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
inst_2 : IsLocalRing R
⊢ Eq (α := Ideal R) (nilradical R) (IsLocalRing.maximalIdeal R)

Tactic:
Ideal.ext fun _ ↦ isNilpotent_iff_mem_maximalIdeal

Premises used: Ideal.ext, Ring.KrullDimLE.isNilpotent_iff_mem_maximalIdeal
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : Ideal.IsMaximal (nilradical R)
⊢ IsLocalRing R

Tactic:
(((Ring.krullDimLE_zero_and_isLocalRing_tfae R).out 3 0 rfl rfl).mp ‹_›).2

Premises used: Ring.krullDimLE_zero_and_isLocalRing_tfae, List.TFAE.out, And.right, Iff.mp, OfNat.ofNat, rfl
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : Ideal.IsMaximal (nilradical R)
⊢ KrullDimLE 0 R

Tactic:
(((Ring.krullDimLE_zero_and_isLocalRing_tfae R).out 3 0 rfl rfl).mp ‹_›).1

Premises used: Ring.krullDimLE_zero_and_isLocalRing_tfae, List.TFAE.out, And.left, Iff.mp, OfNat.ofNat, rfl
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
p : Ideal R
hp : Membership.mem (γ := Set (Ideal R)) (minimalPrimes R) p
S : Type u_2
inst_1 : CommSemiring S
inst_2 : Algebra R S
inst_3 : IsLocalization.AtPrime S p
⊢ KrullDimLE 0 S

Tactic:
have := IsLocalization.subsingleton_primeSpectrum_of_mem_minimalPrimes p hp S
  ⟨Order.krullDim_nonpos_of_subsingleton⟩

Premises used: Order.KrullDimLE.mk, IsLocalization.subsingleton_primeSpectrum_of_mem_minimalPrimes, Order.krullDim_nonpos_of_subsingleton
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommSemiring R
inst_1 : KrullDimLE 0 R
inst_2 : IsReduced R
inst_3 : IsLocalRing R
rw : Eq (α := Ideal R) (IsLocalRing.maximalIdeal R) ⊥
rw₁ : Eq (α := Ideal R) (nilradical R) ⊥
rw₂ : Eq.{u_1 + 1} (α := Ideal R) 0 ⊥
rw₃ rw₄ : Eq.{u_1 + 1} (α := Ideal R) ⊥ ⊥
⊢ IsField R

Tactic:
rw [IsLocalRing.isField_iff_maximalIdeal_eq, ← nilradical_eq_maximalIdeal,
    nilradical_eq_zero, Ideal.zero_eq_bot]

Premises used: nilradical_eq_zero, Ring.KrullDimLE.nilradical_eq_maximalIdeal, IsLocalRing.isField_iff_maximalIdeal_eq, Ideal.zero_eq_bot
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_2
inst : CommRing R
inst_1 : IsReduced R
inst_2 : Nontrivial R
⊢ Subsingleton.{u_2 + 1} (PrimeSpectrum R) ↔ IsField R

Tactic:
⟨fun H ↦ ?_, fun H ↦ letI := H.toField; inferInstance⟩

Premises used: inferInstance, Iff.intro
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_2
inst : CommRing R
inst_1 : IsReduced R
inst_2 : Nontrivial R
refine : ∀ (H : Subsingleton.{u_2 + 1} (PrimeSpectrum R)), IsField R
⊢ Subsingleton.{u_2 + 1} (PrimeSpectrum R) ↔ IsField R

Tactic:
refine ⟨fun H ↦ ?_, fun H ↦ letI := H.toField; inferInstance⟩
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_2
inst : CommRing R
inst_1 : IsReduced R
inst_2 : Nontrivial R
H : Subsingleton.{u_2 + 1} (PrimeSpectrum R)
⊢ Subsingleton.{u_2 + 1} (MaximalSpectrum R)

Tactic:
have : Subsingleton (MaximalSpectrum R) := MaximalSpectrum.toPrimeSpectrum_injective.subsingleton
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_2
inst : CommRing R
inst_1 : IsReduced R
inst_2 : Nontrivial R
H : Subsingleton.{u_2 + 1} (PrimeSpectrum R)
this : Subsingleton.{u_2 + 1} (MaximalSpectrum R)
⊢ IsLocalRing R

Tactic:
have : IsLocalRing R := .of_singleton_maximalSpectrum
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_2
inst : CommRing R
inst_1 : IsReduced R
inst_2 : Nontrivial R
H : Subsingleton.{u_2 + 1} (PrimeSpectrum R)
this_1 : Subsingleton.{u_2 + 1} (MaximalSpectrum R)
this : IsLocalRing R
⊢ IsField R

Tactic:
Ring.KrullDimLE.isField_of_isReduced

Premises used: Ring.KrullDimLE.isField_of_isReduced
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_2
inst : CommRing R
inst_1 : IsReduced R
inst_2 : Nontrivial R
H : Subsingleton.{u_2 + 1} (PrimeSpectrum R)
this_1 : Subsingleton.{u_2 + 1} (MaximalSpectrum R)
this : IsLocalRing R
⊢ IsField R

Tactic:
exact Ring.KrullDimLE.isField_of_isReduced

Premises used: Ring.KrullDimLE.isField_of_isReduced
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
inst_1 : Ring.KrullDimLE 0 R
⊢ I.jacobson = I.radical

Tactic:
simp [jacobson, radical_eq_sInf, Ideal.isMaximal_iff_isPrime]

Premises used: Ideal.jacobson, Ideal.radical_eq_sInf, Ideal.isMaximal_iff_isPrime
-- Module: Mathlib.RingTheory.KrullDimension.Zero
Goal:
R : Type u_1
inst : CommRing R
I_1 : Ideal R
inst_1 : Ring.KrullDimLE 0 R
I : Ideal R
hI : I.IsRadical
rw : I.radical = I
rw₁ rw₂ : I = I
⊢ I.jacobson = I

Tactic:
rw [I.jacobson_eq_radical, hI.radical]

Premises used: Ideal.jacobson_eq_radical, Ideal.IsRadical.radical
-- Module: Mathlib.RingTheory.Lasker
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
IH :
  ∀ s_1 ∈ s,
    ∀ (hs : (s.erase s_1).inf id = N),
      ∃ t ⊆ s.erase s_1, t.inf id = N ∧ ∀ ⦃J : Submodule R M⦄, J ∈ t → ¬(t.erase J).inf id ≤ J
hs : s.inf id = N
pos :
  ∀ (H : ∀ J ∈ s, ¬(s.erase J).inf id ≤ J),
    ∃ t ⊆ s, t.inf id = N ∧ ∀ ⦃J : Submodule R M⦄, J ∈ t → ¬(t.erase J).inf id ≤ J
neg :
  ∀ (H : ∃ J ∈ s, (s.erase J).inf id ≤ J),
    ∃ t ⊆ s, t.inf id = N ∧ ∀ ⦃J : Submodule R M⦄, J ∈ t → ¬(t.erase J).inf id ≤ J
⊢ ∃ t ⊆ s, t.inf id = N ∧ ∀ ⦃J : Submodule R M⦄, J ∈ t → ¬(t.erase J).inf id ≤ J

Tactic:
by_cases! H : ∀ J ∈ s, ¬ (s.erase J).inf id ≤ J
-- Module: Mathlib.RingTheory.Lasker
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
IH :
  ∀ s_1 ∈ s,
    ∀ (hs : (s.erase s_1).inf id = N),
      ∃ t ⊆ s.erase s_1, t.inf id = N ∧ ∀ ⦃J : Submodule R M⦄, J ∈ t → ¬(t.erase J).inf id ≤ J
hs : s.inf id = N
H : ∀ J ∈ s, ¬(s.erase J).inf id ≤ J
⊢ ∃ t ⊆ s, t.inf id = N ∧ ∀ ⦃J : Submodule R M⦄, J ∈ t → ¬(t.erase J).inf id ≤ J

Tactic:
⟨s, Finset.Subset.rfl, hs, H⟩

Premises used: Finset.Subset.rfl, And.intro, Exists.intro
-- Module: Mathlib.RingTheory.Lasker
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
IH :
  ∀ s_1 ∈ s,
    ∀ (hs : (s.erase s_1).inf id = N),
      ∃ t ⊆ s.erase s_1, t.inf id = N ∧ ∀ ⦃J : Submodule R M⦄, J ∈ t → ¬(t.erase J).inf id ≤ J
hs : s.inf id = N
H : ∀ J ∈ s, ¬(s.erase J).inf id ≤ J
⊢ ∃ t ⊆ s, t.inf id = N ∧ ∀ ⦃J : Submodule R M⦄, J ∈ t → ¬(t.erase J).inf id ≤ J

Tactic:
exact ⟨s, Finset.Subset.rfl, hs, H⟩
-- Module: Mathlib.RingTheory.Lasker
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
IH :
  ∀ s_1 ∈ s,
    ∀ (hs : (s.erase s_1).inf id = N),
      ∃ t ⊆ s.erase s_1, t.inf id = N ∧ ∀ ⦃J : Submodule R M⦄, J ∈ t → ¬(t.erase J).inf id ≤ J
hs : s.inf id = N
H : ∃ J ∈ s, (s.erase J).inf id ≤ J
neg :
  ∀ (J : Submodule R M) (hJ : J ∈ s) (hJ' : (s.erase J).inf id ≤ J),
    ∃ t ⊆ s, t.inf id = N ∧ ∀ ