideal/Height.lean          | 20 +++++++++----------
 .../RingTheory/Ideal/KrullsHeightTheorem.lean |  8 ++++----
 .../RingTheory/Ideal/MinimalPrime/Basic.lean  |  8 ++++----
 .../RingTheory/Ideal/MinimalPrime/Colon.lean  |  8 ++++----
 .../Ideal/MinimalPrime/Localization.lean      | 20 +++++++++----------
 .../RingTheory/KrullDimension/Regular.lean    |  2 +-
 7 files changed, 34 insertions(+), 34 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.lean b/Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.lean
index e3dc5170f20330..6ce2cc209d03c4 100644
--- a/Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.lean
+++ b/Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.lean
@@ -131,7 +131,7 @@ variable (R M) in
 lemma minimalPrimes_annihilator_subset_associatedPrimes [IsNoetherianRing R] [Module.Finite R M] :
     (Module.annihilator R M).minimalPrimes ⊆ associatedPrimes R M := by
   intro p hp
-  have prime := Ideal.minimalPrimes_isPrime hp
+  have prime := Ideal.IsPrime.of_mem_minimalPrimes hp
   let Rₚ := Localization.AtPrime p
   have : Nontrivial (LocalizedModule p.primeCompl M) := by
     simpa [← Module.mem_support_iff (p := ⟨p, prime⟩), Module.support_eq_zeroLocus] using
diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index 1c01846ee0b6e2..8e275fc8481417 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -39,7 +39,7 @@ noncomputable def Ideal.primeHeight [hI : I.IsPrime] : ℕ∞ :=
 
 /-- The height of an ideal is defined as the infimum of the heights of its minimal prime ideals. -/
 noncomputable def Ideal.height : ℕ∞ :=
-  ⨅ J ∈ I.minimalPrimes, @Ideal.primeHeight _ _ J (minimalPrimes_isPrime (I := I) ‹_›)
+  ⨅ J ∈ I.minimalPrimes, @Ideal.primeHeight _ _ J (IsPrime.of_mem_minimalPrimes (I := I) ‹_›)
 
 /-- For a prime ideal, its height equals its prime height. -/
 lemma Ideal.height_eq_primeHeight [I.IsPrime] : I.height = I.primeHeight := by
@@ -116,9 +116,9 @@ lemma Ideal.primeHeight_strict_mono {I J : Ideal R} [I.IsPrime] [J.IsPrime]
 theorem Ideal.height_mono {I J : Ideal R} (h : I ≤ J) : I.height ≤ J.height := by
   simp only [height]
   refine le_iInf₂ (fun p hp ↦ ?_)
-  have := Ideal.minimalPrimes_isPrime hp
+  have := Ideal.IsPrime.of_mem_minimalPrimes hp
   obtain ⟨q, hq, e⟩ := Ideal.exists_minimalPrimes_le (h.trans (Ideal.le_of_mem_minimalPrimes hp))
-  haveI := Ideal.minimalPrimes_isPrime hq
+  haveI := Ideal.IsPrime.of_mem_minimalPrimes hq
   exact (iInf₂_le q hq).trans (Ideal.primeHeight_mono e)
 
 @[gcongr]
@@ -130,7 +130,7 @@ lemma Ideal.height_strict_mono_of_is_prime {I J : Ideal R} [I.IsPrime]
     exact I.primeHeight_lt_top
   · rw [← ENat.add_one_le_iff I.primeHeight_ne_top, Ideal.height]
     refine le_iInf₂ (fun K hK ↦ ?_)
-    have := Ideal.minimalPrimes_isPrime hK
+    have := Ideal.IsPrime.of_mem_minimalPrimes hK
     have : I < K := lt_of_lt_of_le h (Ideal.le_of_mem_minimalPrimes hK)
     exact Ideal.primeHeight_add_one_le_of_lt this
 
@@ -146,7 +146,7 @@ lemma Ideal.height_le_ringKrullDim_of_ne_top {I : Ideal R} (h : I ≠ ⊤) :
     I.height ≤ ringKrullDim R := by
   rw [Ideal.height]
   obtain ⟨P, hP⟩ : Nonempty (I.minimalPrimes) := Ideal.nonempty_minimalPrimes h
-  have := Ideal.minimalPrimes_isPrime hP
+  have := Ideal.IsPrime.of_mem_minimalPrimes hP
   refine le_trans ?_ (Ideal.primeHeight_le_ringKrullDim (I := P))
   simpa using iInf₂_le _ hP
 
@@ -184,7 +184,7 @@ lemma Ideal.mem_minimalPrimes_of_height_eq {I J : Ideal R} (e : I ≤ J) [J.IsPr
   obtain ⟨p, h₁, h₂⟩ := Ideal.exists_minimalPrimes_le e
   convert h₁
   refine (eq_of_le_of_not_lt h₂ fun h₃ ↦ ?_).symm
-  have := Ideal.minimalPrimes_isPrime h₁
+  have := Ideal.IsPrime.of_mem_minimalPrimes h₁
   have := finiteHeight_of_le h₂ IsPrime.ne_top'
   exact lt_irrefl _ ((height_strict_mono_of_is_prime h₃).trans_le
     (e'.trans <| height_mono (Ideal.le_of_mem_minimalPrimes h₁)))
@@ -207,7 +207,7 @@ lemma Ideal.height_bot [Nontrivial R] : (⊥ : Ideal R).height = 0 := by
   obtain ⟨p, hp⟩ := Ideal.nonempty_minimalPrimes (R := R) (I := ⊥) top_ne_bot.symm
   simp only [Ideal.height, ENat.iInf_eq_zero]
   refine ⟨p, hp, ?_⟩
-  have := Ideal.minimalPrimes_isPrime hp
+  have := Ideal.IsPrime.of_mem_minimalPrimes hp
   rw [primeHeight_eq_zero_iff.mpr hp]
 
 /-- In a trivial commutative ring, the height of any ideal is `∞`. -/
@@ -288,7 +288,7 @@ lemma RingEquiv.height_comap {S : Type*} [CommRing S] (e : R ≃+* S) (I : Ideal
     rw [← Ideal.comap_coe,
       Ideal.comap_minimalPrimes_eq_of_surjective (f := (↑e : R →+* S)) e.surjective]
     exact e.idealComapOrderIso.injective.mem_set_image.symm
-  · have : J.IsPrime := Ideal.minimalPrimes_isPrime h
+  · have : J.IsPrime := Ideal.IsPrime.of_mem_minimalPrimes h
     simp only [EquivLike.coe_coe, RingEquiv.idealComapOrderIso_apply,
       ← Ideal.height_eq_primeHeight, RingEquiv.height_comap_of_isPrime]
 
@@ -379,7 +379,7 @@ lemma exists_spanRank_le_and_le_height_of_le_height [IsNoetherianRing R] (I : Id
       refine (Ideal.subset_union_prime ⊥ ⊥ ?_).not.mpr ?_
       · rintro K hK - -
         rw [Set.Finite.mem_toFinset] at hK
-        exact Ideal.minimalPrimes_isPrime hK.1
+        exact Ideal.IsPrime.of_mem_minimalPrimes hK.1
       · push Not
         intro K hK e
         have := hr.trans (Ideal.height_mono e)
@@ -395,7 +395,7 @@ lemma exists_spanRank_le_and_le_height_of_le_height [IsNoetherianRing R] (I : Id
       push_cast
       exact add_le_add h₂ ((Submodule.spanRank_span_le_card _).trans (by simp))
     · refine le_iInf₂ (fun p hp ↦ ?_)
-      have := Ideal.minimalPrimes_isPrime hp
+      have := Ideal.IsPrime.of_mem_minimalPrimes hp
       by_cases h : p.height = ⊤
       · rw [← p.height_eq_primeHeight, h]
         exact le_top
diff --git a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
index 897b813b7b05c4..216697afa6012c 100644
--- a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
+++ b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
@@ -106,7 +106,7 @@ lemma Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes_of_isLocalRing
   has height at most 1. -/
 lemma Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes
     (I : Ideal R) [I.IsPrincipal] (p : Ideal R) (hp : p ∈ I.minimalPrimes) : p.height ≤ 1 := by
-  have := Ideal.minimalPrimes_isPrime hp
+  have := Ideal.IsPrime.of_mem_minimalPrimes hp
   let f := algebraMap R (Localization.AtPrime p)
   have := Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes_of_isLocalRing (I.map f) ?_
   · rwa [← IsLocalization.height_comap p.primeCompl,
@@ -117,7 +117,7 @@ lemma Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes
 theorem Ideal.map_height_le_one_of_mem_minimalPrimes {I p : Ideal R} {x : R}
     (hp : p ∈ (I ⊔ span {x}).minimalPrimes) : (p.map (Ideal.Quotient.mk I)).height ≤ 1 :=
   let f := Ideal.Quotient.mk I
-  have : p.IsPrime := Ideal.minimalPrimes_isPrime hp
+  have : p.IsPrime := Ideal.IsPrime.of_mem_minimalPrimes hp
   have hfp : RingHom.ker f ≤ p :=
     I.mk_ker.trans_le (le_sup_left.trans (Ideal.le_of_mem_minimalPrimes hp))
   height_le_one_of_isPrincipal_of_mem_minimalPrimes ((span {x}).map f) (p.map f)
@@ -139,7 +139,7 @@ theorem Ideal.mem_minimalPrimes_span_of_mem_minimalPrimes_span_insert {q p : Ide
   have hf : Function.Surjective f := Quotient.mk_surjective
   have hI'q : span t ≤ q := span_le.mpr htq
   have hI'p : span t ≤ p := hI'q.trans hqp.le
-  have := minimalPrimes_isPrime hp
+  have := IsPrime.of_mem_minimalPrimes hp
   have : (p.map f).IsPrime := map_isPrime_of_surjective hf (by rwa [mk_ker])
   suffices h : (p.map f).height ≤ 1 by
     have h_lt : q.map f < p.map f := (map_mono hqp.le).lt_of_not_ge fun e ↦ hqp.not_ge <| by
@@ -179,7 +179,7 @@ nonrec lemma Ideal.height_le_spanRank_toENat_of_mem_minimalPrimes
   induction hn : s.card using Nat.strong_induction_on generalizing R with
   | h n H =>
     replace hn : s.card ≤ n := hn.le
-    have := Ideal.minimalPrimes_isPrime hp
+    have := Ideal.IsPrime.of_mem_minimalPrimes hp
     cases n with
     | zero =>
       rw [ENat.coe_zero, nonpos_iff_eq_zero, height_eq_primeHeight p,
diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
index 17483edc6cccfd..66e3e15a4b6f1a 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
@@ -51,13 +51,13 @@ lemma minimalPrimes_eq_minimals : minimalPrimes R = {x | Minimal Ideal.IsPrime x
 
 variable {I J}
 
-theorem Ideal.minimalPrimes_isPrime {p : Ideal R} (h : p ∈ I.minimalPrimes) : p.IsPrime :=
+theorem Ideal.IsPrime.of_mem_minimalPrimes {p : Ideal R} (h : p ∈ I.minimalPrimes) : p.IsPrime :=
   h.1.1
 
 theorem Ideal.le_of_mem_minimalPrimes {p : Ideal R} (h : p ∈ I.minimalPrimes) : I ≤ p :=
   h.1.2
 
-theorem Ideal.IsPrime.of_mem_minimalPrimes {p : Ideal R} (h : p ∈ minimalPrimes R) : p.IsPrime :=
+theorem Ideal.IsPrime.of_mem_minimalPrimes' {p : Ideal R} (h : p ∈ minimalPrimes R) : p.IsPrime :=
   h.1.1
 
 theorem Ideal.exists_minimalPrimes_le [J.IsPrime] (e : I ≤ J) : ∃ p ∈ I.minimalPrimes, p ≤ J := by
@@ -123,7 +123,7 @@ theorem Ideal.minimalPrimes_eq_subsingleton (hI : I.IsPrimary) : I.minimalPrimes
   ext J
   constructor
   · exact fun H =>
-      let e := (Ideal.minimalPrimes_isPrime H).radical_le_iff.mpr (Ideal.le_of_mem_minimalPrimes H)
+      let e := (Ideal.IsPrime.of_mem_minimalPrimes H).radical_le_iff.mpr (Ideal.le_of_mem_minimalPrimes H)
       (H.2 ⟨Ideal.isPrime_radical hI, Ideal.le_radical⟩ e).antisymm e
   · rintro (rfl : J = I.radical)
     exact ⟨⟨Ideal.isPrime_radical hI, Ideal.le_radical⟩, fun _ H _ => H.1.radical_le_iff.mpr H.2⟩
@@ -150,7 +150,7 @@ theorem Ideal.minimalPrimes_top : (⊤ : Ideal R).minimalPrimes = ∅ := by
   ext p
   simp only [Set.notMem_empty, iff_false]
   intro h
-  exact (Ideal.minimalPrimes_isPrime h).ne_top (top_le_iff.mp (Ideal.le_of_mem_minimalPrimes h))
+  exact (Ideal.IsPrime.of_mem_minimalPrimes h).ne_top (top_le_iff.mp (Ideal.le_of_mem_minimalPrimes h))
 
 theorem Ideal.minimalPrimes_eq_empty_iff (I : Ideal R) :
     I.minimalPrimes = ∅ ↔ I = ⊤ := by
diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Colon.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Colon.lean
index a9654ed4513af0..547575d16c3f12 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Colon.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Colon.lean
@@ -48,8 +48,8 @@ theorem exists_eq_colon_of_mem_minimalPrimes [IsNoetherianRing R]
     refine ⟨n, hn0, ((h.toFinset.erase I).inf id) ^ n, hn, ?_⟩
     have (K : Ideal R) (hKI : K ≤ I) (hK : K ∈ ann.minimalPrimes) : K = I :=
       le_antisymm hKI (hI.2 hK.1 hKI)
-    simpa [(Ideal.minimalPrimes_isPrime hI).pow_le_iff hn0,
-      (Ideal.minimalPrimes_isPrime hI).inf_le', imp_not_comm, not_imp_not]
+    simpa [(Ideal.IsPrime.of_mem_minimalPrimes hI).pow_le_iff hn0,
+      (Ideal.IsPrime.of_mem_minimalPrimes hI).inf_le', imp_not_comm, not_imp_not]
   obtain ⟨hn0, J, hJ, hJI⟩ := Nat.find_spec key
   -- let `n` be minimal such that there exists an ideal `J` with `I ^ n * J ≤ ann` and `¬ J ≠ I`
   set n := Nat.find key
@@ -75,7 +75,7 @@ theorem exists_eq_colon_of_mem_minimalPrimes [IsNoetherianRing R]
   -- let `z` be the product of these finitely many `f y`'s
   let z := ∏ y ∈ s, f y
   -- then `z ∉ I`
-  have hz : z ∉ I := by simp [z, (Ideal.minimalPrimes_isPrime hI).prod_mem_iff, h]
+  have hz : z ∉ I := by simp [z, (Ideal.IsPrime.of_mem_minimalPrimes hI).prod_mem_iff, h]
   -- and `K ≤ colon N {z • x}`
   have hz' : K ≤ colon N {z • x} := by
     rw [← (map_injective_of_injective K.subtype_injective).eq_iff, map_subtype_top] at hs
@@ -91,7 +91,7 @@ theorem exists_eq_colon_of_mem_minimalPrimes [IsNoetherianRing R]
     simpa only [ann, mem_colon_singleton, mul_comm, mul_smul] using hz' hi
   -- but now `K = I ^ (n - 1) * J` contradicts the minimality of `n`
   have hK : I ^ (n - 1) * (J * Ideal.span {z}) ≤ ann ∧ ¬ J * Ideal.span {z} ≤ I := by
-    rw [← mul_assoc, (Ideal.minimalPrimes_isPrime hI).mul_le, not_or,
+    rw [← mul_assoc, (Ideal.IsPrime.of_mem_minimalPrimes hI).mul_le, not_or,
       Ideal.span_singleton_le_iff_mem]
     exact ⟨hz', hJI, hz⟩
   by_cases hn' : n - 1 = 0
diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean
index a5c4cfb5f300f5..ce937323a986c5 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean
@@ -65,8 +65,8 @@ theorem Ideal.iUnion_minimalPrimes :
   · rintro ⟨y, hy, hx⟩
     obtain ⟨p, hp, hyp⟩ : ∃ p ∈ I.minimalPrimes, y ∉ p := by
       simpa [← Ideal.sInf_minimalPrimes] using hy
-    refine ⟨p, hp, ((Ideal.minimalPrimes_isPrime hp).mem_or_mem ?_).resolve_right hyp⟩
-    exact (Ideal.minimalPrimes_isPrime hp).radical_le_iff.mpr (Ideal.le_of_mem_minimalPrimes hp) hx
+    refine ⟨p, hp, ((Ideal.IsPrime.of_mem_minimalPrimes hp).mem_or_mem ?_).resolve_right hyp⟩
+    exact (Ideal.IsPrime.of_mem_minimalPrimes hp).radical_le_iff.mpr (Ideal.le_of_mem_minimalPrimes hp) hx
 
 theorem Ideal.exists_mul_mem_of_mem_minimalPrimes
     {p : Ideal R} (hp : p ∈ I.minimalPrimes) {x : R} (hx : x ∈ p) :
@@ -97,7 +97,7 @@ lemma Ideal.disjoint_nonZeroDivisors_of_mem_minimalPrimes {p : Ideal R} (hp : p
 
 theorem Ideal.exists_comap_eq_of_mem_minimalPrimes {I : Ideal S} (f : R →+* S) (p)
     (H : p ∈ (I.comap f).minimalPrimes) : ∃ p' : Ideal S, p'.IsPrime ∧ I ≤ p' ∧ p'.comap f = p :=
-  have := Ideal.minimalPrimes_isPrime H
+  have := Ideal.IsPrime.of_mem_minimalPrimes H
   have ⟨p', hIp', hp', le⟩ := exists_ideal_comap_le_prime p I (Ideal.le_of_mem_minimalPrimes H)
   ⟨p', hp', hIp', le.antisymm (H.2 ⟨inferInstance, comap_mono hIp'⟩ le)⟩
 
@@ -113,7 +113,7 @@ theorem Ideal.exists_minimalPrimes_comap_eq {I : Ideal S} (f : R →+* S) (p)
   obtain ⟨p', h₁, h₂, h₃⟩ := Ideal.exists_comap_eq_of_mem_minimalPrimes f p H
   obtain ⟨q, hq, hq'⟩ := Ideal.exists_minimalPrimes_le h₂
   refine ⟨q, hq, Eq.symm ?_⟩
-  have := Ideal.minimalPrimes_isPrime hq
+  have := Ideal.IsPrime.of_mem_minimalPrimes hq
   have := (Ideal.comap_mono hq').trans_eq h₃
   exact (H.2 ⟨inferInstance, Ideal.comap_mono (Ideal.le_of_mem_minimalPrimes hq)⟩ this).antisymm
     this
@@ -130,7 +130,7 @@ variable {R S : Type*} [CommRing R] [CommRing S] {I J : Ideal R}
 
 theorem Ideal.minimalPrimes_comap_of_surjective {f : R →+* S} (hf : Function.Surjective f)
     {I J : Ideal S} (h : J ∈ I.minimalPrimes) : J.comap f ∈ (I.comap f).minimalPrimes := by
-  have := Ideal.minimalPrimes_isPrime h
+  have := Ideal.IsPrime.of_mem_minimalPrimes h
   refine ⟨⟨inferInstance, Ideal.comap_mono (Ideal.le_of_mem_minimalPrimes h)⟩, ?_⟩
   rintro K ⟨hK, e₁⟩ e₂
   have : RingHom.ker f ≤ K := (Ideal.comap_mono bot_le).trans e₁
@@ -179,12 +179,12 @@ theorem IsLocalization.minimalPrimes_map [IsLocalization S A] (J : Ideal R) :
   ext p
   constructor
   · intro hp
-    haveI := Ideal.minimalPrimes_isPrime hp
+    haveI := Ideal.IsPrime.of_mem_minimalPrimes hp
     refine ⟨⟨Ideal.IsPrime.comap _,
       Ideal.map_le_iff_le_comap.mp (Ideal.le_of_mem_minimalPrimes hp)⟩, ?_⟩
     rintro I hI e
     have hI' : Disjoint (S : Set R) I := Set.disjoint_of_subset_right e
-      ((IsLocalization.isPrime_iff_isPrime_disjoint S A _).mp (Ideal.minimalPrimes_isPrime hp)).2
+      ((IsLocalization.isPrime_iff_isPrime_disjoint S A _).mp (Ideal.IsPrime.of_mem_minimalPrimes hp)).2
     refine (Ideal.comap_mono <|
       hp.2 ⟨?_, Ideal.map_mono hI.2⟩ (Ideal.map_le_iff_le_comap.mpr e)).trans_eq ?_
     · exact IsLocalization.isPrime_of_isPrime_disjoint S A I hI.1 hI'
@@ -192,9 +192,9 @@ theorem IsLocalization.minimalPrimes_map [IsLocalization S A] (J : Ideal R) :
   · intro hp
     refine ⟨⟨?_, Ideal.map_le_iff_le_comap.mpr (Ideal.le_of_mem_minimalPrimes hp)⟩, ?_⟩
     · rw [IsLocalization.isPrime_iff_isPrime_disjoint S A, IsLocalization.disjoint_comap_iff S]
-      refine ⟨Ideal.minimalPrimes_isPrime hp, ?_⟩
+      refine ⟨Ideal.IsPrime.of_mem_minimalPrimes hp, ?_⟩
       rintro rfl
-      exact (Ideal.minimalPrimes_isPrime hp).ne_top rfl
+      exact (Ideal.IsPrime.of_mem_minimalPrimes hp).ne_top rfl
     · intro I hI e
       rw [← IsLocalization.map_comap S A I, ← IsLocalization.map_comap S A p]
       haveI := hI.1
@@ -218,7 +218,7 @@ theorem IsLocalization.AtPrime.radical_map_of_mem_minimalPrimes
   · rw [← IsLocalization.comap_le_comap_iff q.primeCompl A,
       AtPrime.map_eq_maximalIdeal q A, AtPrime.comap_maximalIdeal A q]
     apply hIq.2 hJ.1
-    have := (Ideal.minimalPrimes_isPrime hJ).ne_top
+    have := (Ideal.IsPrime.of_mem_minimalPrimes hJ).ne_top
     rw [ne_eq, Ideal.comap_eq_top_iff, ← ne_eq, ← disjoint_comap_iff q.primeCompl A J] at this
     exact Set.disjoint_compl_left_iff_subset.mp this
 
diff --git a/Mathlib/RingTheory/KrullDimension/Regular.lean b/Mathlib/RingTheory/KrullDimension/Regular.lean
index fddb6d4876b442..89bcb36e104007 100644
--- a/Mathlib/RingTheory/KrullDimension/Regular.lean
+++ b/Mathlib/RingTheory/KrullDimension/Regular.lean
@@ -95,7 +95,7 @@ theorem supportDim_quotSMulTop_succ_le_of_notMem_minimalPrimes {x : R}
   let q : LTSeries (support R M) :=
     p.map (Set.MapsTo.restrict id (support R (QuotSMulTop x M)) (support R M) le) (fun _ _ h ↦ h)
   obtain ⟨r, hrm, hr⟩ := exists_minimalPrimes_le (mem_support_iff_of_finite.mp q.head.2)
-  let r : support R M := ⟨⟨r, minimalPrimes_isPrime hrm⟩, mem_support_iff_of_finite.mpr hrm.1.2⟩
+  let r : support R M := ⟨⟨r, IsPrime.of_mem_minimalPrimes hrm⟩, mem_support_iff_of_finite.mpr hrm.1.2⟩
   have hr : r < q.head := lt_of_le_of_ne hr (fun h ↦ hn q.head.1.1 (by rwa [← h]) hp.2)
   exact le_of_eq_of_le (by simp [q]) (le_iSup _ (q.cons r hr))
 

From d6d581af8ce47486c0d00ec17b7db16e7b112917 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Tue, 5 May 2026 20:59:36 +0800
Subject: [PATCH 35/69] fix long line

---
 Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean        | 6 ++++--
 Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean | 6 ++++--
 Mathlib/RingTheory/KrullDimension/Regular.lean          | 3 ++-
 3 files changed, 10 insertions(+), 5 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
index 66e3e15a4b6f1a..c2092dba194aba 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
@@ -123,7 +123,8 @@ theorem Ideal.minimalPrimes_eq_subsingleton (hI : I.IsPrimary) : I.minimalPrimes
   ext J
   constructor
   · exact fun H =>
-      let e := (Ideal.IsPrime.of_mem_minimalPrimes H).radical_le_iff.mpr (Ideal.le_of_mem_minimalPrimes H)
+      let e := (Ideal.IsPrime.of_mem_minimalPrimes H).radical_le_iff.mpr
+        (Ideal.le_of_mem_minimalPrimes H)
       (H.2 ⟨Ideal.isPrime_radical hI, Ideal.le_radical⟩ e).antisymm e
   · rintro (rfl : J = I.radical)
     exact ⟨⟨Ideal.isPrime_radical hI, Ideal.le_radical⟩, fun _ H _ => H.1.radical_le_iff.mpr H.2⟩
@@ -150,7 +151,8 @@ theorem Ideal.minimalPrimes_top : (⊤ : Ideal R).minimalPrimes = ∅ := by
   ext p
   simp only [Set.notMem_empty, iff_false]
   intro h
-  exact (Ideal.IsPrime.of_mem_minimalPrimes h).ne_top (top_le_iff.mp (Ideal.le_of_mem_minimalPrimes h))
+  exact (Ideal.IsPrime.of_mem_minimalPrimes h).ne_top
+    (top_le_iff.mp (Ideal.le_of_mem_minimalPrimes h))
 
 theorem Ideal.minimalPrimes_eq_empty_iff (I : Ideal R) :
     I.minimalPrimes = ∅ ↔ I = ⊤ := by
diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean
index ce937323a986c5..17c15f1cd8f1ca 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean
@@ -66,7 +66,8 @@ theorem Ideal.iUnion_minimalPrimes :
     obtain ⟨p, hp, hyp⟩ : ∃ p ∈ I.minimalPrimes, y ∉ p := by
       simpa [← Ideal.sInf_minimalPrimes] using hy
     refine ⟨p, hp, ((Ideal.IsPrime.of_mem_minimalPrimes hp).mem_or_mem ?_).resolve_right hyp⟩
-    exact (Ideal.IsPrime.of_mem_minimalPrimes hp).radical_le_iff.mpr (Ideal.le_of_mem_minimalPrimes hp) hx
+    exact (Ideal.IsPrime.of_mem_minimalPrimes hp).radical_le_iff.mpr
+      (Ideal.le_of_mem_minimalPrimes hp) hx
 
 theorem Ideal.exists_mul_mem_of_mem_minimalPrimes
     {p : Ideal R} (hp : p ∈ I.minimalPrimes) {x : R} (hx : x ∈ p) :
@@ -184,7 +185,8 @@ theorem IsLocalization.minimalPrimes_map [IsLocalization S A] (J : Ideal R) :
       Ideal.map_le_iff_le_comap.mp (Ideal.le_of_mem_minimalPrimes hp)⟩, ?_⟩
     rintro I hI e
     have hI' : Disjoint (S : Set R) I := Set.disjoint_of_subset_right e
-      ((IsLocalization.isPrime_iff_isPrime_disjoint S A _).mp (Ideal.IsPrime.of_mem_minimalPrimes hp)).2
+      ((IsLocalization.isPrime_iff_isPrime_disjoint S A _).mp
+        (Ideal.IsPrime.of_mem_minimalPrimes hp)).2
     refine (Ideal.comap_mono <|
       hp.2 ⟨?_, Ideal.map_mono hI.2⟩ (Ideal.map_le_iff_le_comap.mpr e)).trans_eq ?_
     · exact IsLocalization.isPrime_of_isPrime_disjoint S A I hI.1 hI'
diff --git a/Mathlib/RingTheory/KrullDimension/Regular.lean b/Mathlib/RingTheory/KrullDimension/Regular.lean
index 89bcb36e104007..d57c56ed221acd 100644
--- a/Mathlib/RingTheory/KrullDimension/Regular.lean
+++ b/Mathlib/RingTheory/KrullDimension/Regular.lean
@@ -95,7 +95,8 @@ theorem supportDim_quotSMulTop_succ_le_of_notMem_minimalPrimes {x : R}
   let q : LTSeries (support R M) :=
     p.map (Set.MapsTo.restrict id (support R (QuotSMulTop x M)) (support R M) le) (fun _ _ h ↦ h)
   obtain ⟨r, hrm, hr⟩ := exists_minimalPrimes_le (mem_support_iff_of_finite.mp q.head.2)
-  let r : support R M := ⟨⟨r, IsPrime.of_mem_minimalPrimes hrm⟩, mem_support_iff_of_finite.mpr hrm.1.2⟩
+  let r : support R M := ⟨⟨r, IsPrime.of_mem_minimalPrimes hrm⟩,
+    mem_support_iff_of_finite.mpr hrm.1.2⟩
   have hr : r < q.head := lt_of_le_of_ne hr (fun h ↦ hn q.head.1.1 (by rwa [← h]) hp.2)
   exact le_of_eq_of_le (by simp [q]) (le_iSup _ (q.cons r hr))
 

From acc4f74109dde1e49b253c32cb915d819448757f Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Tue, 5 May 2026 21:01:23 +0800
Subject: [PATCH 36/69] fix

---
 Mathlib/RingTheory/Ideal/Height.lean | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index 36c505bf3fb0c3..6f2dd53e83c68f 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -447,7 +447,7 @@ lemma Ideal.sup_height_isPrime_eq_ringKrullDim [Nontrivial R] :
     · exact ⟨⊥, by simp [I_top]⟩
     · obtain ⟨P, hP⟩ := Set.nonempty_coe_sort.mp (nonempty_minimalPrimes I_top)
       refine ⟨P, iSup_pos (α := ℕ∞) I_top ▸ le_iSup_of_le (hP.left.left) ?_⟩
-      have := Ideal.minimalPrimes_isPrime hP
+      have := Ideal.IsPrime.of_mem_minimalPrimes hP
       exact iInf_le_of_le P (iInf_le_of_le hP (ge_of_eq (Ideal.height_eq_primeHeight P)))
 
 @[deprecated "Use `Ideal.sup_height_isPrime_eq_ringKrullDim` instead." (since := "2026-04-02")]

From b31ee5880f2fd280a82b31c23ca513fd6fab9652 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Tue, 5 May 2026 21:03:34 +0800
Subject: [PATCH 37/69] fix

---
 Mathlib/RingTheory/Ideal/Height.lean | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index f2f1e2b7fb8f18..c9e74f4b73c5d4 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -51,7 +51,7 @@ lemma Ideal.height_eq_order_height_of_isPrime [I.IsPrime] : I.height =
 
 lemma Ideal.height_eq_inf_minimalPrimes : I.height = ⨅ J ∈ I.minimalPrimes, J.height := by
   apply iInf_congr (fun p ↦ iInf_congr fun hp ↦ ?_)
-  have := I.minimalPrimes_isPrime hp
+  have := I.IsPrime.of_mem_minimalPrimes hp
   exact (Ideal.height_eq_primeHeight _).symm
 
 /-- An ideal has finite height if it is either the unit ideal or its height is finite.
@@ -383,7 +383,7 @@ theorem IsLocalization.height_comap (S : Submonoid R) {A : Type*} [CommRing A] [
   rw [(J.comap _).height_eq_inf_minimalPrimes, J.height_eq_inf_minimalPrimes]
   simp only [IsLocalization.minimalPrimes_comap S A, iInf_image]
   apply iInf_congr (fun p ↦ iInf_congr fun hp ↦ ?_)
-  have := Ideal.minimalPrimes_isPrime hp
+  have := Ideal.IsPrime.of_mem_minimalPrimes hp
   exact IsLocalization.height_comap_eq_of_isPrime S _
 
 theorem IsLocalization.AtPrime.ringKrullDim_eq_height (I : Ideal R) [I.IsPrime] (A : Type*)

From d6f76edc2953f113321ab2ed061c593c3849e4ad Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Tue, 5 May 2026 22:54:29 +0800
Subject: [PATCH 38/69] fix

---
 Mathlib/RingTheory/Ideal/Height.lean | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index c9e74f4b73c5d4..c181a39af13470 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -51,7 +51,7 @@ lemma Ideal.height_eq_order_height_of_isPrime [I.IsPrime] : I.height =
 
 lemma Ideal.height_eq_inf_minimalPrimes : I.height = ⨅ J ∈ I.minimalPrimes, J.height := by
   apply iInf_congr (fun p ↦ iInf_congr fun hp ↦ ?_)
-  have := I.IsPrime.of_mem_minimalPrimes hp
+  have := Ideal.IsPrime.of_mem_minimalPrimes hp
   exact (Ideal.height_eq_primeHeight _).symm
 
 /-- An ideal has finite height if it is either the unit ideal or its height is finite.

From a2c2b50a638c5e585e8591c7b46abd29980009d5 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Thu, 7 May 2026 20:58:51 +0800
Subject: [PATCH 39/69] private primeHeight

---
 Mathlib/RingTheory/Ideal/Height.lean | 28 +++++++++++++++-------------
 1 file changed, 15 insertions(+), 13 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index c181a39af13470..4a6a1837757911 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -34,15 +34,16 @@ open Ideal
 
 /-- The height of a prime ideal is defined as the supremum of the lengths of strictly decreasing
 chains of prime ideals below it. -/
-noncomputable def Ideal.primeHeight [hI : I.IsPrime] : ℕ∞ :=
+private noncomputable def Ideal.primeHeight [hI : I.IsPrime] : ℕ∞ :=
   Order.height (⟨I, hI⟩ : PrimeSpectrum R)
 
 /-- The height of an ideal is defined as the infimum of the heights of its minimal prime ideals. -/
+@[no_expose]
 noncomputable def Ideal.height : ℕ∞ :=
   ⨅ J ∈ I.minimalPrimes, @Ideal.primeHeight _ _ J (IsPrime.of_mem_minimalPrimes (I := I) ‹_›)
 
 /-- For a prime ideal, its height equals its prime height. -/
-lemma Ideal.height_eq_primeHeight [I.IsPrime] : I.height = I.primeHeight := by
+private lemma Ideal.height_eq_primeHeight [I.IsPrime] : I.height = I.primeHeight := by
   simp [height, primeHeight, Ideal.minimalPrimes_eq_subsingleton_self]
 
 lemma Ideal.height_eq_order_height_of_isPrime [I.IsPrime] : I.height =
@@ -77,7 +78,7 @@ lemma Ideal.height_ne_top_of_isPrime {I : Ideal R} [I.FiniteHeight] [I.IsPrime]
   Ideal.height_ne_top ‹I.IsPrime›.ne_top
 
 @[deprecated "Use `Ideal.height_ne_top_of_isPrime` instead." (since := "2026-04-04")]
-lemma Ideal.primeHeight_ne_top (I : Ideal R) [I.FiniteHeight] [I.IsPrime] :
+private lemma Ideal.primeHeight_ne_top (I : Ideal R) [I.FiniteHeight] [I.IsPrime] :
     I.primeHeight ≠ ⊤ := by
   rw [← I.height_eq_primeHeight]
   exact Ideal.height_ne_top ‹I.IsPrime›.ne_top
@@ -87,7 +88,7 @@ lemma Ideal.height_lt_top_of_isPrime {I : Ideal R} [I.FiniteHeight] [I.IsPrime]
   Ideal.height_lt_top ‹I.IsPrime›.ne_top
 
 @[deprecated "Use `Ideal.height_lt_top_of_isPrime` instead." (since := "2026-04-04")]
-lemma Ideal.primeHeight_lt_top (I : Ideal R) [I.FiniteHeight] [I.IsPrime] :
+private lemma Ideal.primeHeight_lt_top (I : Ideal R) [I.FiniteHeight] [I.IsPrime] :
     I.primeHeight < ⊤ := by
   rw [← I.height_eq_primeHeight]
   exact Ideal.height_lt_top ‹I.IsPrime›.ne_top
@@ -109,7 +110,7 @@ private lemma Ideal.height_mono_of_isPrime {I J : Ideal R} [I.IsPrime] [J.IsPrim
   exact h
 
 @[deprecated "Use `Ideal.height_mono_of_isPrime` instead." (since := "2026-04-04")]
-lemma Ideal.primeHeight_mono {I J : Ideal R} [I.IsPrime] [J.IsPrime] (h : I ≤ J) :
+private lemma Ideal.primeHeight_mono {I J : Ideal R} [I.IsPrime] [J.IsPrime] (h : I ≤ J) :
     I.primeHeight ≤ J.primeHeight := by
   simpa [Ideal.height_eq_primeHeight] using Ideal.height_mono_of_isPrime h
 
@@ -119,7 +120,7 @@ lemma Ideal.height_add_one_le_of_lt_of_isPrime {I J : Ideal R} [I.IsPrime] [J.Is
   exact Order.height_add_one_le h
 
 @[deprecated "Use `Ideal.height_add_one_le_of_lt_of_isPrime` instead." (since := "2026-04-04")]
-lemma Ideal.primeHeight_add_one_le_of_lt {I J : Ideal R} [I.IsPrime] [J.IsPrime]
+private lemma Ideal.primeHeight_add_one_le_of_lt {I J : Ideal R} [I.IsPrime] [J.IsPrime]
     (h : I < J) : I.primeHeight + 1 ≤ J.primeHeight := by
   simpa [Ideal.height_eq_primeHeight] using Ideal.height_add_one_le_of_lt_of_isPrime h
 
@@ -160,7 +161,7 @@ alias Ideal.height_strict_mono_of_isPrime_of_is_prime :=
 
 @[deprecated "Use `Ideal.height_strict_mono_of_isPrime_of_isPrime` instead."
   (since := "2026-04-02")]
-lemma Ideal.primeHeight_strict_mono {I J : Ideal R} [I.IsPrime] [J.IsPrime] (h : I < J)
+private lemma Ideal.primeHeight_strict_mono {I J : Ideal R} [I.IsPrime] [J.IsPrime] (h : I < J)
     [J.FiniteHeight] : I.primeHeight < J.primeHeight := by
   simpa [← Ideal.height_eq_primeHeight] using Ideal.height_strict_mono_of_isPrime_of_isPrime h
 
@@ -257,7 +258,7 @@ theorem Ideal.isMaximal_of_height_eq_ringKrullDim {I : Ideal R} [I.IsPrime]
   · exact hM' ▸ hM
 
 @[deprecated "Use `Ideal.isMaximal_of_height_eq_ringKrullDim` instead." (since := "2026-04-02")]
-theorem Ideal.isMaximal_of_primeHeight_eq_ringKrullDim {I : Ideal R} [I.IsPrime]
+private theorem Ideal.isMaximal_of_primeHeight_eq_ringKrullDim {I : Ideal R} [I.IsPrime]
     [FiniteRingKrullDim R] (e : I.primeHeight = ringKrullDim R) : I.IsMaximal :=
   Ideal.isMaximal_of_height_eq_ringKrullDim (by simpa [Ideal.height_eq_primeHeight])
 
@@ -285,8 +286,9 @@ theorem Ideal.height_eq_ringKrullDim_iff [FiniteRingKrullDim R] [IsLocalRing R]
     exact IsLocalRing.maximalIdeal_height_eq_ringKrullDim
 
 @[deprecated "Use `Ideal.height_eq_ringKrullDim_iff` instead." (since := "2026-04-02")]
-theorem Ideal.primeHeight_eq_ringKrullDim_iff [FiniteRingKrullDim R] [IsLocalRing R] {I : Ideal R}
-    [I.IsPrime] : Ideal.primeHeight I = ringKrullDim R ↔ I = IsLocalRing.maximalIdeal R := by
+private theorem Ideal.primeHeight_eq_ringKrullDim_iff [FiniteRingKrullDim R] [IsLocalRing R]
+    {I : Ideal R} [I.IsPrime] :
+    Ideal.primeHeight I = ringKrullDim R ↔ I = IsLocalRing.maximalIdeal R := by
   rw [← Ideal.height_eq_primeHeight, Ideal.height_eq_ringKrullDim_iff]
 
 lemma Ideal.height_le_iff {p : Ideal R} {n : ℕ} [p.IsPrime] :
@@ -407,7 +409,7 @@ lemma IsLocalization.height_map_of_disjoint {S : Type*} [CommRing S] [Algebra R
   exact WithBot.coe_eq_coe.mp this
 
 @[deprecated "Use `mem_minimalPrimes_of_height_eq` instead." (since := "2026-04-02")]
-lemma mem_minimalPrimes_of_primeHeight_eq_height {I J : Ideal R} [J.IsPrime] (e : I ≤ J)
+private lemma mem_minimalPrimes_of_primeHeight_eq_height {I J : Ideal R} [J.IsPrime] (e : I ≤ J)
     (e' : J.primeHeight = I.height) [J.FiniteHeight] : J ∈ I.minimalPrimes := by
   rw [← J.height_eq_primeHeight] at e'
   exact mem_minimalPrimes_of_height_eq e (e' ▸ le_refl _)
@@ -490,7 +492,7 @@ lemma Ideal.sup_height_isPrime_eq_ringKrullDim [Nontrivial R] :
       exact iInf_le_of_le P (iInf_le_of_le hP (ge_of_eq (Ideal.height_eq_primeHeight P)))
 
 @[deprecated "Use `Ideal.sup_height_isPrime_eq_ringKrullDim` instead." (since := "2026-04-02")]
-lemma Ideal.sup_primeHeight_eq_ringKrullDim [Nontrivial R] :
+private lemma Ideal.sup_primeHeight_eq_ringKrullDim [Nontrivial R] :
     ↑(⨆ (I : Ideal R) (_ : I.IsPrime), I.primeHeight) = ringKrullDim R := by
   simp [← Ideal.height_eq_primeHeight, Ideal.sup_height_isPrime_eq_ringKrullDim]
 
@@ -508,7 +510,7 @@ lemma Ideal.sup_height_of_maximal_eq_ringKrullDim [Nontrivial R] :
       exact ⟨M, iSup_mono' (fun hI ↦ ⟨hM, height_mono hIM⟩)⟩
 
 @[deprecated "Use `Ideal.sup_height_of_maximal_eq_ringKrullDim` instead." (since := "2026-04-02")]
-lemma Ideal.sup_primeHeight_of_maximal_eq_ringKrullDim [Nontrivial R] :
+private lemma Ideal.sup_primeHeight_of_maximal_eq_ringKrullDim [Nontrivial R] :
     ↑(⨆ (I : Ideal R) (_ : I.IsMaximal), I.primeHeight) = ringKrullDim R := by
   simp_rw [← Ideal.height_eq_primeHeight, Ideal.sup_height_of_maximal_eq_ringKrullDim]
 

From 7f19e330f1e0564925a618ef935fc47572548e2c Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Thu, 7 May 2026 21:20:03 +0800
Subject: [PATCH 40/69] fix

---
 Mathlib/AlgebraicGeometry/Properties.lean  | 2 +-
 Mathlib/RingTheory/KrullDimension/PID.lean | 2 +-
 2 files changed, 2 insertions(+), 2 deletions(-)

diff --git a/Mathlib/AlgebraicGeometry/Properties.lean b/Mathlib/AlgebraicGeometry/Properties.lean
index 764b571ba938f5..0344da531bd33b 100644
--- a/Mathlib/AlgebraicGeometry/Properties.lean
+++ b/Mathlib/AlgebraicGeometry/Properties.lean
@@ -354,7 +354,7 @@ lemma coheight_eq_of_isOpenImmersion {U X : Scheme} {x : U} (f : U ⟶ X) [IsOpe
 open Order in
 lemma idealHeight_eq_coheight (R : CommRingCat) (x : Spec R) :
     x.asIdeal.height = coheight x := by
-  rw [Ideal.height_eq_primeHeight x.asIdeal, Ideal.primeHeight,
+  rw [Ideal.height_eq_order_height_of_isPrime,
     ← Order.coheight_orderIso (specOrderIsoPrimeSpectrum R), ← height_ofDual,
     specOrderIsoPrimeSpectrum_apply, OrderDual.ofDual_toDual]
   rfl
diff --git a/Mathlib/RingTheory/KrullDimension/PID.lean b/Mathlib/RingTheory/KrullDimension/PID.lean
index 74c56d4b0bde82..c87e2986505260 100644
--- a/Mathlib/RingTheory/KrullDimension/PID.lean
+++ b/Mathlib/RingTheory/KrullDimension/PID.lean
@@ -51,5 +51,5 @@ lemma IsPrincipalIdealRing.height_eq_one_of_isMaximal {R : Type*} [CommRing R] [
   · suffices h : (m.height : WithBot ℕ∞) ≤ 1 by norm_cast at h
     rw [← IsPrincipalIdealRing.ringKrullDim_eq_one _ h]
     exact Ideal.height_le_ringKrullDim_of_ne_top Ideal.IsPrime.ne_top'
-  · rw [Order.one_le_iff_pos, Ideal.height_eq_primeHeight, Ideal.primeHeight, Order.height_pos]
+  · rw [Order.one_le_iff_pos, Ideal.height_eq_order_height_of_isPrime, Order.height_pos]
     exact not_isMin_of_lt (b := ⊥) (Ideal.bot_lt_of_maximal m h)

From c6df49a131394480eb54518017eb19428e9a6f93 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Fri, 8 May 2026 11:47:32 +0800
Subject: [PATCH 41/69] add predicate IsMinimalPrime

---
 Mathlib/RingTheory/Ideal/Height.lean          | 35 ++++++++--------
 .../RingTheory/Ideal/KrullsHeightTheorem.lean | 27 +++++++------
 .../RingTheory/Ideal/MinimalPrime/Basic.lean  | 38 +++++++++---------
 .../RingTheory/Ideal/MinimalPrime/Colon.lean  | 10 ++---
 .../Ideal/MinimalPrime/Localization.lean      | 40 ++++++++-----------
 Mathlib/RingTheory/KrullDimension/Zero.lean   |  4 +-
 .../RingTheory/Spectrum/Prime/Topology.lean   |  2 +-
 7 files changed, 73 insertions(+), 83 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index 8e275fc8481417..74ea36cc589984 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -39,7 +39,7 @@ noncomputable def Ideal.primeHeight [hI : I.IsPrime] : ℕ∞ :=
 
 /-- The height of an ideal is defined as the infimum of the heights of its minimal prime ideals. -/
 noncomputable def Ideal.height : ℕ∞ :=
-  ⨅ J ∈ I.minimalPrimes, @Ideal.primeHeight _ _ J (IsPrime.of_mem_minimalPrimes (I := I) ‹_›)
+  ⨅ J ∈ I.minimalPrimes, @Ideal.primeHeight _ _ J ‹J ∈ I.minimalPrimes›.isPrime
 
 /-- For a prime ideal, its height equals its prime height. -/
 lemma Ideal.height_eq_primeHeight [I.IsPrime] : I.height = I.primeHeight := by
@@ -116,9 +116,9 @@ lemma Ideal.primeHeight_strict_mono {I J : Ideal R} [I.IsPrime] [J.IsPrime]
 theorem Ideal.height_mono {I J : Ideal R} (h : I ≤ J) : I.height ≤ J.height := by
   simp only [height]
   refine le_iInf₂ (fun p hp ↦ ?_)
-  have := Ideal.IsPrime.of_mem_minimalPrimes hp
-  obtain ⟨q, hq, e⟩ := Ideal.exists_minimalPrimes_le (h.trans (Ideal.le_of_mem_minimalPrimes hp))
-  haveI := Ideal.IsPrime.of_mem_minimalPrimes hq
+  have := hp.isPrime
+  obtain ⟨q, hq, e⟩ := Ideal.exists_minimalPrimes_le (h.trans hp.le)
+  haveI := hq.isPrime
   exact (iInf₂_le q hq).trans (Ideal.primeHeight_mono e)
 
 @[gcongr]
@@ -130,9 +130,8 @@ lemma Ideal.height_strict_mono_of_is_prime {I J : Ideal R} [I.IsPrime]
     exact I.primeHeight_lt_top
   · rw [← ENat.add_one_le_iff I.primeHeight_ne_top, Ideal.height]
     refine le_iInf₂ (fun K hK ↦ ?_)
-    have := Ideal.IsPrime.of_mem_minimalPrimes hK
-    have : I < K := lt_of_lt_of_le h (Ideal.le_of_mem_minimalPrimes hK)
-    exact Ideal.primeHeight_add_one_le_of_lt this
+    have := hK.isPrime
+    exact Ideal.primeHeight_add_one_le_of_lt (lt_of_lt_of_le h hK.le)
 
 /-- A prime ideal of finite height is equal to any ideal that contains it with no greater height. -/
 lemma Ideal.eq_of_le_of_height_le [I.IsPrime] [I.FiniteHeight]
@@ -146,7 +145,7 @@ lemma Ideal.height_le_ringKrullDim_of_ne_top {I : Ideal R} (h : I ≠ ⊤) :
     I.height ≤ ringKrullDim R := by
   rw [Ideal.height]
   obtain ⟨P, hP⟩ : Nonempty (I.minimalPrimes) := Ideal.nonempty_minimalPrimes h
-  have := Ideal.IsPrime.of_mem_minimalPrimes hP
+  have := hP.isPrime
   refine le_trans ?_ (Ideal.primeHeight_le_ringKrullDim (I := P))
   simpa using iInf₂_le _ hP
 
@@ -184,16 +183,14 @@ lemma Ideal.mem_minimalPrimes_of_height_eq {I J : Ideal R} (e : I ≤ J) [J.IsPr
   obtain ⟨p, h₁, h₂⟩ := Ideal.exists_minimalPrimes_le e
   convert h₁
   refine (eq_of_le_of_not_lt h₂ fun h₃ ↦ ?_).symm
-  have := Ideal.IsPrime.of_mem_minimalPrimes h₁
+  have := h₁.isPrime
   have := finiteHeight_of_le h₂ IsPrime.ne_top'
-  exact lt_irrefl _ ((height_strict_mono_of_is_prime h₃).trans_le
-    (e'.trans <| height_mono (Ideal.le_of_mem_minimalPrimes h₁)))
+  exact lt_irrefl _ ((height_strict_mono_of_is_prime h₃).trans_le (e'.trans <| height_mono h₁.le))
 
 /-- A prime ideal has height zero if and only if it is minimal -/
 lemma Ideal.primeHeight_eq_zero_iff {I : Ideal R} [I.IsPrime] :
     primeHeight I = 0 ↔ I ∈ minimalPrimes R := by
-  rw [Ideal.primeHeight, Order.height_eq_zero, minimalPrimes, Ideal.minimalPrimes]
-  simp only [bot_le, and_true, Set.mem_setOf_eq, Minimal, IsMin]
+  rw [Ideal.primeHeight, Order.height_eq_zero, minimalPrimes_eq_minimals]
   constructor
   · intro h
     refine ⟨inferInstance, ?_⟩
@@ -288,7 +285,7 @@ lemma RingEquiv.height_comap {S : Type*} [CommRing S] (e : R ≃+* S) (I : Ideal
     rw [← Ideal.comap_coe,
       Ideal.comap_minimalPrimes_eq_of_surjective (f := (↑e : R →+* S)) e.surjective]
     exact e.idealComapOrderIso.injective.mem_set_image.symm
-  · have : J.IsPrime := Ideal.IsPrime.of_mem_minimalPrimes h
+  · have : J.IsPrime := h.isPrime
     simp only [EquivLike.coe_coe, RingEquiv.idealComapOrderIso_apply,
       ← Ideal.height_eq_primeHeight, RingEquiv.height_comap_of_isPrime]
 
@@ -379,7 +376,7 @@ lemma exists_spanRank_le_and_le_height_of_le_height [IsNoetherianRing R] (I : Id
       refine (Ideal.subset_union_prime ⊥ ⊥ ?_).not.mpr ?_
       · rintro K hK - -
         rw [Set.Finite.mem_toFinset] at hK
-        exact Ideal.IsPrime.of_mem_minimalPrimes hK.1
+        exact hK.1.isPrime
       · push Not
         intro K hK e
         have := hr.trans (Ideal.height_mono e)
@@ -395,7 +392,7 @@ lemma exists_spanRank_le_and_le_height_of_le_height [IsNoetherianRing R] (I : Id
       push_cast
       exact add_le_add h₂ ((Submodule.spanRank_span_le_card _).trans (by simp))
     · refine le_iInf₂ (fun p hp ↦ ?_)
-      have := Ideal.IsPrime.of_mem_minimalPrimes hp
+      have := hp.isPrime
       by_cases h : p.height = ⊤
       · rw [← p.height_eq_primeHeight, h]
         exact le_top
@@ -412,10 +409,10 @@ lemma exists_spanRank_le_and_le_height_of_le_height [IsNoetherianRing R] (I : Id
           apply le_antisymm
           · rwa [← p.height_eq_primeHeight]
           · rwa [← e]
-        refine ⟨mem_minimalPrimes_of_primeHeight_eq_height
-          (le_sup_left.trans (Ideal.le_of_mem_minimalPrimes hp)) this.symm, ?_⟩
+        refine ⟨mem_minimalPrimes_of_primeHeight_eq_height (le_sup_left.trans hp.le) this.symm, ?_⟩
         rwa [p.height_eq_primeHeight, eq_comm]
-      · exact (Ideal.le_of_mem_minimalPrimes hp) <| Ideal.mem_sup_right <| mem_span_singleton_self x
+      · apply hp.le
+        exact Ideal.mem_sup_right <| mem_span_singleton_self x
 
 /-- In a nontrivial commutative ring `R`, the supremum of heights of all ideals is equal to the
 Krull dimension of `R`. -/
diff --git a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
index 216697afa6012c..4727a66d44e03c 100644
--- a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
+++ b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
@@ -83,7 +83,7 @@ lemma Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes_of_isLocalRing
     ← (IsLocalization.orderEmbedding q.primeCompl (Localization.AtPrime q)).map_rel_iff]
   refine Submodule.le_of_le_smul_of_le_jacobson_bot (I := I) (IsNoetherian.noetherian _) ?_ ?_
   · rw [IsLocalRing.jacobson_eq_maximalIdeal]
-    exacts [Ideal.le_of_mem_minimalPrimes hp, bot_ne_top]
+    exacts [hp.le, bot_ne_top]
   · replace hn := congr(Ideal.comap (Ideal.Quotient.mk I) $(hn _ n.le_succ))
     simp only [qs, OrderHom.coe_mk, ← RingHom.ker_eq_comap_bot, Ideal.mk_ker,
       Ideal.comap_map_of_surjective _ Ideal.Quotient.mk_surjective] at hn
@@ -106,7 +106,7 @@ lemma Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes_of_isLocalRing
   has height at most 1. -/
 lemma Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes
     (I : Ideal R) [I.IsPrincipal] (p : Ideal R) (hp : p ∈ I.minimalPrimes) : p.height ≤ 1 := by
-  have := Ideal.IsPrime.of_mem_minimalPrimes hp
+  have := hp.isPrime
   let f := algebraMap R (Localization.AtPrime p)
   have := Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes_of_isLocalRing (I.map f) ?_
   · rwa [← IsLocalization.height_comap p.primeCompl,
@@ -117,12 +117,12 @@ lemma Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes
 theorem Ideal.map_height_le_one_of_mem_minimalPrimes {I p : Ideal R} {x : R}
     (hp : p ∈ (I ⊔ span {x}).minimalPrimes) : (p.map (Ideal.Quotient.mk I)).height ≤ 1 :=
   let f := Ideal.Quotient.mk I
-  have : p.IsPrime := Ideal.IsPrime.of_mem_minimalPrimes hp
+  have : p.IsPrime := hp.isPrime
   have hfp : RingHom.ker f ≤ p :=
-    I.mk_ker.trans_le (le_sup_left.trans (Ideal.le_of_mem_minimalPrimes hp))
+    I.mk_ker.trans_le (le_sup_left.trans hp.le)
   height_le_one_of_isPrincipal_of_mem_minimalPrimes ((span {x}).map f) (p.map f)
     ⟨⟨map_isPrime_of_surjective Quotient.mk_surjective hfp,
-      map_mono (le_sup_right.trans (Ideal.le_of_mem_minimalPrimes hp))⟩,
+      map_mono (le_sup_right.trans hp.le)⟩,
         fun _ ⟨hr, hxr⟩ hrp ↦ map_le_iff_le_comap.mpr <| hp.2 ⟨hr.comap f, sup_le_iff.mpr
           ⟨I.mk_ker.symm.trans_le <| ker_le_comap (Ideal.Quotient.mk I), le_comap_of_map_le hxr⟩⟩ <|
             (comap_mono hrp).trans <| Eq.le <|
@@ -139,7 +139,8 @@ theorem Ideal.mem_minimalPrimes_span_of_mem_minimalPrimes_span_insert {q p : Ide
   have hf : Function.Surjective f := Quotient.mk_surjective
   have hI'q : span t ≤ q := span_le.mpr htq
   have hI'p : span t ≤ p := hI'q.trans hqp.le
-  have := IsPrime.of_mem_minimalPrimes hp
+  have := hp.isPrime
+  have lep := hp.le
   have : (p.map f).IsPrime := map_isPrime_of_surjective hf (by rwa [mk_ker])
   suffices h : (p.map f).height ≤ 1 by
     have h_lt : q.map f < p.map f := (map_mono hqp.le).lt_of_not_ge fun e ↦ hqp.not_ge <| by
@@ -154,8 +155,7 @@ theorem Ideal.mem_minimalPrimes_span_of_mem_minimalPrimes_span_insert {q p : Ide
     rwa [comap_map_of_surjective f hf, ← RingHom.ker_eq_comap_bot,
       mk_ker, sup_eq_left.mpr hI'q] at this
   refine height_le_one_of_isPrincipal_of_mem_minimalPrimes ((span {x}).map f) (p.map f) ⟨⟨this,
-    map_mono <| span_le.mpr <| Set.singleton_subset_iff.mpr <| (Ideal.le_of_mem_minimalPrimes hp) <|
-      subset_span <| .inl rfl⟩,
+    map_mono <| span_le.mpr <| Set.singleton_subset_iff.mpr <| lep <| subset_span <| .inl rfl⟩,
     fun r ⟨hr, hxr⟩ hrp ↦ map_le_iff_le_comap.mpr (hp.2 ⟨hr.comap f, ?_⟩ ?_)⟩
   · rw [span_le, Set.insert_subset_iff]
     have := map_le_iff_le_comap.mp hxr (subset_span rfl)
@@ -179,7 +179,7 @@ nonrec lemma Ideal.height_le_spanRank_toENat_of_mem_minimalPrimes
   induction hn : s.card using Nat.strong_induction_on generalizing R with
   | h n H =>
     replace hn : s.card ≤ n := hn.le
-    have := Ideal.IsPrime.of_mem_minimalPrimes hp
+    have := hp.isPrime
     cases n with
     | zero =>
       rw [ENat.coe_zero, nonpos_iff_eq_zero, height_eq_primeHeight p,
@@ -207,7 +207,7 @@ nonrec lemma Ideal.height_le_spanRank_toENat_of_mem_minimalPrimes
           ⟨le_sup_left, x, mem_sup_right (mem_span_singleton_self _), hxq⟩).trans_le hJ)
           ((le_maximalIdeal hJ'.ne_top).lt_of_not_ge h)
       have h : (s' : Set R) ⊆ (q ⊔ span {x}).radical := by
-        have := (Ideal.le_of_mem_minimalPrimes hp).trans this
+        have := hp.le.trans this
         rw [span_le, Finset.coe_insert, Set.insert_subset_iff] at this
         exact this.2
       obtain ⟨t, ht, hspan⟩ := exists_subset_radical_span_sup_of_subset_radical_sup _ _ _ h
@@ -324,8 +324,8 @@ lemma Ideal.height_le_height_add_spanFinrank_of_le {I p : Ideal R} [p.IsPrime] (
   let p' := p.map (algebraMap R (R ⧸ I))
   have : p'.IsPrime := isPrime_map_quotientMk_of_isPrime hrp
   obtain ⟨s, hps, hs⟩ := exists_finset_card_eq_height_of_isNoetherianRing p'
-  have hsp' : (s : Set (R ⧸ I)) ⊆ (p' : Set _) :=
-    fun _ hx ↦ (Ideal.le_of_mem_minimalPrimes hps) (subset_span hx)
+  have lep' := hps.le
+  have hsp' : (s : Set (R ⧸ I)) ⊆ (p' : Set _) := fun _ hx ↦ lep' (subset_span hx)
   have : Set.SurjOn (Ideal.Quotient.mk I) p s := by
     refine Set.SurjOn.mono subset_rfl hsp' fun x hx ↦ ?_
     obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x
@@ -421,8 +421,9 @@ lemma Ideal.height_le_height_add_of_liesOver [IsNoetherianRing S] (p : Ideal R)
   obtain ⟨s, hp, heq⟩ := p.exists_finset_card_eq_height_of_isNoetherianRing
   let P' := P.map (Ideal.Quotient.mk <| p.map (algebraMap R S))
   obtain ⟨s', hP', heq'⟩ := P'.exists_finset_card_eq_height_of_isNoetherianRing
+  have leP' := hP'.le
   have hsP'sub : (s' : Set <| S ⧸ (Ideal.map (algebraMap R S) p)) ⊆ (P' : Set <| S ⧸ _) :=
-    fun x hx ↦ (Ideal.le_of_mem_minimalPrimes hP') (Ideal.subset_span hx)
+    fun x hx ↦ leP' (Ideal.subset_span hx)
   have : Set.SurjOn (Ideal.Quotient.mk (p.map (algebraMap R S))) P s' := by
     refine Set.SurjOn.mono subset_rfl hsP'sub fun x hx ↦ ?_
     obtain ⟨y, rfl⟩ := Ideal.Quotient.mk_surjective x
diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
index c2092dba194aba..71da00fef48909 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
@@ -36,14 +36,20 @@ section
 
 variable {R S : Type*} [CommSemiring R] [CommSemiring S] (I J : Ideal R)
 
+protected def Ideal.IsMinimalPrime (p : Ideal R) : Prop := Minimal (fun q ↦ q.IsPrime ∧ I ≤ q) p
+
+lemma Ideal.IsMinimalPrime.isPrime {p : Ideal R} (h : I.IsMinimalPrime p) : p.IsPrime := h.1.1
+
+lemma Ideal.IsMinimalPrime.le {p : Ideal R} (h : I.IsMinimalPrime p) : I ≤ p := h.1.2
+
 /-- `I.minimalPrimes` is the set of ideals that are minimal primes over `I`. -/
-protected def Ideal.minimalPrimes : Set (Ideal R) :=
-  {p | Minimal (fun q ↦ q.IsPrime ∧ I ≤ q) p}
+protected abbrev Ideal.minimalPrimes : Set (Ideal R) :=
+  {p | I.IsMinimalPrime p}
 
 variable (R) in
 /-- `minimalPrimes R` is the set of minimal primes of `R`.
 This is defined as `Ideal.minimalPrimes ⊥`. -/
-def minimalPrimes : Set (Ideal R) :=
+abbrev minimalPrimes : Set (Ideal R) :=
   Ideal.minimalPrimes ⊥
 
 lemma minimalPrimes_eq_minimals : minimalPrimes R = {x | Minimal Ideal.IsPrime x} :=
@@ -51,13 +57,8 @@ lemma minimalPrimes_eq_minimals : minimalPrimes R = {x | Minimal Ideal.IsPrime x
 
 variable {I J}
 
-theorem Ideal.IsPrime.of_mem_minimalPrimes {p : Ideal R} (h : p ∈ I.minimalPrimes) : p.IsPrime :=
-  h.1.1
-
-theorem Ideal.le_of_mem_minimalPrimes {p : Ideal R} (h : p ∈ I.minimalPrimes) : I ≤ p :=
-  h.1.2
-
-theorem Ideal.IsPrime.of_mem_minimalPrimes' {p : Ideal R} (h : p ∈ minimalPrimes R) : p.IsPrime :=
+theorem Ideal.IsPrime.of_mem_minimalPrimes {p : Ideal R} (h : p ∈ minimalPrimes R) :
+    p.IsPrime :=
   h.1.1
 
 theorem Ideal.exists_minimalPrimes_le [J.IsPrime] (e : I ≤ J) : ∃ p ∈ I.minimalPrimes, p ≤ J := by
@@ -122,17 +123,15 @@ variable {R S : Type*} [CommSemiring R] [CommSemiring S] {I J : Ideal R}
 theorem Ideal.minimalPrimes_eq_subsingleton (hI : I.IsPrimary) : I.minimalPrimes = {I.radical} := by
   ext J
   constructor
-  · exact fun H =>
-      let e := (Ideal.IsPrime.of_mem_minimalPrimes H).radical_le_iff.mpr
-        (Ideal.le_of_mem_minimalPrimes H)
-      (H.2 ⟨Ideal.isPrime_radical hI, Ideal.le_radical⟩ e).antisymm e
+  · intro H
+    have le := H.out.isPrime.radical_le_iff.mpr H.le
+    exact (H.2 ⟨Ideal.isPrime_radical hI, Ideal.le_radical⟩ le).antisymm le
   · rintro (rfl : J = I.radical)
     exact ⟨⟨Ideal.isPrime_radical hI, Ideal.le_radical⟩, fun _ H _ => H.1.radical_le_iff.mpr H.2⟩
 
 theorem Ideal.minimalPrimes_eq_subsingleton_self [I.IsPrime] : I.minimalPrimes = {I} := by
   ext J
-  refine ⟨fun H => (H.2 ⟨inferInstance, rfl.le⟩ (Ideal.le_of_mem_minimalPrimes H)).antisymm
-    (Ideal.le_of_mem_minimalPrimes H), ?_⟩
+  refine ⟨fun H => (H.2 ⟨inferInstance, rfl.le⟩ H.le).antisymm H.le, ?_⟩
   rintro (rfl : J = I)
   exact ⟨⟨inferInstance, rfl.le⟩, fun _ h _ => h.2⟩
 
@@ -151,8 +150,7 @@ theorem Ideal.minimalPrimes_top : (⊤ : Ideal R).minimalPrimes = ∅ := by
   ext p
   simp only [Set.notMem_empty, iff_false]
   intro h
-  exact (Ideal.IsPrime.of_mem_minimalPrimes h).ne_top
-    (top_le_iff.mp (Ideal.le_of_mem_minimalPrimes h))
+  exact h.isPrime.ne_top (top_le_iff.mp h.le)
 
 theorem Ideal.minimalPrimes_eq_empty_iff (I : Ideal R) :
     I.minimalPrimes = ∅ ↔ I = ⊤ := by
@@ -172,7 +170,7 @@ lemma Ideal.mem_minimalPrimes_sup {R : Type*} [CommRing R] {p I J : Ideal R} [p.
   refine ⟨⟨‹_›, ?_⟩, fun q ⟨_, hq⟩ hqp ↦ ?_⟩
   · rw [sup_le_iff]
     refine ⟨hle, ?_⟩
-    simpa [hle] using Ideal.comap_mono (f := Ideal.Quotient.mk I) (Ideal.le_of_mem_minimalPrimes h)
+    simpa [hle] using Ideal.comap_mono (f := Ideal.Quotient.mk I) h.le
   · rw [sup_le_iff] at hq
     have h2 : p.map (Quotient.mk I) ≤ q.map (Quotient.mk I) :=
       h.2 ⟨isPrime_map_quotientMk_of_isPrime hq.1, map_mono hq.2⟩ (map_mono hqp)
@@ -192,7 +190,7 @@ lemma Ideal.map_sup_mem_minimalPrimes_of_map_quotientMk_mem_minimalPrimes
   refine ⟨⟨inferInstance, sup_le_iff.mpr ?_⟩, fun q ⟨_, hleq⟩ hqle ↦ ?_⟩
   · refine ⟨?_, hJP⟩
     rw [Ideal.map_le_iff_le_comap, ← Ideal.under_def, ← Ideal.over_def P p]
-    exact Ideal.le_of_mem_minimalPrimes hI
+    exact hI.le
   · simp only [sup_le_iff] at hleq
     have h1 : p.map (algebraMap R S) ≤ q := by
       rw [Ideal.map_le_iff_le_comap]
diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Colon.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Colon.lean
index 547575d16c3f12..d7db15b1f87253 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Colon.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Colon.lean
@@ -48,8 +48,7 @@ theorem exists_eq_colon_of_mem_minimalPrimes [IsNoetherianRing R]
     refine ⟨n, hn0, ((h.toFinset.erase I).inf id) ^ n, hn, ?_⟩
     have (K : Ideal R) (hKI : K ≤ I) (hK : K ∈ ann.minimalPrimes) : K = I :=
       le_antisymm hKI (hI.2 hK.1 hKI)
-    simpa [(Ideal.IsPrime.of_mem_minimalPrimes hI).pow_le_iff hn0,
-      (Ideal.IsPrime.of_mem_minimalPrimes hI).inf_le', imp_not_comm, not_imp_not]
+    simpa [hI.isPrime.pow_le_iff hn0, hI.isPrime.inf_le', imp_not_comm, not_imp_not]
   obtain ⟨hn0, J, hJ, hJI⟩ := Nat.find_spec key
   -- let `n` be minimal such that there exists an ideal `J` with `I ^ n * J ≤ ann` and `¬ J ≠ I`
   set n := Nat.find key
@@ -75,7 +74,7 @@ theorem exists_eq_colon_of_mem_minimalPrimes [IsNoetherianRing R]
   -- let `z` be the product of these finitely many `f y`'s
   let z := ∏ y ∈ s, f y
   -- then `z ∉ I`
-  have hz : z ∉ I := by simp [z, (Ideal.IsPrime.of_mem_minimalPrimes hI).prod_mem_iff, h]
+  have hz : z ∉ I := by simp [z, hI.isPrime.prod_mem_iff, h]
   -- and `K ≤ colon N {z • x}`
   have hz' : K ≤ colon N {z • x} := by
     rw [← (map_injective_of_injective K.subtype_injective).eq_iff, map_subtype_top] at hs
@@ -91,12 +90,11 @@ theorem exists_eq_colon_of_mem_minimalPrimes [IsNoetherianRing R]
     simpa only [ann, mem_colon_singleton, mul_comm, mul_smul] using hz' hi
   -- but now `K = I ^ (n - 1) * J` contradicts the minimality of `n`
   have hK : I ^ (n - 1) * (J * Ideal.span {z}) ≤ ann ∧ ¬ J * Ideal.span {z} ≤ I := by
-    rw [← mul_assoc, (Ideal.IsPrime.of_mem_minimalPrimes hI).mul_le, not_or,
-      Ideal.span_singleton_le_iff_mem]
+    rw [← mul_assoc, hI.isPrime.mul_le, not_or, Ideal.span_singleton_le_iff_mem]
     exact ⟨hz', hJI, hz⟩
   by_cases hn' : n - 1 = 0
   · simp [K, show n = 1 by grind] at hz'
-    exact (hK.2 (hz'.trans (Ideal.le_of_mem_minimalPrimes hI))).elim
+    exact (hK.2 (hz'.trans hI.le)).elim
   · grind [Nat.find_min' key ⟨hn', J * Ideal.span {z}, hK⟩]
 
 end Submodule
diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean
index 17c15f1cd8f1ca..0e24ee5ad176e0 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean
@@ -65,9 +65,8 @@ theorem Ideal.iUnion_minimalPrimes :
   · rintro ⟨y, hy, hx⟩
     obtain ⟨p, hp, hyp⟩ : ∃ p ∈ I.minimalPrimes, y ∉ p := by
       simpa [← Ideal.sInf_minimalPrimes] using hy
-    refine ⟨p, hp, ((Ideal.IsPrime.of_mem_minimalPrimes hp).mem_or_mem ?_).resolve_right hyp⟩
-    exact (Ideal.IsPrime.of_mem_minimalPrimes hp).radical_le_iff.mpr
-      (Ideal.le_of_mem_minimalPrimes hp) hx
+    refine ⟨p, hp, (hp.isPrime.mem_or_mem ?_).resolve_right hyp⟩
+    exact hp.isPrime.radical_le_iff.mpr hp.le hx
 
 theorem Ideal.exists_mul_mem_of_mem_minimalPrimes
     {p : Ideal R} (hp : p ∈ I.minimalPrimes) {x : R} (hx : x ∈ p) :
@@ -98,8 +97,8 @@ lemma Ideal.disjoint_nonZeroDivisors_of_mem_minimalPrimes {p : Ideal R} (hp : p
 
 theorem Ideal.exists_comap_eq_of_mem_minimalPrimes {I : Ideal S} (f : R →+* S) (p)
     (H : p ∈ (I.comap f).minimalPrimes) : ∃ p' : Ideal S, p'.IsPrime ∧ I ≤ p' ∧ p'.comap f = p :=
-  have := Ideal.IsPrime.of_mem_minimalPrimes H
-  have ⟨p', hIp', hp', le⟩ := exists_ideal_comap_le_prime p I (Ideal.le_of_mem_minimalPrimes H)
+  have := H.isPrime
+  have ⟨p', hIp', hp', le⟩ := exists_ideal_comap_le_prime p I H.le
   ⟨p', hp', hIp', le.antisymm (H.2 ⟨inferInstance, comap_mono hIp'⟩ le)⟩
 
 @[stacks 00FK] theorem Ideal.exists_comap_eq_of_mem_minimalPrimes_of_injective {f : R →+* S}
@@ -114,10 +113,9 @@ theorem Ideal.exists_minimalPrimes_comap_eq {I : Ideal S} (f : R →+* S) (p)
   obtain ⟨p', h₁, h₂, h₃⟩ := Ideal.exists_comap_eq_of_mem_minimalPrimes f p H
   obtain ⟨q, hq, hq'⟩ := Ideal.exists_minimalPrimes_le h₂
   refine ⟨q, hq, Eq.symm ?_⟩
-  have := Ideal.IsPrime.of_mem_minimalPrimes hq
+  have := hq.isPrime
   have := (Ideal.comap_mono hq').trans_eq h₃
-  exact (H.2 ⟨inferInstance, Ideal.comap_mono (Ideal.le_of_mem_minimalPrimes hq)⟩ this).antisymm
-    this
+  exact (H.2 ⟨inferInstance, Ideal.comap_mono hq.le⟩ this).antisymm this
 
 theorem Ideal.minimalPrimes_comap_subset (f : R →+* S) (J : Ideal S) :
     (J.comap f).minimalPrimes ⊆ Ideal.comap f '' J.minimalPrimes :=
@@ -131,8 +129,8 @@ variable {R S : Type*} [CommRing R] [CommRing S] {I J : Ideal R}
 
 theorem Ideal.minimalPrimes_comap_of_surjective {f : R →+* S} (hf : Function.Surjective f)
     {I J : Ideal S} (h : J ∈ I.minimalPrimes) : J.comap f ∈ (I.comap f).minimalPrimes := by
-  have := Ideal.IsPrime.of_mem_minimalPrimes h
-  refine ⟨⟨inferInstance, Ideal.comap_mono (Ideal.le_of_mem_minimalPrimes h)⟩, ?_⟩
+  have := h.isPrime
+  refine ⟨⟨inferInstance, Ideal.comap_mono h.le⟩, ?_⟩
   rintro K ⟨hK, e₁⟩ e₂
   have : RingHom.ker f ≤ K := (Ideal.comap_mono bot_le).trans e₁
   rw [← sup_eq_left.mpr this, RingHom.ker_eq_comap_bot, ← Ideal.comap_map_of_surjective f hf]
@@ -161,8 +159,7 @@ lemma Ideal.minimalPrimes_map_of_surjective {S : Type*} [CommRing S] {f : R →+
   rw [← Ideal.comap_minimalPrimes_eq_of_surjective hf, ← Set.image_comp,
     Ideal.comap_map_of_surjective f hf, Set.image_congr, Set.image_id, RingHom.ker]
   intro x hx
-  exact (Ideal.comap_map_of_surjective f hf _).trans
-    (sup_eq_left.mpr <| le_sup_right.trans (Ideal.le_of_mem_minimalPrimes hx))
+  exact (Ideal.comap_map_of_surjective f hf _).trans (sup_eq_left.mpr <| le_sup_right.trans hx.le)
 
 theorem Ideal.minimalPrimes_eq_comap :
     I.minimalPrimes = Ideal.comap (Ideal.Quotient.mk I) '' minimalPrimes (R ⧸ I) := by
@@ -180,27 +177,24 @@ theorem IsLocalization.minimalPrimes_map [IsLocalization S A] (J : Ideal R) :
   ext p
   constructor
   · intro hp
-    haveI := Ideal.IsPrime.of_mem_minimalPrimes hp
-    refine ⟨⟨Ideal.IsPrime.comap _,
-      Ideal.map_le_iff_le_comap.mp (Ideal.le_of_mem_minimalPrimes hp)⟩, ?_⟩
+    have := hp.isPrime
+    refine ⟨⟨Ideal.IsPrime.comap _, Ideal.map_le_iff_le_comap.mp hp.le⟩, ?_⟩
     rintro I hI e
     have hI' : Disjoint (S : Set R) I := Set.disjoint_of_subset_right e
-      ((IsLocalization.isPrime_iff_isPrime_disjoint S A _).mp
-        (Ideal.IsPrime.of_mem_minimalPrimes hp)).2
+      ((IsLocalization.isPrime_iff_isPrime_disjoint S A _).mp hp.isPrime).2
     refine (Ideal.comap_mono <|
       hp.2 ⟨?_, Ideal.map_mono hI.2⟩ (Ideal.map_le_iff_le_comap.mpr e)).trans_eq ?_
     · exact IsLocalization.isPrime_of_isPrime_disjoint S A I hI.1 hI'
     · exact IsLocalization.comap_map_of_isPrime_disjoint S A hI.1 hI'
   · intro hp
-    refine ⟨⟨?_, Ideal.map_le_iff_le_comap.mpr (Ideal.le_of_mem_minimalPrimes hp)⟩, ?_⟩
+    refine ⟨⟨?_, Ideal.map_le_iff_le_comap.mpr hp.le⟩, ?_⟩
     · rw [IsLocalization.isPrime_iff_isPrime_disjoint S A, IsLocalization.disjoint_comap_iff S]
-      refine ⟨Ideal.IsPrime.of_mem_minimalPrimes hp, ?_⟩
+      refine ⟨hp.isPrime, ?_⟩
       rintro rfl
-      exact (Ideal.IsPrime.of_mem_minimalPrimes hp).ne_top rfl
+      exact hp.isPrime.ne_top rfl
     · intro I hI e
       rw [← IsLocalization.map_comap S A I, ← IsLocalization.map_comap S A p]
-      haveI := hI.1
-      exact Ideal.map_mono (hp.2 ⟨Ideal.IsPrime.comap _, Ideal.map_le_iff_le_comap.mp hI.2⟩
+      exact Ideal.map_mono (hp.2 ⟨hI.1.comap _, Ideal.map_le_iff_le_comap.mp hI.2⟩
         (Ideal.comap_mono e))
 
 theorem IsLocalization.minimalPrimes_comap [IsLocalization S A] (J : Ideal A) :
@@ -220,7 +214,7 @@ theorem IsLocalization.AtPrime.radical_map_of_mem_minimalPrimes
   · rw [← IsLocalization.comap_le_comap_iff q.primeCompl A,
       AtPrime.map_eq_maximalIdeal q A, AtPrime.comap_maximalIdeal A q]
     apply hIq.2 hJ.1
-    have := (Ideal.IsPrime.of_mem_minimalPrimes hJ).ne_top
+    have := hJ.isPrime.ne_top
     rw [ne_eq, Ideal.comap_eq_top_iff, ← ne_eq, ← disjoint_comap_iff q.primeCompl A J] at this
     exact Set.disjoint_compl_left_iff_subset.mp this
 
diff --git a/Mathlib/RingTheory/KrullDimension/Zero.lean b/Mathlib/RingTheory/KrullDimension/Zero.lean
index dd5e85a37710ff..09dac318663204 100644
--- a/Mathlib/RingTheory/KrullDimension/Zero.lean
+++ b/Mathlib/RingTheory/KrullDimension/Zero.lean
@@ -34,7 +34,9 @@ lemma Ring.KrullDimLE.mem_minimalPrimes_iff_le_of_isPrime {I J : Ideal R} [I.IsP
 variable (R) in
 lemma Ring.KrullDimLE.minimalPrimes_eq_setOf_isPrime :
     minimalPrimes R = { I | I.IsPrime } := by
-  ext; simp [minimalPrimes, mem_minimalPrimes_iff]
+  ext
+  rw [mem_minimalPrimes_iff]
+  simp
 
 variable (R) in
 lemma Ring.KrullDimLE.minimalPrimes_eq_setOf_isMaximal :
diff --git a/Mathlib/RingTheory/Spectrum/Prime/Topology.lean b/Mathlib/RingTheory/Spectrum/Prime/Topology.lean
index 3b956aa9811add..b8b39eaaea65e2 100644
--- a/Mathlib/RingTheory/Spectrum/Prime/Topology.lean
+++ b/Mathlib/RingTheory/Spectrum/Prime/Topology.lean
@@ -958,7 +958,7 @@ lemma denseRange_comap_iff_minimalPrimes :
   · intro H I hI
     have : I ∈ (RingHom.ker f).minimalPrimes := by
       rw [denseRange_comap_iff_ker_le_nilRadical] at H
-      simp only [minimalPrimes, Ideal.minimalPrimes, Set.mem_setOf] at hI ⊢
+      simp only [Set.mem_setOf, Ideal.IsMinimalPrime] at hI ⊢
       convert hI using 2 with p
       exact ⟨fun h ↦ ⟨h.1, bot_le⟩, fun h ↦ ⟨h.1, H.trans (h.1.radical_le_iff.mpr bot_le)⟩⟩
     obtain ⟨p, hp, _, rfl⟩ := Ideal.exists_comap_eq_of_mem_minimalPrimes f (I := ⊥) I this

From b7ad058bedcfb99078b5cb38333ac8c0569c9888 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Fri, 8 May 2026 11:53:08 +0800
Subject: [PATCH 42/69] fix

---
 Mathlib/RingTheory/Ideal/Height.lean | 32 ++++++++++++++--------------
 1 file changed, 16 insertions(+), 16 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index b6fa067a4f0448..3884bc3955c77d 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -52,7 +52,7 @@ lemma Ideal.height_eq_order_height_of_isPrime [I.IsPrime] : I.height =
 
 lemma Ideal.height_eq_inf_minimalPrimes : I.height = ⨅ J ∈ I.minimalPrimes, J.height := by
   apply iInf_congr (fun p ↦ iInf_congr fun hp ↦ ?_)
-  have := Ideal.IsPrime.of_mem_minimalPrimes hp
+  have := hp.isPrime
   exact (Ideal.height_eq_primeHeight _).symm
 
 /-- An ideal has finite height if it is either the unit ideal or its height is finite.
@@ -132,9 +132,9 @@ theorem Ideal.height_top : (⊤ : Ideal R).height = ⊤ := by
 theorem Ideal.height_mono {I J : Ideal R} (h : I ≤ J) : I.height ≤ J.height := by
   simp only [I.height_eq_inf_minimalPrimes, J.height_eq_inf_minimalPrimes]
   refine le_iInf₂ (fun p hp ↦ ?_)
-  have := Ideal.IsPrime.of_mem_minimalPrimes hp
-  obtain ⟨q, hq, e⟩ := Ideal.exists_minimalPrimes_le (h.trans (Ideal.le_of_mem_minimalPrimes hp))
-  have := Ideal.IsPrime.of_mem_minimalPrimes hq
+  have := hp.isPrime
+  obtain ⟨q, hq, e⟩ := Ideal.exists_minimalPrimes_le (h.trans hp.le)
+  have := hq.isPrime
   exact (iInf₂_le q hq).trans (Ideal.height_mono_of_isPrime e)
 
 @[gcongr]
@@ -145,8 +145,8 @@ lemma Ideal.height_strict_mono_of_isPrime {I J : Ideal R} [I.IsPrime]
     exact I.height_lt_top IsPrime.ne_top'
   · rw [← ENat.add_one_le_iff (I.height_ne_top IsPrime.ne_top'), J.height_eq_inf_minimalPrimes]
     refine le_iInf₂ (fun K hK ↦ ?_)
-    have := Ideal.IsPrime.of_mem_minimalPrimes hK
-    have : I < K := lt_of_lt_of_le h (Ideal.le_of_mem_minimalPrimes hK)
+    have := hK.isPrime
+    have : I < K := lt_of_lt_of_le h hK.le
     exact Ideal.height_add_one_le_of_lt_of_isPrime this
 
 lemma Ideal.height_strict_mono_of_isPrime_of_isPrime {I J : Ideal R} [I.IsPrime] [J.IsPrime]
@@ -183,7 +183,7 @@ lemma Ideal.height_le_ringKrullDim_of_ne_top {I : Ideal R} (h : I ≠ ⊤) :
     I.height ≤ ringKrullDim R := by
   obtain ⟨P, hP⟩ : Nonempty (I.minimalPrimes) := Ideal.nonempty_minimalPrimes h
   rw [I.height_eq_inf_minimalPrimes]
-  have := Ideal.IsPrime.of_mem_minimalPrimes hP
+  have := hP.isPrime
   refine (WithBot.coe_le_coe.mpr (iInf₂_le _ hP)).trans P.height_le_ringKrullDim_of_isPrime
 
 /-- If `R` has finite Krull dimension, there exists a maximal ideal `m` with `ht m = dim R`. -/
@@ -219,12 +219,11 @@ lemma Ideal.mem_minimalPrimes_of_height_eq {I J : Ideal R} (e : I ≤ J) [J.IsPr
   have := h₁.isPrime
   have := finiteHeight_of_le h₂ IsPrime.ne_top'
   exact lt_irrefl _ ((height_strict_mono_of_isPrime h₃).trans_le
-    (e'.trans <| height_mono (Ideal.le_of_mem_minimalPrimes h₁)))
+    (e'.trans <| height_mono h₁.le))
 
 /-- A prime ideal has height zero if and only if it is minimal -/
 lemma Ideal.height_eq_zero_iff {I : Ideal R} [I.IsPrime] : height I = 0 ↔ I ∈ minimalPrimes R := by
-  rw [Ideal.height_eq_primeHeight, Ideal.primeHeight, Order.height_eq_zero, minimalPrimes]
-  simp only [Ideal.minimalPrimes, bot_le, and_true, Set.mem_setOf_eq, Minimal, IsMin]
+  rw [Ideal.height_eq_order_height_of_isPrime, Order.height_eq_zero, minimalPrimes_eq_minimals]
   refine ⟨fun h ↦ ⟨‹_›, ?_⟩, fun ⟨hI, hI'⟩ b hb ↦ hI' b.isPrime hb⟩
   by_contra! ⟨P, ⟨hP₁, ⟨hP₂, hP₃⟩⟩⟩
   exact hP₃ (h (b := ⟨P, hP₁⟩) hP₂)
@@ -385,7 +384,7 @@ theorem IsLocalization.height_comap (S : Submonoid R) {A : Type*} [CommRing A] [
   rw [(J.comap _).height_eq_inf_minimalPrimes, J.height_eq_inf_minimalPrimes]
   simp only [IsLocalization.minimalPrimes_comap S A, iInf_image]
   apply iInf_congr (fun p ↦ iInf_congr fun hp ↦ ?_)
-  have := Ideal.IsPrime.of_mem_minimalPrimes hp
+  have := hp.isPrime
   exact IsLocalization.height_comap_eq_of_isPrime S _
 
 theorem IsLocalization.AtPrime.ringKrullDim_eq_height (I : Ideal R) [I.IsPrime] (A : Type*)
@@ -448,14 +447,15 @@ lemma exists_spanRank_le_and_le_height_of_le_height [IsNoetherianRing R] (I : Id
       by_cases h : p.height = ⊤
       · exact le_of_le_of_eq le_top h.symm
       have : p.FiniteHeight := ⟨Or.inr h⟩
-      have := Ideal.height_mono (le_sup_left.trans (Ideal.le_of_mem_minimalPrimes hp))
+      have := Ideal.height_mono (le_sup_left.trans hp.le)
       suffices h : (r : ℕ∞) ≠ p.height by
         exact Order.add_one_le_of_lt (lt_of_le_of_ne (h₃.trans this) h)
       intro e
       apply hx₂ p
-      · refine ⟨mem_minimalPrimes_of_height_eq
-          (le_sup_left.trans (Ideal.le_of_mem_minimalPrimes hp)) (le_of_eq_of_le e.symm h₃), e.symm⟩
-      · exact (Ideal.le_of_mem_minimalPrimes hp) <| Ideal.mem_sup_right <| mem_span_singleton_self x
+      · refine ⟨mem_minimalPrimes_of_height_eq (le_sup_left.trans hp.le) (e.symm.trans_le h₃),
+          e.symm⟩
+      · apply hp.le
+        exact Ideal.mem_sup_right <| mem_span_singleton_self x
 
 /-- In a nontrivial commutative ring `R`, the supremum of heights of all ideals is equal to the
 Krull dimension of `R`. -/
@@ -488,7 +488,7 @@ lemma Ideal.sup_height_isPrime_eq_ringKrullDim [Nontrivial R] :
     · exact ⟨⊥, by simp [I_top]⟩
     · obtain ⟨P, hP⟩ := Set.nonempty_coe_sort.mp (nonempty_minimalPrimes I_top)
       refine ⟨P, iSup_pos (α := ℕ∞) I_top ▸ le_iSup_of_le (hP.left.left) ?_⟩
-      have := Ideal.IsPrime.of_mem_minimalPrimes hP
+      have := hP.isPrime
       exact iInf_le_of_le P (iInf_le_of_le hP (ge_of_eq (Ideal.height_eq_primeHeight P)))
 
 @[deprecated "Use `Ideal.sup_height_isPrime_eq_ringKrullDim` instead." (since := "2026-04-02")]

From b2163caa9f91fe5818f54473b76c500bf9c22bb1 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Fri, 8 May 2026 11:54:58 +0800
Subject: [PATCH 43/69] fix

---
 Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.lean | 5 ++---
 Mathlib/RingTheory/KrullDimension/Regular.lean             | 3 +--
 2 files changed, 3 insertions(+), 5 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.lean b/Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.lean
index 6ce2cc209d03c4..a5cc113e02ede6 100644
--- a/Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.lean
+++ b/Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.lean
@@ -131,11 +131,10 @@ variable (R M) in
 lemma minimalPrimes_annihilator_subset_associatedPrimes [IsNoetherianRing R] [Module.Finite R M] :
     (Module.annihilator R M).minimalPrimes ⊆ associatedPrimes R M := by
   intro p hp
-  have prime := Ideal.IsPrime.of_mem_minimalPrimes hp
+  have prime := hp.isPrime
   let Rₚ := Localization.AtPrime p
   have : Nontrivial (LocalizedModule p.primeCompl M) := by
-    simpa [← Module.mem_support_iff (p := ⟨p, prime⟩), Module.support_eq_zeroLocus] using
-      Ideal.le_of_mem_minimalPrimes hp
+    simpa [← Module.mem_support_iff (p := ⟨p, prime⟩), Module.support_eq_zeroLocus] using hp.le
   rcases associatedPrimes.nonempty Rₚ (LocalizedModule p.primeCompl M) with ⟨q, hq⟩
   have q_prime : q.IsPrime := IsAssociatedPrime.isPrime hq
   simp only [← preimage_comap_associatedPrimes_eq_associatedPrimes_of_isLocalizedModule p.primeCompl
diff --git a/Mathlib/RingTheory/KrullDimension/Regular.lean b/Mathlib/RingTheory/KrullDimension/Regular.lean
index d57c56ed221acd..389bb3119f3f4a 100644
--- a/Mathlib/RingTheory/KrullDimension/Regular.lean
+++ b/Mathlib/RingTheory/KrullDimension/Regular.lean
@@ -95,8 +95,7 @@ theorem supportDim_quotSMulTop_succ_le_of_notMem_minimalPrimes {x : R}
   let q : LTSeries (support R M) :=
     p.map (Set.MapsTo.restrict id (support R (QuotSMulTop x M)) (support R M) le) (fun _ _ h ↦ h)
   obtain ⟨r, hrm, hr⟩ := exists_minimalPrimes_le (mem_support_iff_of_finite.mp q.head.2)
-  let r : support R M := ⟨⟨r, IsPrime.of_mem_minimalPrimes hrm⟩,
-    mem_support_iff_of_finite.mpr hrm.1.2⟩
+  let r : support R M := ⟨⟨r, hrm.isPrime⟩, mem_support_iff_of_finite.mpr hrm.1.2⟩
   have hr : r < q.head := lt_of_le_of_ne hr (fun h ↦ hn q.head.1.1 (by rwa [← h]) hp.2)
   exact le_of_eq_of_le (by simp [q]) (le_iSup _ (q.cons r hr))
 

From 8535ebb694ebed8cca9be7e2f468a990c81f56e0 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Fri, 8 May 2026 12:31:40 +0800
Subject: [PATCH 44/69] add doc

---
 Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean | 1 +
 1 file changed, 1 insertion(+)

diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
index 71da00fef48909..38dc8a47448940 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
@@ -36,6 +36,7 @@ section
 
 variable {R S : Type*} [CommSemiring R] [CommSemiring S] (I J : Ideal R)
 
+/-- The predicate for describing an ideal is minimal prime over certain ideal `I`. -/
 protected def Ideal.IsMinimalPrime (p : Ideal R) : Prop := Minimal (fun q ↦ q.IsPrime ∧ I ≤ q) p
 
 lemma Ideal.IsMinimalPrime.isPrime {p : Ideal R} (h : I.IsMinimalPrime p) : p.IsPrime := h.1.1

From a594879c62a28b75e05d654a5bfb0b851c6dc49f Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Fri, 8 May 2026 12:34:34 +0800
Subject: [PATCH 45/69] add deprecation

---
 Mathlib/RingTheory/Ideal/Height.lean             | 2 +-
 Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean | 7 +++++--
 2 files changed, 6 insertions(+), 3 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index 74ea36cc589984..b4df06a82cc35c 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -204,7 +204,7 @@ lemma Ideal.height_bot [Nontrivial R] : (⊥ : Ideal R).height = 0 := by
   obtain ⟨p, hp⟩ := Ideal.nonempty_minimalPrimes (R := R) (I := ⊥) top_ne_bot.symm
   simp only [Ideal.height, ENat.iInf_eq_zero]
   refine ⟨p, hp, ?_⟩
-  have := Ideal.IsPrime.of_mem_minimalPrimes hp
+  have := Ideal.IsPrime.of_mem_minimalPrimes' hp
   rw [primeHeight_eq_zero_iff.mpr hp]
 
 /-- In a trivial commutative ring, the height of any ideal is `∞`. -/
diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
index 38dc8a47448940..b2513268de7475 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
@@ -58,8 +58,11 @@ lemma minimalPrimes_eq_minimals : minimalPrimes R = {x | Minimal Ideal.IsPrime x
 
 variable {I J}
 
-theorem Ideal.IsPrime.of_mem_minimalPrimes {p : Ideal R} (h : p ∈ minimalPrimes R) :
-    p.IsPrime :=
+@[deprecated "Use `Ideal.IsMinimalPrime.isPrime` instead." (since := "2026-05-08")]
+theorem Ideal.IsPrime.of_mem_minimalPrimes {p : Ideal R} (h : p ∈ minimalPrimes R) : p.IsPrime :=
+  h.1.1
+
+theorem Ideal.IsPrime.of_mem_minimalPrimes' {p : Ideal R} (h : p ∈ minimalPrimes R) : p.IsPrime :=
   h.1.1
 
 theorem Ideal.exists_minimalPrimes_le [J.IsPrime] (e : I ≤ J) : ∃ p ∈ I.minimalPrimes, p ≤ J := by

From 3300967b327394d52e0a5ab55bff5f5e5dbff2d0 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Fri, 8 May 2026 12:36:42 +0800
Subject: [PATCH 46/69] add doc

---
 Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean | 1 +
 1 file changed, 1 insertion(+)

diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
index b2513268de7475..01fda98a1c78a5 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
@@ -62,6 +62,7 @@ variable {I J}
 theorem Ideal.IsPrime.of_mem_minimalPrimes {p : Ideal R} (h : p ∈ minimalPrimes R) : p.IsPrime :=
   h.1.1
 
+/-- Variant of `Ideal.IsMinimalPrime.isPrime` for the set of all minimal primes of `R`. -/
 theorem Ideal.IsPrime.of_mem_minimalPrimes' {p : Ideal R} (h : p ∈ minimalPrimes R) : p.IsPrime :=
   h.1.1
 

From 8babb5d47ce4b5c5d7b614a1ceafb7bf7e2700e3 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Tue, 12 May 2026 19:17:58 +0800
Subject: [PATCH 47/69] fix

---
 Mathlib/RingTheory/Ideal/Height.lean | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index 0c8eba31adff54..c7927db78b7fc0 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -204,7 +204,7 @@ lemma Ideal.one_le_height_span_singleton_of_mem_nonZeroDivisors
     {x : R} (hx : x ∈ nonZeroDivisors R) : 1 ≤ (span {x}).height := by
   dsimp [Ideal.height]
   refine le_iInf₂ fun q hq => ?_
-  have : q.IsPrime := minimalPrimes_isPrime hq
+  have : q.IsPrime := hq.isPrime
   rw [ENat.one_le_iff_ne_zero, Ne, primeHeight_eq_zero_iff]
   intro hmin
   exact absurd hx <| notMem_nonZeroDivisors_of_mem_mem_minimalPrimes

From 421559a2f4d1a574bbd9e3edbe427df1cc63cb22 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Tue, 12 May 2026 20:25:05 +0800
Subject: [PATCH 48/69] fix

---
 Mathlib/RingTheory/Ideal/Height.lean | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index fbb4dcc5f32a9a..ae7af5328b7489 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -236,10 +236,10 @@ private lemma Ideal.primeHeight_eq_zero_iff {I : Ideal R} [I.IsPrime] :
 /-- If `x` is a non-zero-divisor, then `span {x}` has height at least 1. -/
 lemma Ideal.one_le_height_span_singleton_of_mem_nonZeroDivisors
     {x : R} (hx : x ∈ nonZeroDivisors R) : 1 ≤ (span {x}).height := by
-  dsimp [Ideal.height]
+  rw [Ideal.height_eq_inf_minimalPrimes]
   refine le_iInf₂ fun q hq => ?_
   have : q.IsPrime := hq.isPrime
-  rw [ENat.one_le_iff_ne_zero, Ne, primeHeight_eq_zero_iff]
+  rw [ENat.one_le_iff_ne_zero, Ne, height_eq_zero_iff]
   intro hmin
   exact absurd hx <| notMem_nonZeroDivisors_of_mem_mem_minimalPrimes
     (hq.1.2 <| Ideal.mem_span_singleton.mpr <| dvd_refl x) hmin

From 0d7c0df9ed5af5a9634a645903b5265a4f06b322 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Tue, 12 May 2026 22:33:28 +0800
Subject: [PATCH 49/69] fix deprecation

---
 Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
index 01fda98a1c78a5..3b43ccc3fbd9af 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
@@ -59,11 +59,11 @@ lemma minimalPrimes_eq_minimals : minimalPrimes R = {x | Minimal Ideal.IsPrime x
 variable {I J}
 
 @[deprecated "Use `Ideal.IsMinimalPrime.isPrime` instead." (since := "2026-05-08")]
-theorem Ideal.IsPrime.of_mem_minimalPrimes {p : Ideal R} (h : p ∈ minimalPrimes R) : p.IsPrime :=
+theorem Ideal.minimalPrimes_isPrime {p : Ideal R} (h : p ∈ minimalPrimes R) : p.IsPrime :=
   h.1.1
 
 /-- Variant of `Ideal.IsMinimalPrime.isPrime` for the set of all minimal primes of `R`. -/
-theorem Ideal.IsPrime.of_mem_minimalPrimes' {p : Ideal R} (h : p ∈ minimalPrimes R) : p.IsPrime :=
+theorem Ideal.IsPrime.of_mem_minimalPrimes {p : Ideal R} (h : p ∈ minimalPrimes R) : p.IsPrime :=
   h.1.1
 
 theorem Ideal.exists_minimalPrimes_le [J.IsPrime] (e : I ≤ J) : ∃ p ∈ I.minimalPrimes, p ≤ J := by

From cb6481677045f0350919c42f643e3ee20ef87a4b Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Tue, 12 May 2026 22:50:49 +0800
Subject: [PATCH 50/69] adapt to changes in under

---
 Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean
index 39dff232b8beb1..0815cca79f45dc 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean
@@ -194,12 +194,12 @@ theorem IsLocalization.minimalPrimes_map [IsLocalization S A] (J : Ideal R) :
     · exact IsLocalization.under_map_of_isPrime_disjoint S A hI.1 hI'
   · intro hp
     refine ⟨⟨?_, Ideal.map_le_iff_le_comap.mpr hp.le⟩, ?_⟩
-    · rw [IsLocalization.isPrime_iff_isPrime_disjoint S A, IsLocalization.disjoint_comap_iff S]
+    · rw [IsLocalization.isPrime_iff_isPrime_disjoint S A, IsLocalization.disjoint_under_iff S]
       refine ⟨hp.isPrime, ?_⟩
       rintro rfl
       exact hp.isPrime.ne_top rfl
     · intro I hI e
-      rw [← IsLocalization.map_comap S A I, ← IsLocalization.map_comap S A p]
+      rw [← IsLocalization.map_under S A I, ← IsLocalization.map_under S A p]
       exact Ideal.map_mono (hp.2 ⟨hI.1.comap _, Ideal.map_le_iff_le_comap.mp hI.2⟩
         (Ideal.comap_mono e))
 
@@ -221,7 +221,7 @@ theorem IsLocalization.AtPrime.radical_map_of_mem_minimalPrimes
       AtPrime.map_eq_maximalIdeal q A, AtPrime.under_maximalIdeal A q]
     apply hIq.2 hJ.1
     have := hJ.isPrime.ne_top
-    rw [ne_eq, Ideal.comap_eq_top_iff, ← ne_eq, ← disjoint_comap_iff q.primeCompl A J] at this
+    rw [ne_eq, Ideal.comap_eq_top_iff, ← ne_eq, ← disjoint_under_iff q.primeCompl A J] at this
     exact Set.disjoint_compl_left_iff_subset.mp this
 
 end

From e2c53ab4f6c106d071b6484987880f158c668cbf Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Tue, 12 May 2026 22:55:19 +0800
Subject: [PATCH 51/69] fix naming

---
 Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean | 10 +++++-----
 1 file changed, 5 insertions(+), 5 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
index 3b43ccc3fbd9af..35f8173fffb045 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
@@ -36,16 +36,16 @@ section
 
 variable {R S : Type*} [CommSemiring R] [CommSemiring S] (I J : Ideal R)
 
-/-- The predicate for describing an ideal is minimal prime over certain ideal `I`. -/
-protected def Ideal.IsMinimalPrime (p : Ideal R) : Prop := Minimal (fun q ↦ q.IsPrime ∧ I ≤ q) p
+/-- `IsMinimalPrime I p` says that `p` is a minimal prime over `I`. -/
+def _root_.IsMinimalPrime (p : Ideal R) : Prop := Minimal (fun q ↦ q.IsPrime ∧ I ≤ q) p
 
-lemma Ideal.IsMinimalPrime.isPrime {p : Ideal R} (h : I.IsMinimalPrime p) : p.IsPrime := h.1.1
+lemma IsMinimalPrime.isPrime {p : Ideal R} (h : IsMinimalPrime I p) : p.IsPrime := h.1.1
 
-lemma Ideal.IsMinimalPrime.le {p : Ideal R} (h : I.IsMinimalPrime p) : I ≤ p := h.1.2
+lemma IsMinimalPrime.le {p : Ideal R} (h : IsMinimalPrime I p) : I ≤ p := h.1.2
 
 /-- `I.minimalPrimes` is the set of ideals that are minimal primes over `I`. -/
 protected abbrev Ideal.minimalPrimes : Set (Ideal R) :=
-  {p | I.IsMinimalPrime p}
+  {p | IsMinimalPrime I p}
 
 variable (R) in
 /-- `minimalPrimes R` is the set of minimal primes of `R`.

From 6ac19f6121dc655ce060a7b9ec0812eca886d625 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Tue, 12 May 2026 23:11:18 +0800
Subject: [PATCH 52/69] fix

---
 Mathlib/RingTheory/Ideal/Height.lean            | 2 +-
 Mathlib/RingTheory/Spectrum/Prime/Topology.lean | 2 +-
 2 files changed, 2 insertions(+), 2 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index 8936db344d4d88..295d7761f4a4cb 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -215,7 +215,7 @@ lemma Ideal.height_bot [Nontrivial R] : (⊥ : Ideal R).height = 0 := by
   obtain ⟨p, hp⟩ := Ideal.nonempty_minimalPrimes (R := R) (I := ⊥) top_ne_bot.symm
   simp only [Ideal.height, ENat.iInf_eq_zero]
   refine ⟨p, hp, ?_⟩
-  have := Ideal.IsPrime.of_mem_minimalPrimes' hp
+  have := Ideal.IsPrime.of_mem_minimalPrimes hp
   rw [primeHeight_eq_zero_iff.mpr hp]
 
 /-- In a trivial commutative ring, the height of any ideal is `∞`. -/
diff --git a/Mathlib/RingTheory/Spectrum/Prime/Topology.lean b/Mathlib/RingTheory/Spectrum/Prime/Topology.lean
index 49a274abcc0296..3028bf73db9325 100644
--- a/Mathlib/RingTheory/Spectrum/Prime/Topology.lean
+++ b/Mathlib/RingTheory/Spectrum/Prime/Topology.lean
@@ -958,7 +958,7 @@ lemma denseRange_comap_iff_minimalPrimes :
   · intro H I hI
     have : I ∈ (RingHom.ker f).minimalPrimes := by
       rw [denseRange_comap_iff_ker_le_nilRadical] at H
-      simp only [Set.mem_setOf, Ideal.IsMinimalPrime] at hI ⊢
+      simp only [Set.mem_setOf, IsMinimalPrime] at hI ⊢
       convert hI using 2 with p
       exact ⟨fun h ↦ ⟨h.1, bot_le⟩, fun h ↦ ⟨h.1, H.trans (h.1.radical_le_iff.mpr bot_le)⟩⟩
     obtain ⟨p, hp, _, rfl⟩ := Ideal.exists_comap_eq_of_mem_minimalPrimes f (I := ⊥) I this

From 5354bc478ae2e5bf7b9d1d459aab24a1bcb8084f Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Tue, 12 May 2026 23:35:50 +0800
Subject: [PATCH 53/69] fix

---
 Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
index a5d97348bcdd98..a240f420f3323a 100644
--- a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
+++ b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
@@ -109,7 +109,7 @@ lemma Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes
   have := hp.isPrime
   let f := algebraMap R (Localization.AtPrime p)
   have := Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes_of_isLocalRing (I.map f) ?_
-  · rwa [← IsLocalization.height_under p.primeCompl,
+  · rwa [← IsLocalization.height_under p.primeCompl, ← Ideal.under_def,
       Localization.AtPrime.under_maximalIdeal] at this
   · rwa [IsLocalization.minimalPrimes_map p.primeCompl (Localization.AtPrime p) I,
       Set.mem_preimage, Localization.AtPrime.under_maximalIdeal]

From 4dcf91f03ee441eeab3ebfb7cde82f4334853f02 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Wed, 13 May 2026 00:07:10 +0800
Subject: [PATCH 54/69] refactor name

---
 .../RingTheory/Ideal/MinimalPrime/Basic.lean  | 19 ++++++++++++++-----
 .../RingTheory/Spectrum/Prime/Topology.lean   |  2 +-
 2 files changed, 15 insertions(+), 6 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
index 35f8173fffb045..0393b187d7d7f9 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
@@ -37,21 +37,30 @@ section
 variable {R S : Type*} [CommSemiring R] [CommSemiring S] (I J : Ideal R)
 
 /-- `IsMinimalPrime I p` says that `p` is a minimal prime over `I`. -/
-def _root_.IsMinimalPrime (p : Ideal R) : Prop := Minimal (fun q ↦ q.IsPrime ∧ I ≤ q) p
+protected def Ideal.IsMinimalPrime (p : Ideal R) : Prop := Minimal (fun q ↦ q.IsPrime ∧ I ≤ q) p
 
-lemma IsMinimalPrime.isPrime {p : Ideal R} (h : IsMinimalPrime I p) : p.IsPrime := h.1.1
+variable {I} in
+lemma Ideal.IsMinimalPrime.isPrime {p : Ideal R} (h : I.IsMinimalPrime p) : p.IsPrime := h.1.1
 
