awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : KrullDimLE 0 R
x : Ideal R
⊢ Membership.mem (γ := Set (Ideal R)) (minimalPrimes R) x ↔
    Membership.mem (γ := Set (Ideal R)) {I : Ideal R | I.IsMaximal} x

Tactic:
simp [minimalPrimes_eq_setOf_isPrime, Ideal.isMaximal_iff_isPrime]

Premises used: Ring.KrullDimLE.minimalPrimes_eq_setOf_isPrime, Ideal.isMaximal_iff_isPrime
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : Ring.KrullDimLE 0 R
I : Ideal R
inst_2 : Subsingleton R
⊢ Ring.KrullDimLE 0 R

Tactic:
inferInstance

Premises used: inferInstance
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : KrullDimLE 0 R
inst_2 : IsDomain R
by_contra : ∀ (h : ¬IsField R), False
⊢ IsField R

Tactic:
by_contra h
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : KrullDimLE 0 R
inst_2 : IsDomain R
h : ¬IsField R
obtain : ∀ (p : Ideal R) (hp : p ≠ ⊥) (h : p.IsPrime), False
⊢ False

Tactic:
obtain ⟨p, hp, h⟩ := Ring.not_isField_iff_exists_prime.mp h

Premises used: Ring.not_isField_iff_exists_prime, Iff.mp
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : KrullDimLE 0 R
inst_2 : IsDomain R
h_1 : ¬IsField R
p : Ideal R
hp : p ≠ ⊥
h : p.IsPrime
⊢ False

Tactic:
hp.symm (Ideal.isPrime_bot.isMaximal'.eq_of_le h.ne_top bot_le)

Premises used: Ne.symm, Ideal.isPrime_bot, Ideal.IsMaximal.eq_of_le, Ideal.IsPrime.ne_top, Ideal.IsPrime.isMaximal', bot_le
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : KrullDimLE 0 R
inst_2 : IsDomain R
h_1 : ¬IsField R
p : Ideal R
hp : p ≠ ⊥
h : p.IsPrime
⊢ False

Tactic:
exact hp.symm (Ideal.isPrime_bot.isMaximal'.eq_of_le h.ne_top bot_le)

Premises used: Ne.symm, Ideal.isPrime_bot, Ideal.IsMaximal.eq_of_le, Ideal.IsPrime.ne_top, Ideal.IsPrime.isMaximal', bot_le
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : Nontrivial R
rw : ringKrullDim R ≤ 0 → 0 ≤ ringKrullDim R
⊢ Ring.KrullDimLE 0 R ↔ ringKrullDim R = 0

Tactic:
rw [Ring.KrullDimLE, Order.krullDimLE_iff, le_antisymm_iff, ← ringKrullDim, Nat.cast_zero,
    iff_self_and]

Premises used: iff_self_and, le_antisymm_iff, ringKrullDim, Nat.cast_zero, Ring.KrullDimLE, Order.krullDimLE_iff
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : Nontrivial R
⊢ ringKrullDim R ≤ 0 → 0 ≤ ringKrullDim R

Tactic:
fun _ ↦ ringKrullDim_nonneg_of_nontrivial

Premises used: ringKrullDim_nonneg_of_nontrivial
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : Nontrivial R
⊢ ringKrullDim R ≤ 0 → 0 ≤ ringKrullDim R

Tactic:
exact fun _ ↦ ringKrullDim_nonneg_of_nontrivial

Premises used: ringKrullDim_nonneg_of_nontrivial
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
⊢ KrullDimLE 0 R ∧ IsLocalRing R

Tactic:
1

Premises used: sorryAx, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Bool.false
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
⊢ ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x

Tactic:
3

Premises used: sorryAx, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Bool.false
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
⊢ ∀ (a : KrullDimLE 0 R ∧ IsLocalRing R) (x : R), IsNilpotent x ↔ ¬IsUnit x

Tactic:
1 → 3

Premises used: sorryAx, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Bool.false
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
h₁ : KrullDimLE 0 R
h₂ : IsLocalRing R
x : R
change : Membership.mem (γ := Ideal R) (nilradical R) x ↔ Membership.mem (γ := Ideal R) (IsLocalRing.maximalIdeal R) x
⊢ IsNilpotent x ↔ ¬IsUnit x

Tactic:
change x ∈ nilradical R ↔ x ∈ IsLocalRing.maximalIdeal R

Premises used: IsLocalRing.maximalIdeal, nilradical
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
h₁ : KrullDimLE 0 R
h₂ : IsLocalRing R
x : R
rw :
  Membership.mem (γ := Ideal R) (sInf {J : Ideal R | 0 ≤ J ∧ J.IsPrime}) x ↔
    Membership.mem (γ := Ideal R) (IsLocalRing.maximalIdeal R) x
⊢ Membership.mem (γ := Ideal R) (nilradical R) x ↔ Membership.mem (γ := Ideal R) (IsLocalRing.maximalIdeal R) x

Tactic:
rw [nilradical, Ideal.radical_eq_sInf]

Premises used: Ideal.radical_eq_sInf, nilradical
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
h₁ : KrullDimLE 0 R
h₂ : IsLocalRing R
x : R
⊢ Membership.mem (γ := Ideal R) (sInf {J : Ideal R | 0 ≤ J ∧ J.IsPrime}) x ↔
    Membership.mem (γ := Ideal R) (IsLocalRing.maximalIdeal R) x

Tactic:
simp [← Ideal.isMaximal_iff_isPrime, IsLocalRing.isMaximal_iff]

Premises used: IsLocalRing.isMaximal_iff, Ideal.isMaximal_iff_isPrime
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
⊢ ∀ (a : KrullDimLE 0 R ∧ IsLocalRing R) (x : R), IsNilpotent x ↔ ¬IsUnit x

Tactic:
tfae_have 1 → 3 := by
    intro ⟨h₁, h₂⟩ x
    change x ∈ nilradical R ↔ x ∈ IsLocalRing.maximalIdeal R
    rw [nilradical, Ideal.radical_eq_sInf]
    simp [← Ideal.isMaximal_iff_isPrime, IsLocalRing.isMaximal_iff]
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
tfae_1_to_3 : ∀ (a : KrullDimLE 0 R ∧ IsLocalRing R) (x : R), IsNilpotent x ↔ ¬IsUnit x
⊢ ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x

Tactic:
3

Premises used: sorryAx, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Bool.false
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
tfae_1_to_3 : ∀ (a : KrullDimLE 0 R ∧ IsLocalRing R) (x : R), IsNilpotent x ↔ ¬IsUnit x
⊢ Ideal.IsMaximal (nilradical R)

Tactic:
4

Premises used: sorryAx, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Bool.false
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
tfae_1_to_3 : ∀ (a : KrullDimLE 0 R ∧ IsLocalRing R) (x : R), IsNilpotent x ↔ ¬IsUnit x
⊢ ∀ (a : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x), Ideal.IsMaximal (nilradical R)

Tactic:
3 → 4

Premises used: sorryAx, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Bool.false
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
tfae_1_to_3 : ∀ (a : KrullDimLE 0 R ∧ IsLocalRing R) (x : R), IsNilpotent x ↔ ¬IsUnit x
⊢ ∀ (a : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x), Ideal.IsMaximal (nilradical R)

Tactic:
fun H ↦ ⟨fun e ↦ ?_, fun I hI ↦ ?_⟩

Premises used: Ideal.IsMaximal.mk, And.intro
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
tfae_1_to_3 : ∀ (a : KrullDimLE 0 R ∧ IsLocalRing R) (x : R), IsNilpotent x ↔ ¬IsUnit x
refine_1 : ∀ (H : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x) (e : Eq (α := Ideal R) (nilradical R) ⊤), False
refine_2 : ∀ (H : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x) (I : Ideal R) (hI : nilradical R < I), I = ⊤
⊢ ∀ (a : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x), Ideal.IsMaximal (nilradical R)

Tactic:
refine fun H ↦ ⟨fun e ↦ ?_, fun I hI ↦ ?_⟩
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
tfae_1_to_3 : ∀ (a : KrullDimLE 0 R ∧ IsLocalRing R) (x : R), IsNilpotent x ↔ ¬IsUnit x
H : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x
e : Eq (α := Ideal R) (nilradical R) ⊤
refine_1 : ∀ (n : ℕ) (hn : Membership.mem (γ := Ideal R) 0 (HPow.hPow (α := R) 1 n)), False
⊢ False

Tactic:
obtain ⟨n, hn⟩ := (Ideal.eq_top_iff_one _).mp e

Premises used: Ideal.eq_top_iff_one, Iff.mp
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
tfae_1_to_3 : ∀ (a : KrullDimLE 0 R ∧ IsLocalRing R) (x : R), IsNilpotent x ↔ ¬IsUnit x
H : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x
e : Eq (α := Ideal R) (nilradical R) ⊤
n : ℕ
hn : Membership.mem (γ := Ideal R) 0 (HPow.hPow (α := R) 1 n)
⊢ Eq.{u_1 + 1} (α := R) 1 0

Tactic:
simpa using hn
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
tfae_1_to_3 : ∀ (a : KrullDimLE 0 R ∧ IsLocalRing R) (x : R), IsNilpotent x ↔ ¬IsUnit x
H : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x
e : Eq (α := Ideal R) (nilradical R) ⊤
n : ℕ
hn : Membership.mem (γ := Ideal R) 0 (HPow.hPow (α := R) 1 n)
⊢ False

Tactic:
exact (H 0).mp .zero ((show (1 : R) = 0 by simpa using hn) ▸ isUnit_one)

Premises used: Eq.rec, isUnit_one, Iff.mp, OfNat.ofNat, IsNilpotent.zero
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
tfae_1_to_3 : ∀ (a : KrullDimLE 0 R ∧ IsLocalRing R) (x : R), IsNilpotent x ↔ ¬IsUnit x
H : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x
I : Ideal R
hI : nilradical R < I
refine_2 : ∀ (x : R) (hx : x ∈ I) (hx' : x ∉ nilradical R), I = ⊤
⊢ I = ⊤

Tactic:
obtain ⟨x, hx, hx'⟩ := (SetLike.lt_iff_le_and_exists.mp hI).2

Premises used: SetLike.lt_iff_le_and_exists, And.right, Iff.mp
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
tfae_1_to_3 : ∀ (a : KrullDimLE 0 R ∧ IsLocalRing R) (x : R), IsNilpotent x ↔ ¬IsUnit x
H : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x
I : Ideal R
hI : nilradical R < I
x : R
hx : x ∈ I
hx' : x ∉ nilradical R
⊢ I = ⊤

Tactic:
Ideal.eq_top_of_isUnit_mem _ hx (not_not.mp ((H x).not.mp hx'))

Premises used: Classical.not_not, Ideal.eq_top_of_isUnit_mem, Iff.not, Iff.mp
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
tfae_1_to_3 : ∀ (a : KrullDimLE 0 R ∧ IsLocalRing R) (x : R), IsNilpotent x ↔ ¬IsUnit x
H : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x
I : Ideal R
hI : nilradical R < I
x : R
hx : x ∈ I
hx' : x ∉ nilradical R
⊢ I = ⊤

Tactic:
exact Ideal.eq_top_of_isUnit_mem _ hx (not_not.mp ((H x).not.mp hx'))

Premises used: Classical.not_not, Ideal.eq_top_of_isUnit_mem, Iff.not, Iff.mp
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
tfae_1_to_3 : ∀ (a : KrullDimLE 0 R ∧ IsLocalRing R) (x : R), IsNilpotent x ↔ ¬IsUnit x
⊢ ∀ (a : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x), Ideal.IsMaximal (nilradical R)

Tactic:
tfae_have 3 → 4 := by
    refine fun H ↦ ⟨fun e ↦ ?_, fun I hI ↦ ?_⟩
    · obtain ⟨n, hn⟩ := (Ideal.eq_top_iff_one _).mp e
      exact (H 0).mp .zero ((show (1 : R) = 0 by simpa using hn) ▸ isUnit_one)
    · obtain ⟨x, hx, hx'⟩ := (SetLike.lt_iff_le_and_exists.mp hI).2
      exact Ideal.eq_top_of_isUnit_mem _ hx (not_not.mp ((H x).not.mp hx'))
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
tfae_1_to_3 : ∀ (a : KrullDimLE 0 R ∧ IsLocalRing R) (x : R), IsNilpotent x ↔ ¬IsUnit x
tfae_3_to_4 : ∀ (a : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x), Ideal.IsMaximal (nilradical R)
⊢ Ideal.IsMaximal (nilradical R)

Tactic:
4

Premises used: sorryAx, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Bool.false
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
tfae_1_to_3 : ∀ (a : KrullDimLE 0 R ∧ IsLocalRing R) (x : R), IsNilpotent x ↔ ¬IsUnit x
tfae_3_to_4 : ∀ (a : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x), Ideal.IsMaximal (nilradical R)
⊢ ∃! I : Ideal R, I.IsPrime

Tactic:
2

Premises used: sorryAx, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Bool.false
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
tfae_1_to_3 : ∀ (a : KrullDimLE 0 R ∧ IsLocalRing R) (x : R), IsNilpotent x ↔ ¬IsUnit x
tfae_3_to_4 : ∀ (a : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x), Ideal.IsMaximal (nilradical R)
⊢ ∀ (a : Ideal.IsMaximal (nilradical R)), ∃! I : Ideal R, I.IsPrime

Tactic:
4 → 2

Premises used: sorryAx, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Bool.false
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
tfae_1_to_3 : ∀ (a : KrullDimLE 0 R ∧ IsLocalRing R) (x : R), IsNilpotent x ↔ ¬IsUnit x
tfae_3_to_4 : ∀ (a : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x), Ideal.IsMaximal (nilradical R)
⊢ ∀ (a : Ideal.IsMaximal (nilradical R)), ∃! I : Ideal R, I.IsPrime

Tactic:
tfae_have 4 → 2 := fun H ↦ ⟨_, H.isPrime, fun p (hp : p.IsPrime) ↦
      (H.eq_of_le hp.ne_top (nilradical_le_prime p)).symm⟩
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
tfae_1_to_3 : ∀ (a : KrullDimLE 0 R ∧ IsLocalRing R) (x : R), IsNilpotent x ↔ ¬IsUnit x
tfae_3_to_4 : ∀ (a : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x), Ideal.IsMaximal (nilradical R)
tfae_4_to_2 : ∀ (a : Ideal.IsMaximal (nilradical R)), ∃! I : Ideal R, I.IsPrime
⊢ ∃! I : Ideal R, I.IsPrime

Tactic:
2

Premises used: sorryAx, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Bool.false
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
tfae_1_to_3 : ∀ (a : KrullDimLE 0 R ∧ IsLocalRing R) (x : R), IsNilpotent x ↔ ¬IsUnit x
tfae_3_to_4 : ∀ (a : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x), Ideal.IsMaximal (nilradical R)
tfae_4_to_2 : ∀ (a : Ideal.IsMaximal (nilradical R)), ∃! I : Ideal R, I.IsPrime
⊢ KrullDimLE 0 R ∧ IsLocalRing R

Tactic:
1

Premises used: sorryAx, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Bool.false
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
tfae_1_to_3 : ∀ (a : KrullDimLE 0 R ∧ IsLocalRing R) (x : R), IsNilpotent x ↔ ¬IsUnit x
tfae_3_to_4 : ∀ (a : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x), Ideal.IsMaximal (nilradical R)
tfae_4_to_2 : ∀ (a : Ideal.IsMaximal (nilradical R)), ∃! I : Ideal R, I.IsPrime
⊢ ∀ (a : ∃! I : Ideal R, I.IsPrime), KrullDimLE 0 R ∧ IsLocalRing R

Tactic:
2 → 1

Premises used: sorryAx, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Bool.false
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
tfae_1_to_3 : ∀ (a : KrullDimLE 0 R ∧ IsLocalRing R) (x : R), IsNilpotent x ↔ ¬IsUnit x
tfae_3_to_4 : ∀ (a : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x), Ideal.IsMaximal (nilradical R)
tfae_4_to_2 : ∀ (a : Ideal.IsMaximal (nilradical R)), ∃! I : Ideal R, I.IsPrime
P : Ideal R
hP₁ : P.IsPrime
hP₂ : ∀ (y : Ideal R), (fun (I : Ideal R) ↦ I.IsPrime) y → y = P
obtain : ∀ (P : Ideal R) (hP₃ : P.IsMaximal), KrullDimLE 0 R ∧ IsLocalRing R
⊢ KrullDimLE 0 R ∧ IsLocalRing R

Tactic:
obtain ⟨P, hP₃, -⟩ := P.exists_le_maximal hP₁.ne_top

Premises used: Ideal.exists_le_maximal, Ideal.IsPrime.ne_top
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
tfae_1_to_3 : ∀ (a : KrullDimLE 0 R ∧ IsLocalRing R) (x : R), IsNilpotent x ↔ ¬IsUnit x
tfae_3_to_4 : ∀ (a : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x), Ideal.IsMaximal (nilradical R)
tfae_4_to_2 : ∀ (a : Ideal.IsMaximal (nilradical R)), ∃! I : Ideal R, I.IsPrime
P_1 : Ideal R
hP₁ : P_1.IsPrime
hP₂ : ∀ (y : Ideal R), (fun (I : Ideal R) ↦ I.IsPrime) y → y = P_1
P : Ideal R
hP₃ : P.IsMaximal
⊢ P.IsPrime

Tactic:
obtain rfl := hP₂ P hP₃.isPrime

Premises used: Ideal.IsMaximal.isPrime, rfl
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
tfae_1_to_3 : ∀ (a : KrullDimLE 0 R ∧ IsLocalRing R) (x : R), IsNilpotent x ↔ ¬IsUnit x
tfae_3_to_4 : ∀ (a : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x), Ideal.IsMaximal (nilradical R)
tfae_4_to_2 : ∀ (a : Ideal.IsMaximal (nilradical R)), ∃! I : Ideal R, I.IsPrime
P_1 : Ideal R
hP₁ : P_1.IsPrime
hP₂ : ∀ (y : Ideal R), (fun (I : Ideal R) ↦ I.IsPrime) y → y = P_1
P : Ideal R
hP₃ : P.IsMaximal
hP₄ : P.IsPrime
⊢ ∀ (y : Ideal R), (fun (I : Ideal R) ↦ I.IsPrime) y → y = P

Tactic:
obtain rfl := hP₂ P hP₃.isPrime

Premises used: Ideal.IsMaximal.isPrime, rfl
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
tfae_1_to_3 : ∀ (a : KrullDimLE 0 R ∧ IsLocalRing R) (x : R), IsNilpotent x ↔ ¬IsUnit x
tfae_3_to_4 : ∀ (a : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x), Ideal.IsMaximal (nilradical R)
tfae_4_to_2 : ∀ (a : Ideal.IsMaximal (nilradical R)), ∃! I : Ideal R, I.IsPrime
P : Ideal R
hP₃ : P.IsMaximal
hP₁ : P.IsPrime
hP₂ : ∀ (y : Ideal R), (fun (I : Ideal R) ↦ I.IsPrime) y → y = P
⊢ KrullDimLE 0 R ∧ IsLocalRing R

Tactic:
⟨.mk₀ fun Q h ↦ hP₂ Q h ▸ hP₃, .of_unique_max_ideal ⟨P, hP₃, fun Q h ↦ hP₂ Q h.isPrime⟩⟩

Premises used: Ring.KrullDimLE.mk₀, IsLocalRing.of_unique_max_ideal, Eq.rec, Ideal.IsMaximal.isPrime, And.intro, Exists.intro, Eq.symm
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
tfae_1_to_3 : ∀ (a : KrullDimLE 0 R ∧ IsLocalRing R) (x : R), IsNilpotent x ↔ ¬IsUnit x
tfae_3_to_4 : ∀ (a : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x), Ideal.IsMaximal (nilradical R)
tfae_4_to_2 : ∀ (a : Ideal.IsMaximal (nilradical R)), ∃! I : Ideal R, I.IsPrime
P : Ideal R
hP₃ : P.IsMaximal
hP₁ : P.IsPrime
hP₂ : ∀ (y : Ideal R), (fun (I : Ideal R) ↦ I.IsPrime) y → y = P
⊢ KrullDimLE 0 R ∧ IsLocalRing R

Tactic:
exact ⟨.mk₀ fun Q h ↦ hP₂ Q h ▸ hP₃, .of_unique_max_ideal ⟨P, hP₃, fun Q h ↦ hP₂ Q h.isPrime⟩⟩
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
tfae_1_to_3 : ∀ (a : KrullDimLE 0 R ∧ IsLocalRing R) (x : R), IsNilpotent x ↔ ¬IsUnit x
tfae_3_to_4 : ∀ (a : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x), Ideal.IsMaximal (nilradical R)
tfae_4_to_2 : ∀ (a : Ideal.IsMaximal (nilradical R)), ∃! I : Ideal R, I.IsPrime
⊢ ∀ (a : ∃! I : Ideal R, I.IsPrime), KrullDimLE 0 R ∧ IsLocalRing R

Tactic:
tfae_have 2 → 1 := by
    rintro ⟨P, hP₁, hP₂⟩
    obtain ⟨P, hP₃, -⟩ := P.exists_le_maximal hP₁.ne_top
    obtain rfl := hP₂ P hP₃.isPrime
    exact ⟨.mk₀ fun Q h ↦ hP₂ Q h ▸ hP₃, .of_unique_max_ideal ⟨P, hP₃, fun Q h ↦ hP₂ Q h.isPrime⟩⟩
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
tfae_1_to_3 : ∀ (a : KrullDimLE 0 R ∧ IsLocalRing R) (x : R), IsNilpotent x ↔ ¬IsUnit x
tfae_3_to_4 : ∀ (a : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x), Ideal.IsMaximal (nilradical R)
tfae_4_to_2 : ∀ (a : Ideal.IsMaximal (nilradical R)), ∃! I : Ideal R, I.IsPrime
tfae_2_to_1 : ∀ (a : ∃! I : Ideal R, I.IsPrime), KrullDimLE 0 R ∧ IsLocalRing R
⊢ [KrullDimLE 0 R ∧ IsLocalRing R, ∃! I : Ideal R, I.IsPrime, ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x,
      Ideal.IsMaximal (nilradical R)].TFAE

Tactic:
tfae_finish
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : Ring.KrullDimLE 0 R
inst_2 : IsLocalRing R
M : Submonoid R
⊢ ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x

Tactic:
have := ((Ring.krullDimLE_zero_and_isLocalRing_tfae R).out 0 2 rfl rfl).mp ⟨‹_›, ‹_›⟩
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : Ring.KrullDimLE 0 R
inst_2 : IsLocalRing R
M : Submonoid R
this : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x
⊢ M ≤ IsUnit.submonoid R ↔ 0 ∉ M

Tactic:
⟨fun h₁ h₂ ↦ not_isUnit_zero (h₁ h₂),
    fun H x hx ↦ (this x).not_left.mp fun ⟨n, hn⟩ ↦ H (hn ▸ pow_mem hx n)⟩

Premises used: False, Eq.rec, not_isUnit_zero, Iff.not_left, Iff.intro, pow_mem, Iff.mp, _private.0.le_isUnit_iff_zero_notMem.match_1
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : Ring.KrullDimLE 0 R
inst_2 : IsLocalRing R
M : Submonoid R
this : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x
⊢ M ≤ IsUnit.submonoid R ↔ 0 ∉ M

Tactic:
exact ⟨fun h₁ h₂ ↦ not_isUnit_zero (h₁ h₂),
    fun H x hx ↦ (this x).not_left.mp fun ⟨n, hn⟩ ↦ H (hn ▸ pow_mem hx n)⟩
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : KrullDimLE 0 R
inst_2 : IsLocalRing R
⊢ ∃! I : Ideal R, I.IsPrime

Tactic:
((Ring.krullDimLE_zero_and_isLocalRing_tfae R).out 0 1 rfl rfl).mp ⟨‹_›, ‹_›⟩

Premises used: Ring.krullDimLE_zero_and_isLocalRing_tfae, List.TFAE.out, And.intro, Iff.mp, OfNat.ofNat, rfl
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : KrullDimLE 0 R
inst_2 : IsLocalRing R
J : Ideal R
inst_3 : J.IsPrime
⊢ J = IsLocalRing.maximalIdeal R

Tactic:
(((Ring.krullDimLE_zero_and_isLocalRing_tfae R).out 0 1 rfl rfl).mp ⟨‹_›, ‹_›⟩).unique
    ‹_› inferInstance

Premises used: Ring.krullDimLE_zero_and_isLocalRing_tfae, inferInstance, List.TFAE.out, And.intro, Iff.mp, ExistsUnique.unique, OfNat.ofNat, rfl
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : KrullDimLE 0 R
inst_2 : IsLocalRing R
I : Ideal R
hI : I ≠ ⊤
rw : Eq (α := Ideal R) (sInf {J : Ideal R | I ≤ J ∧ J.IsPrime}) (IsLocalRing.maximalIdeal R)
⊢ I.radical = IsLocalRing.maximalIdeal R

Tactic:
rw [Ideal.radical_eq_sInf]

Premises used: Ideal.radical_eq_sInf
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : KrullDimLE 0 R
inst_2 : IsLocalRing R
I : Ideal R
hI : I ≠ ⊤
⊢ Eq (α := Ideal R) (sInf {J : Ideal R | I ≤ J ∧ J.IsPrime}) (IsLocalRing.maximalIdeal R)

Tactic:
(sInf_le ?_).antisymm (le_sInf ?_)

Premises used: LE.le.antisymm, sInf_le, le_sInf
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : KrullDimLE 0 R
inst_2 : IsLocalRing R
I : Ideal R
hI : I ≠ ⊤
refine_1 : Membership.mem (γ := Set (Ideal R)) {J : Ideal R | I ≤ J ∧ J.IsPrime} (IsLocalRing.maximalIdeal R)
refine_2 :
  ∀ (b : Ideal R),
    Membership.mem (γ := Set (Ideal R)) {J : Ideal R | I ≤ J ∧ J.IsPrime} b → IsLocalRing.maximalIdeal R ≤ b
⊢ Eq (α := Ideal R) (sInf {J : Ideal R | I ≤ J ∧ J.IsPrime}) (IsLocalRing.maximalIdeal R)

Tactic:
refine (sInf_le ?_).antisymm (le_sInf ?_)

Premises used: LE.le.antisymm, sInf_le, le_sInf
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : KrullDimLE 0 R
inst_2 : IsLocalRing R
I : Ideal R
hI : I ≠ ⊤
⊢ Membership.mem (γ := Set (Ideal R)) {J : Ideal R | I ≤ J ∧ J.IsPrime} (IsLocalRing.maximalIdeal R)

Tactic:
⟨IsLocalRing.le_maximalIdeal hI, inferInstance⟩

Premises used: Set.Mem, setOf, inferInstance, And.intro, IsLocalRing.le_maximalIdeal
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : KrullDimLE 0 R
inst_2 : IsLocalRing R
I : Ideal R
hI : I ≠ ⊤
⊢ Membership.mem (γ := Set (Ideal R)) {J : Ideal R | I ≤ J ∧ J.IsPrime} (IsLocalRing.maximalIdeal R)

Tactic:
exact ⟨IsLocalRing.le_maximalIdeal hI, inferInstance⟩
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : KrullDimLE 0 R
inst_2 : IsLocalRing R
I : Ideal R
hI : I ≠ ⊤
J : Ideal R
h₁ : I ≤ J
h₂ : J.IsPrime
⊢ IsLocalRing.maximalIdeal R ≤ J

Tactic:
(Ring.KrullDimLE.eq_maximalIdeal_of_isPrime J).ge

Premises used: Ring.KrullDimLE.eq_maximalIdeal_of_isPrime, Eq.ge
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : KrullDimLE 0 R
inst_2 : IsLocalRing R
I : Ideal R
hI : I ≠ ⊤
J : Ideal R
h₁ : I ≤ J
h₂ : J.IsPrime
⊢ IsLocalRing.maximalIdeal R ≤ J

Tactic:
exact (Ring.KrullDimLE.eq_maximalIdeal_of_isPrime J).ge

Premises used: Ring.KrullDimLE.eq_maximalIdeal_of_isPrime, Eq.ge
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : KrullDimLE 0 R
inst_2 : IsLocalRing R
⊢ Subsingleton.{u_1 + 1} (PrimeSpectrum R)

Tactic:
⟨fun x y ↦ PrimeSpectrum.ext <|
    (eq_maximalIdeal_of_isPrime x.1).trans (eq_maximalIdeal_of_isPrime y.1).symm⟩

Premises used: Ring.KrullDimLE.eq_maximalIdeal_of_isPrime, PrimeSpectrum.ext, Subsingleton.intro, Eq.symm, PrimeSpectrum.asIdeal, Eq.trans
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : KrullDimLE 0 R
inst_2 : IsLocalRing R
x : R
⊢ IsNilpotent x ↔ Membership.mem (γ := Ideal R) (IsLocalRing.maximalIdeal R) x

Tactic:
((Ring.krullDimLE_zero_and_isLocalRing_tfae R).out 0 2 rfl rfl).mp ⟨‹_›, ‹_›⟩ x

Premises used: Set.Mem, Ring.krullDimLE_zero_and_isLocalRing_tfae, setOf, nonunits, List.TFAE.out, And.intro, Iff.mp, OfNat.ofNat, Not, rfl
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : KrullDimLE 0 R
inst_2 : IsLocalRing R
x : R
⊢ IsNilpotent x ↔ x ∈ nonunits R

Tactic:
isNilpotent_iff_mem_maximalIdeal

Premises used: Set.Mem, setOf, nonunits, Ring.KrullDimLE.isNilpotent_iff_mem_maximalIdeal, Not