-lemma IsMinimalPrime.le {p : Ideal R} (h : IsMinimalPrime I p) : I ≤ p := h.1.2
+variable {I} in
+lemma Ideal.IsMinimalPrime.le {p : Ideal R} (h : I.IsMinimalPrime p) : I ≤ p := h.1.2
+
+abbrev IsMinimalPrime (p : Ideal R) : Prop := (⊥ : Ideal R).IsMinimalPrime p
+
+lemma IsMinimalPrime.isPrime {p : Ideal R} (h : IsMinimalPrime p) : p.IsPrime := h.1.1
+
+lemma IsMinimalPrime.iff_minimal (p : Ideal R) : IsMinimalPrime p ↔ Minimal Ideal.IsPrime p := by
+  simp [Ideal.IsMinimalPrime]
 
 /-- `I.minimalPrimes` is the set of ideals that are minimal primes over `I`. -/
 protected abbrev Ideal.minimalPrimes : Set (Ideal R) :=
-  {p | IsMinimalPrime I p}
+  {p | I.IsMinimalPrime p}
 
 variable (R) in
 /-- `minimalPrimes R` is the set of minimal primes of `R`.
 This is defined as `Ideal.minimalPrimes ⊥`. -/
 abbrev minimalPrimes : Set (Ideal R) :=
-  Ideal.minimalPrimes ⊥
+  {p | IsMinimalPrime p}
 
 lemma minimalPrimes_eq_minimals : minimalPrimes R = {x | Minimal Ideal.IsPrime x} :=
   congr_arg Minimal (by simp)
diff --git a/Mathlib/RingTheory/Spectrum/Prime/Topology.lean b/Mathlib/RingTheory/Spectrum/Prime/Topology.lean
index 3028bf73db9325..49a274abcc0296 100644
--- a/Mathlib/RingTheory/Spectrum/Prime/Topology.lean
+++ b/Mathlib/RingTheory/Spectrum/Prime/Topology.lean
@@ -958,7 +958,7 @@ lemma denseRange_comap_iff_minimalPrimes :
   · intro H I hI
     have : I ∈ (RingHom.ker f).minimalPrimes := by
       rw [denseRange_comap_iff_ker_le_nilRadical] at H
-      simp only [Set.mem_setOf, IsMinimalPrime] at hI ⊢
+      simp only [Set.mem_setOf, Ideal.IsMinimalPrime] at hI ⊢
       convert hI using 2 with p
       exact ⟨fun h ↦ ⟨h.1, bot_le⟩, fun h ↦ ⟨h.1, H.trans (h.1.radical_le_iff.mpr bot_le)⟩⟩
     obtain ⟨p, hp, _, rfl⟩ := Ideal.exists_comap_eq_of_mem_minimalPrimes f (I := ⊥) I this

From 32775b4e69dcec47c814214230ddaafc3234b26e Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Wed, 13 May 2026 00:07:21 +0800
Subject: [PATCH 55/69] golf

---
 Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean | 6 ++----
 1 file changed, 2 insertions(+), 4 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
index dc12af9d5751eb..30a166eceeb779 100644
--- a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
+++ b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
@@ -338,8 +338,7 @@ lemma Ideal.height_le_height_add_spanFinrank_of_le {I p : Ideal R} [p.IsPrime] (
   let p' := p.map (algebraMap R (R ⧸ I))
   have : p'.IsPrime := isPrime_map_quotientMk_of_isPrime hrp
   obtain ⟨s, hps, hs⟩ := exists_finset_card_eq_height_of_isNoetherianRing p'
-  have lep' := hps.le
-  have hsp' : (s : Set (R ⧸ I)) ⊆ (p' : Set _) := fun _ hx ↦ lep' (subset_span hx)
+  have hsp' : (s : Set (R ⧸ I)) ⊆ (p' : Set _) := fun _ hx ↦ hps.le (subset_span hx)
   have : Set.SurjOn (Ideal.Quotient.mk I) p s := by
     refine Set.SurjOn.mono subset_rfl hsp' fun x hx ↦ ?_
     obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x
@@ -435,9 +434,8 @@ lemma Ideal.height_le_height_add_of_liesOver [IsNoetherianRing S] (p : Ideal R)
   obtain ⟨s, hp, heq⟩ := p.exists_finset_card_eq_height_of_isNoetherianRing
   let P' := P.map (Ideal.Quotient.mk <| p.map (algebraMap R S))
   obtain ⟨s', hP', heq'⟩ := P'.exists_finset_card_eq_height_of_isNoetherianRing
-  have leP' := hP'.le
   have hsP'sub : (s' : Set <| S ⧸ (Ideal.map (algebraMap R S) p)) ⊆ (P' : Set <| S ⧸ _) :=
-    fun x hx ↦ leP' (Ideal.subset_span hx)
+    fun x hx ↦ hP'.le (Ideal.subset_span hx)
   have : Set.SurjOn (Ideal.Quotient.mk (p.map (algebraMap R S))) P s' := by
     refine Set.SurjOn.mono subset_rfl hsP'sub fun x hx ↦ ?_
     obtain ⟨y, rfl⟩ := Ideal.Quotient.mk_surjective x

From 9b9ebbcf4e7e4857df034dca1d69c85096fa1deb Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Wed, 13 May 2026 00:26:35 +0800
Subject: [PATCH 56/69] add doc

---
 Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean | 1 +
 1 file changed, 1 insertion(+)

diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
index 0393b187d7d7f9..4086b4f40074d3 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
@@ -45,6 +45,7 @@ lemma Ideal.IsMinimalPrime.isPrime {p : Ideal R} (h : I.IsMinimalPrime p) : p.Is
 variable {I} in
 lemma Ideal.IsMinimalPrime.le {p : Ideal R} (h : I.IsMinimalPrime p) : I ≤ p := h.1.2
 
+/-- `IsMinimalPrime p` says that `p` is a minimal prime of the ring. -/
 abbrev IsMinimalPrime (p : Ideal R) : Prop := (⊥ : Ideal R).IsMinimalPrime p
 
 lemma IsMinimalPrime.isPrime {p : Ideal R} (h : IsMinimalPrime p) : p.IsPrime := h.1.1

From 126d2d9e2bd216b1b685a5589869d8fe7d00c2f8 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Wed, 13 May 2026 10:23:57 +0800
Subject: [PATCH 57/69] golf

---
 Mathlib/RingTheory/Ideal/Height.lean              | 7 ++-----
 Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean | 6 ++----
 Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean  | 3 +--
 3 files changed, 5 insertions(+), 11 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index 295d7761f4a4cb..15d71475c438d0 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -214,9 +214,7 @@ lemma Ideal.one_le_height_span_singleton_of_mem_nonZeroDivisors
 lemma Ideal.height_bot [Nontrivial R] : (⊥ : Ideal R).height = 0 := by
   obtain ⟨p, hp⟩ := Ideal.nonempty_minimalPrimes (R := R) (I := ⊥) top_ne_bot.symm
   simp only [Ideal.height, ENat.iInf_eq_zero]
-  refine ⟨p, hp, ?_⟩
-  have := Ideal.IsPrime.of_mem_minimalPrimes hp
-  rw [primeHeight_eq_zero_iff.mpr hp]
+  exact ⟨p, hp, haveI := hp.isPrime; primeHeight_eq_zero_iff.mpr hp⟩
 
 /-- In a trivial commutative ring, the height of any ideal is `∞`. -/
 @[simp, nontriviality]
@@ -428,8 +426,7 @@ lemma exists_spanRank_le_and_le_height_of_le_height [IsNoetherianRing R] (I : Id
           · rwa [← e]
         refine ⟨mem_minimalPrimes_of_primeHeight_eq_height (le_sup_left.trans hp.le) this.symm, ?_⟩
         rwa [p.height_eq_primeHeight, eq_comm]
-      · apply hp.le
-        exact Ideal.mem_sup_right <| mem_span_singleton_self x
+      · exact hp.le <| Ideal.mem_sup_right <| mem_span_singleton_self x
 
 /-- In a nontrivial commutative ring `R`, the supremum of heights of all ideals is equal to the
 Krull dimension of `R`. -/
diff --git a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
index 30a166eceeb779..4371f9bdc6a4f3 100644
--- a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
+++ b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
@@ -118,8 +118,7 @@ theorem Ideal.map_height_le_one_of_mem_minimalPrimes {I p : Ideal R} {x : R}
     (hp : p ∈ (I ⊔ span {x}).minimalPrimes) : (p.map (Ideal.Quotient.mk I)).height ≤ 1 :=
   let f := Ideal.Quotient.mk I
   have : p.IsPrime := hp.isPrime
-  have hfp : RingHom.ker f ≤ p :=
-    I.mk_ker.trans_le (le_sup_left.trans hp.le)
+  have hfp : RingHom.ker f ≤ p := I.mk_ker.trans_le (le_sup_left.trans hp.le)
   height_le_one_of_isPrincipal_of_mem_minimalPrimes ((span {x}).map f) (p.map f)
     ⟨⟨map_isPrime_of_surjective Quotient.mk_surjective hfp,
       map_mono (le_sup_right.trans hp.le)⟩,
@@ -154,7 +153,6 @@ theorem Ideal.mem_minimalPrimes_span_of_mem_minimalPrimes_span_insert {q p : Ide
   have hI'q : span t ≤ q := span_le.mpr htq
   have hI'p : span t ≤ p := hI'q.trans hqp.le
   have := hp.isPrime
-  have lep := hp.le
   have : (p.map f).IsPrime := map_isPrime_of_surjective hf (by rwa [mk_ker])
   suffices h : (p.map f).height ≤ 1 by
     have h_lt : q.map f < p.map f := (map_mono hqp.le).lt_of_not_ge fun e ↦ hqp.not_ge <| by
@@ -169,7 +167,7 @@ theorem Ideal.mem_minimalPrimes_span_of_mem_minimalPrimes_span_insert {q p : Ide
     rwa [comap_map_of_surjective f hf, ← RingHom.ker_eq_comap_bot,
       mk_ker, sup_eq_left.mpr hI'q] at this
   refine height_le_one_of_isPrincipal_of_mem_minimalPrimes ((span {x}).map f) (p.map f) ⟨⟨this,
-    map_mono <| span_le.mpr <| Set.singleton_subset_iff.mpr <| lep <| subset_span <| .inl rfl⟩,
+    map_mono <| span_le.mpr <| Set.singleton_subset_iff.mpr <| hp.le <| subset_span <| .inl rfl⟩,
     fun r ⟨hr, hxr⟩ hrp ↦ map_le_iff_le_comap.mpr (hp.2 ⟨hr.comap f, ?_⟩ ?_)⟩
   · rw [span_le, Set.insert_subset_iff]
     have := map_le_iff_le_comap.mp hxr (subset_span rfl)
diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
index 4086b4f40074d3..5ed12cb4e7f562 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
@@ -184,8 +184,7 @@ lemma Ideal.mem_minimalPrimes_sup {R : Type*} [CommRing R] {p I J : Ideal R} [p.
     p ∈ (I ⊔ J).minimalPrimes := by
   refine ⟨⟨‹_›, ?_⟩, fun q ⟨_, hq⟩ hqp ↦ ?_⟩
   · rw [sup_le_iff]
-    refine ⟨hle, ?_⟩
-    simpa [hle] using Ideal.comap_mono (f := Ideal.Quotient.mk I) h.le
+    refine ⟨hle, by simpa [hle] using Ideal.comap_mono (f := Ideal.Quotient.mk I) h.le⟩
   · rw [sup_le_iff] at hq
     have h2 : p.map (Quotient.mk I) ≤ q.map (Quotient.mk I) :=
       h.2 ⟨isPrime_map_quotientMk_of_isPrime hq.1, map_mono hq.2⟩ (map_mono hqp)

From 4d9a85d4b5d4a5d570eb86ceabc46db49a520b63 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Wed, 13 May 2026 12:01:55 +0800
Subject: [PATCH 58/69] remove unused lemma

---
 Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean | 4 ----
 1 file changed, 4 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
index 5ed12cb4e7f562..1d4dea8b6ad45e 100644
--- a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
+++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean
@@ -72,10 +72,6 @@ variable {I J}
 theorem Ideal.minimalPrimes_isPrime {p : Ideal R} (h : p ∈ minimalPrimes R) : p.IsPrime :=
   h.1.1
 
-/-- Variant of `Ideal.IsMinimalPrime.isPrime` for the set of all minimal primes of `R`. -/
-theorem Ideal.IsPrime.of_mem_minimalPrimes {p : Ideal R} (h : p ∈ minimalPrimes R) : p.IsPrime :=
-  h.1.1
-
 theorem Ideal.exists_minimalPrimes_le [J.IsPrime] (e : I ≤ J) : ∃ p ∈ I.minimalPrimes, p ≤ J := by
   set S := { p : (Ideal R)ᵒᵈ | Ideal.IsPrime p ∧ I ≤ OrderDual.ofDual p }
   suffices h : ∃ m, OrderDual.toDual J ≤ m ∧ Maximal (· ∈ S) m by

From 6d6d52cb765aa9b98688c0f0d6c1bf6c53b6ac49 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Wed, 13 May 2026 13:57:25 +0800
Subject: [PATCH 59/69] alternative fix

---
 Mathlib/RingTheory/KrullDimension/Zero.lean | 3 +--
 1 file changed, 1 insertion(+), 2 deletions(-)

diff --git a/Mathlib/RingTheory/KrullDimension/Zero.lean b/Mathlib/RingTheory/KrullDimension/Zero.lean
index 09dac318663204..0e8c6265d81628 100644
--- a/Mathlib/RingTheory/KrullDimension/Zero.lean
+++ b/Mathlib/RingTheory/KrullDimension/Zero.lean
@@ -35,8 +35,7 @@ variable (R) in
 lemma Ring.KrullDimLE.minimalPrimes_eq_setOf_isPrime :
     minimalPrimes R = { I | I.IsPrime } := by
   ext
-  rw [mem_minimalPrimes_iff]
-  simp
+  exact Ideal.mem_minimalPrimes_iff_isPrime
 
 variable (R) in
 lemma Ring.KrullDimLE.minimalPrimes_eq_setOf_isMaximal :

From 2784124e2b43f71d634e0c69df8e7162156067d5 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Wed, 13 May 2026 14:05:28 +0800
Subject: [PATCH 60/69] golf

---
 Mathlib/RingTheory/Ideal/Height.lean | 7 +++----
 1 file changed, 3 insertions(+), 4 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index 691e673b121164..01eaed34275dd8 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -247,10 +247,9 @@ lemma Ideal.one_le_height_span_singleton_of_mem_nonZeroDivisors
 @[simp]
 lemma Ideal.height_bot [Nontrivial R] : (⊥ : Ideal R).height = 0 := by
   obtain ⟨p, hp⟩ := Ideal.nonempty_minimalPrimes (R := R) (I := ⊥) top_ne_bot.symm
-  simp only [Ideal.height, ENat.iInf_eq_zero]
-  refine ⟨p, hp, ?_⟩
-  have := hp.isPrime
-  rw [← Ideal.height_eq_primeHeight, height_eq_zero_iff.mpr hp]
+  rw [Ideal.height_eq_inf_minimalPrimes]
+  simp only [ENat.iInf_eq_zero]
+  refine ⟨p, hp, haveI := hp.isPrime; height_eq_zero_iff.mpr hp⟩
 
 /-- In a trivial commutative ring, the height of any ideal is `∞`. -/
 @[simp, nontriviality]

From 81e88ddf4fdaafe9d7587e796b66ba4e1ac48075 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Mon, 18 May 2026 10:20:12 +0800
Subject: [PATCH 61/69] use under

---
 Mathlib/RingTheory/Ideal/Height.lean | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index 01eaed34275dd8..4f176d3abfa20f 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -390,7 +390,7 @@ private theorem IsLocalization.primeHeight_comap (S : Submonoid R) {A : Type*} [
   simpa [Ideal.height_eq_primeHeight] using IsLocalization.height_under_eq_of_isPrime S J
 
 theorem IsLocalization.height_under (S : Submonoid R) {A : Type*} [CommRing A] [Algebra R A]
-    [IsLocalization S A] (J : Ideal A) : (J.comap (algebraMap R A)).height = J.height := by
+    [IsLocalization S A] (J : Ideal A) : (J.under R).height = J.height := by
   rw [(J.comap _).height_eq_inf_minimalPrimes, J.height_eq_inf_minimalPrimes]
   simp only [IsLocalization.minimalPrimes_comap S A, iInf_image]
   apply iInf_congr (fun p ↦ iInf_congr fun hp ↦ ?_)

From ff9fc4cd7bf0dba3251c1c64ea2cc3515fc7f39f Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Mon, 18 May 2026 10:53:47 +0800
Subject: [PATCH 62/69] fix

---
 Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
index 3cfc6d99f0a15d..801771d7d490fb 100644
--- a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
+++ b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
@@ -109,7 +109,7 @@ lemma Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes
   have := hp.isPrime
   let f := algebraMap R (Localization.AtPrime p)
   have := Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes_of_isLocalRing (I.map f) ?_
-  · rwa [← IsLocalization.height_under p.primeCompl, ← Ideal.under_def,
+  · rwa [← IsLocalization.height_under p.primeCompl,
       Localization.AtPrime.under_maximalIdeal] at this
   · rwa [IsLocalization.minimalPrimes_map p.primeCompl (Localization.AtPrime p) I,
       Set.mem_preimage, Localization.AtPrime.under_maximalIdeal]

From 2506602e98567bc76f5169638f8c00591204d6f6 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Thu, 21 May 2026 21:29:21 +0800
Subject: [PATCH 63/69] remove expose

---
 Mathlib/RingTheory/Ideal/Height.lean | 3 +--
 1 file changed, 1 insertion(+), 2 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index 4f176d3abfa20f..1f094d5c801f6b 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -26,7 +26,7 @@ In this file, we define the height of a prime ideal and the height of an ideal.
 
 -/
 
-@[expose] public section
+public section
 
 variable {R : Type*} [CommRing R] (I : Ideal R)
 
@@ -38,7 +38,6 @@ private noncomputable def Ideal.primeHeight [hI : I.IsPrime] : ℕ∞ :=
   Order.height (⟨I, hI⟩ : PrimeSpectrum R)
 
 /-- The height of an ideal is defined as the infimum of the heights of its minimal prime ideals. -/
-@[no_expose]
 noncomputable def Ideal.height : ℕ∞ :=
   ⨅ J ∈ I.minimalPrimes, @Ideal.primeHeight _ _ J ‹J ∈ I.minimalPrimes›.isPrime
 

From 34942ddfbd32b9458db2ce24adc0cce0dc4154b7 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Thu, 21 May 2026 21:33:35 +0800
Subject: [PATCH 64/69] refactor PID proof

---
 Mathlib/RingTheory/KrullDimension/PID.lean | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/Mathlib/RingTheory/KrullDimension/PID.lean b/Mathlib/RingTheory/KrullDimension/PID.lean
index c87e2986505260..370302852f63ef 100644
--- a/Mathlib/RingTheory/KrullDimension/PID.lean
+++ b/Mathlib/RingTheory/KrullDimension/PID.lean
@@ -51,5 +51,5 @@ lemma IsPrincipalIdealRing.height_eq_one_of_isMaximal {R : Type*} [CommRing R] [
   · suffices h : (m.height : WithBot ℕ∞) ≤ 1 by norm_cast at h
     rw [← IsPrincipalIdealRing.ringKrullDim_eq_one _ h]
     exact Ideal.height_le_ringKrullDim_of_ne_top Ideal.IsPrime.ne_top'
-  · rw [Order.one_le_iff_pos, Ideal.height_eq_order_height_of_isPrime, Order.height_pos]
-    exact not_isMin_of_lt (b := ⊥) (Ideal.bot_lt_of_maximal m h)
+  · apply le_of_eq_of_le _ (Ideal.height_add_one_le_of_lt_of_isPrime (Ideal.bot_lt_of_maximal m h))
+    simp

From 1937f84e11fef50ce3c17d6a159024a03a9a8488 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Thu, 21 May 2026 21:43:29 +0800
Subject: [PATCH 65/69] refactor order height lemma

---
 Mathlib/AlgebraicGeometry/Properties.lean |  3 +--
 Mathlib/RingTheory/Ideal/Height.lean      | 22 ++++++++++++----------
 2 files changed, 13 insertions(+), 12 deletions(-)

diff --git a/Mathlib/AlgebraicGeometry/Properties.lean b/Mathlib/AlgebraicGeometry/Properties.lean
index 0344da531bd33b..e84461e05f0eac 100644
--- a/Mathlib/AlgebraicGeometry/Properties.lean
+++ b/Mathlib/AlgebraicGeometry/Properties.lean
@@ -354,10 +354,9 @@ lemma coheight_eq_of_isOpenImmersion {U X : Scheme} {x : U} (f : U ⟶ X) [IsOpe
 open Order in
 lemma idealHeight_eq_coheight (R : CommRingCat) (x : Spec R) :
     x.asIdeal.height = coheight x := by
-  rw [Ideal.height_eq_order_height_of_isPrime,
+  rw [PrimeSpectrum.height_eq_order_height_of_isPrime,
     ← Order.coheight_orderIso (specOrderIsoPrimeSpectrum R), ← height_ofDual,
     specOrderIsoPrimeSpectrum_apply, OrderDual.ofDual_toDual]
-  rfl
 
 open Order in
 @[stacks 02IZ]
diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index 1f094d5c801f6b..77aa9044de4621 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -45,9 +45,9 @@ noncomputable def Ideal.height : ℕ∞ :=
 private lemma Ideal.height_eq_primeHeight [I.IsPrime] : I.height = I.primeHeight := by
   simp [height, primeHeight, Ideal.minimalPrimes_eq_subsingleton_self]
 
-lemma Ideal.height_eq_order_height_of_isPrime [I.IsPrime] : I.height =
-    Order.height (⟨I, ‹I.IsPrime›⟩ : PrimeSpectrum R) :=
-  I.height_eq_primeHeight
+lemma PrimeSpectrum.height_eq_order_height_of_isPrime (p : PrimeSpectrum R) :
+    p.1.height = Order.height p :=
+  p.1.height_eq_primeHeight
 
 lemma Ideal.height_eq_inf_minimalPrimes : I.height = ⨅ J ∈ I.minimalPrimes, J.height := by
   apply iInf_congr (fun p ↦ iInf_congr fun hp ↦ ?_)
@@ -104,7 +104,7 @@ lemma Ideal.exists_ltSeries_length_eq_height (p : Ideal R) [p.IsPrime] [p.Finite
 
 private lemma Ideal.height_mono_of_isPrime {I J : Ideal R} [I.IsPrime] [J.IsPrime] (h : I ≤ J) :
     I.height ≤ J.height := by
-  simp only [Ideal.height_eq_order_height_of_isPrime]
+  simp only [Ideal.height_eq_primeHeight, Ideal.primeHeight]
   gcongr
   exact h
 
@@ -115,7 +115,7 @@ private lemma Ideal.primeHeight_mono {I J : Ideal R} [I.IsPrime] [J.IsPrime] (h
 
 lemma Ideal.height_add_one_le_of_lt_of_isPrime {I J : Ideal R} [I.IsPrime] [J.IsPrime] (h : I < J) :
     I.height + 1 ≤ J.height := by
-  simp only [Ideal.height_eq_order_height_of_isPrime]
+  simp only [Ideal.height_eq_primeHeight, Ideal.primeHeight]
   exact Order.height_add_one_le h
 
 @[deprecated "Use `Ideal.height_add_one_le_of_lt_of_isPrime` instead." (since := "2026-04-04")]
@@ -166,7 +166,7 @@ private lemma Ideal.primeHeight_strict_mono {I J : Ideal R} [I.IsPrime] [J.IsPri
 
 lemma Ideal.height_le_ringKrullDim_of_isPrime {I : Ideal R} [I.IsPrime] :
     I.height ≤ ringKrullDim R := by
-  rw [I.height_eq_order_height_of_isPrime]
+  rw [Ideal.height_eq_primeHeight, Ideal.primeHeight]
   exact Order.height_le_krullDim _
 
 /-- A prime ideal of finite height is equal to any ideal that contains it with no greater height. -/
@@ -222,7 +222,8 @@ lemma Ideal.mem_minimalPrimes_of_height_eq {I J : Ideal R} (e : I ≤ J) [J.IsPr
 
 /-- A prime ideal has height zero if and only if it is minimal -/
 lemma Ideal.height_eq_zero_iff {I : Ideal R} [I.IsPrime] : height I = 0 ↔ I ∈ minimalPrimes R := by
-  rw [Ideal.height_eq_order_height_of_isPrime, Order.height_eq_zero, minimalPrimes_eq_minimals]
+  rw [Ideal.height_eq_primeHeight, Ideal.primeHeight, Order.height_eq_zero,
+    minimalPrimes_eq_minimals]
   refine ⟨fun h ↦ ⟨‹_›, ?_⟩, fun ⟨hI, hI'⟩ b hb ↦ hI' b.isPrime hb⟩
   by_contra! ⟨P, ⟨hP₁, ⟨hP₂, hP₃⟩⟩⟩
   exact hP₃ (h (b := ⟨P, hP₁⟩) hP₂)
@@ -274,7 +275,7 @@ private theorem Ideal.isMaximal_of_primeHeight_eq_ringKrullDim {I : Ideal R} [I.
 @[simp]
 theorem IsLocalRing.maximalIdeal_height_eq_ringKrullDim [IsLocalRing R] :
     (IsLocalRing.maximalIdeal R).height = ringKrullDim R := by
-  rw [Ideal.height_eq_order_height_of_isPrime]
+  rw [Ideal.height_eq_primeHeight, Ideal.primeHeight]
   exact Order.height_top_eq_krullDim
 
 @[deprecated "Use `IsLocalRing.maximalIdeal_height_eq_ringKrullDim` instead."
@@ -368,8 +369,9 @@ lemma ringKrullDim_le_iff_isMaximal_height_le {R : Type*} [CommRing R] (n : With
 private theorem IsLocalization.height_under_eq_of_isPrime (S : Submonoid R) {A : Type*} [CommRing A]
     [Algebra R A] [IsLocalization S A] (J : Ideal A) [J.IsPrime] :
     (J.comap (algebraMap R A)).height = J.height := by
-  rw [eq_comm, Ideal.height_eq_order_height_of_isPrime, Ideal.height_eq_order_height_of_isPrime,
-    ← WithBot.coe_inj, Order.height_eq_krullDim_Iic, Order.height_eq_krullDim_Iic]
+  rw [eq_comm, Ideal.height_eq_primeHeight, Ideal.height_eq_primeHeight, Ideal.primeHeight,
+    Ideal.primeHeight, ← WithBot.coe_inj, Order.height_eq_krullDim_Iic,
+    Order.height_eq_krullDim_Iic]
   let e := IsLocalization.orderIsoOfPrime S A
   have H (p : Ideal R) (hp : p ≤ J.comap (algebraMap R A)) : Disjoint (S : Set R) p :=
     Set.disjoint_of_subset_right hp (e ⟨_, ‹J.IsPrime›⟩).2.2

From 825663aadcc93ff798f5bc9c1d9f0b5af84144ff Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Thu, 21 May 2026 21:45:00 +0800
Subject: [PATCH 66/69] fix doc for primeHeight

---
 Mathlib/RingTheory/Ideal/Height.lean | 4 ----
 1 file changed, 4 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index 77aa9044de4621..0cd51f3d9e2e96 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -17,10 +17,6 @@ In this file, we define the height of a prime ideal and the height of an ideal.
 
 ## Main definitions
 
-* `Ideal.primeHeight` : The height of a prime ideal $\mathfrak{p}$. We define it as the supremum of
-  the lengths of strictly decreasing chains of prime ideals below it. This definition is implemented
-  via `Order.height`.
-
 * `Ideal.height` : The height of an ideal. We defined it as the infimum of the `primeHeight` of the
   minimal prime ideals of I.
 

From 8cae9271f2ccf43a296910ceaaef085a7452a8cb Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Thu, 21 May 2026 21:51:44 +0800
Subject: [PATCH 67/69] fix naming

---
 Mathlib/RingTheory/Ideal/Height.lean | 10 +++++-----
 1 file changed, 5 insertions(+), 5 deletions(-)

diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index 0cd51f3d9e2e96..8e5fde64376212 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -487,7 +487,7 @@ lemma Ideal.sup_height_eq_ringKrullDim [Nontrivial R] :
 
 /-- In a nontrivial commutative ring `R`, the supremum of heights of all prime ideals is
 equal to the Krull dimension of `R`. -/
-lemma Ideal.sup_height_isPrime_eq_ringKrullDim [Nontrivial R] :
+lemma Ideal.sup_isPrime_height_eq_ringKrullDim [Nontrivial R] :
     ↑(⨆ (I : Ideal R) (_ : I.IsPrime), I.height) = ringKrullDim R := by
   rw [← sup_height_eq_ringKrullDim, WithBot.coe_inj]
   apply le_antisymm
@@ -503,11 +503,11 @@ lemma Ideal.sup_height_isPrime_eq_ringKrullDim [Nontrivial R] :
 @[deprecated "Use `Ideal.sup_height_isPrime_eq_ringKrullDim` instead." (since := "2026-04-02")]
 private lemma Ideal.sup_primeHeight_eq_ringKrullDim [Nontrivial R] :
     ↑(⨆ (I : Ideal R) (_ : I.IsPrime), I.primeHeight) = ringKrullDim R := by
-  simp [← Ideal.height_eq_primeHeight, Ideal.sup_height_isPrime_eq_ringKrullDim]
+  simp [← Ideal.height_eq_primeHeight, Ideal.sup_isPrime_height_eq_ringKrullDim]
 
 /-- In a nontrivial commutative ring `R`, the supremum of heights of all maximal ideals is
 equal to the Krull dimension of `R`. -/
-lemma Ideal.sup_height_of_maximal_eq_ringKrullDim [Nontrivial R] :
+lemma Ideal.sup_isMaximal_height_eq_ringKrullDim [Nontrivial R] :
     ↑(⨆ (I : Ideal R) (_ : I.IsMaximal), I.height) = ringKrullDim R := by
   rw [← Ideal.sup_height_eq_ringKrullDim, WithBot.coe_inj]
   apply le_antisymm
@@ -521,7 +521,7 @@ lemma Ideal.sup_height_of_maximal_eq_ringKrullDim [Nontrivial R] :
 @[deprecated "Use `Ideal.sup_height_of_maximal_eq_ringKrullDim` instead." (since := "2026-04-02")]
 private lemma Ideal.sup_primeHeight_of_maximal_eq_ringKrullDim [Nontrivial R] :
     ↑(⨆ (I : Ideal R) (_ : I.IsMaximal), I.primeHeight) = ringKrullDim R := by
-  simp_rw [← Ideal.height_eq_primeHeight, Ideal.sup_height_of_maximal_eq_ringKrullDim]
+  simp_rw [← Ideal.height_eq_primeHeight, Ideal.sup_isMaximal_height_eq_ringKrullDim]
 
 section isLocalization
 
@@ -536,7 +536,7 @@ lemma Ring.krullDimLE_of_isLocalization_maximal {n : ℕ}
     Ring.KrullDimLE n R := by
   simp_rw [Ring.krullDimLE_iff] at h ⊢
   nontriviality R
-  rw [← Ideal.sup_height_of_maximal_eq_ringKrullDim]
+  rw [← Ideal.sup_isMaximal_height_eq_ringKrullDim]
   refine (WithBot.coe_le_coe).mpr (iSup₂_le_iff.mpr fun P hP ↦ ?_)
   rw [← WithBot.coe_le_coe, ← IsLocalization.AtPrime.ringKrullDim_eq_height P (Rₚ P)]
   exact h P

From a4420aadfb2136d7730a656e21d91119ea7ea869 Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Thu, 21 May 2026 21:59:36 +0800
Subject: [PATCH 68/69] temp fix

---
 Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
index 801771d7d490fb..537bb2b4f995f2 100644
--- a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
+++ b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
@@ -480,7 +480,8 @@ lemma Ideal.height_eq_height_add_of_liesOver_of_hasGoingDown [IsNoetherianRing S
       simp [hlq, map_le_iff_le_comap, LiesOver.over (p := p) (P := P)]
   obtain ⟨lp', hlp'len, hlp', _⟩ := exists_ltSeries_of_hasGoingDown lp l'.head.asIdeal
   have : (lp'.smash l' hlp').length = lp.length + lq.length := by simp [hlp'len, l']
-  rw [← hlenp, ← hlenq, ← Nat.cast_add, ← this, Ideal.height_eq_order_height_of_isPrime]
+  rw [← hlenp, ← hlenq, ← Nat.cast_add, ← this,
+    (⟨P, ‹_›⟩ : PrimeSpectrum S).height_eq_order_height_of_isPrime]
   apply Order.length_le_height
   simp [hlq, l', ← PrimeSpectrum.asIdeal_le_asIdeal, map_le_iff_le_comap,
     LiesOver.over (p := p) (P := P)]

From 45b8864a30cc386c4e0b9a7a11a0272e119f306a Mon Sep 17 00:00:00 2001
From: Thmoas-Guan <150537269+Thmoas-Guan@users.noreply.github.com>
Date: Thu, 21 May 2026 22:03:04 +0800
Subject: [PATCH 69/69] fix naming

---
 Mathlib/AlgebraicGeometry/Properties.lean         | 2 +-
 Mathlib/RingTheory/Ideal/Height.lean              | 6 +++---
 Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean | 3 +--
 3 files changed, 5 insertions(+), 6 deletions(-)

diff --git a/Mathlib/AlgebraicGeometry/Properties.lean b/Mathlib/AlgebraicGeometry/Properties.lean
index e84461e05f0eac..c37783d7f1fde8 100644
--- a/Mathlib/AlgebraicGeometry/Properties.lean
+++ b/Mathlib/AlgebraicGeometry/Properties.lean
@@ -354,7 +354,7 @@ lemma coheight_eq_of_isOpenImmersion {U X : Scheme} {x : U} (f : U ⟶ X) [IsOpe
 open Order in
 lemma idealHeight_eq_coheight (R : CommRingCat) (x : Spec R) :
     x.asIdeal.height = coheight x := by
-  rw [PrimeSpectrum.height_eq_order_height_of_isPrime,
+  rw [PrimeSpectrum.height_eq_orderHeight,
     ← Order.coheight_orderIso (specOrderIsoPrimeSpectrum R), ← height_ofDual,
     specOrderIsoPrimeSpectrum_apply, OrderDual.ofDual_toDual]
 
diff --git a/Mathlib/RingTheory/Ideal/Height.lean b/Mathlib/RingTheory/Ideal/Height.lean
index 8e5fde64376212..52d06e6ec73e24 100644
--- a/Mathlib/RingTheory/Ideal/Height.lean
+++ b/Mathlib/RingTheory/Ideal/Height.lean
@@ -41,9 +41,9 @@ noncomputable def Ideal.height : ℕ∞ :=
 private lemma Ideal.height_eq_primeHeight [I.IsPrime] : I.height = I.primeHeight := by
   simp [height, primeHeight, Ideal.minimalPrimes_eq_subsingleton_self]
 
-lemma PrimeSpectrum.height_eq_order_height_of_isPrime (p : PrimeSpectrum R) :
-    p.1.height = Order.height p :=
-  p.1.height_eq_primeHeight
+lemma PrimeSpectrum.height_eq_orderHeight (p : PrimeSpectrum R) :
+    p.asIdeal.height = Order.height p :=
+  p.asIdeal.height_eq_primeHeight
 
 lemma Ideal.height_eq_inf_minimalPrimes : I.height = ⨅ J ∈ I.minimalPrimes, J.height := by
   apply iInf_congr (fun p ↦ iInf_congr fun hp ↦ ?_)
diff --git a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
index 537bb2b4f995f2..5aa2e8787d83d3 100644
--- a/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
+++ b/Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean
@@ -480,8 +480,7 @@ lemma Ideal.height_eq_height_add_of_liesOver_of_hasGoingDown [IsNoetherianRing S
       simp [hlq, map_le_iff_le_comap, LiesOver.over (p := p) (P := P)]
   obtain ⟨lp', hlp'len, hlp', _⟩ := exists_ltSeries_of_hasGoingDown lp l'.head.asIdeal
   have : (lp'.smash l' hlp').length = lp.length + lq.length := by simp [hlp'len, l']
-  rw [← hlenp, ← hlenq, ← Nat.cast_add, ← this,
-    (⟨P, ‹_›⟩ : PrimeSpectrum S).height_eq_order_height_of_isPrime]
+  rw [← hlenp, ← hlenq, ← Nat.cast_add, ← this, (⟨P, ‹_›⟩ : PrimeSpectrum S).height_eq_orderHeight]
   apply Order.length_le_height
   simp [hlq, l', ← PrimeSpectrum.asIdeal_le_asIdeal, map_le_iff_le_comap,
     LiesOver.over (p := p) (P := P)]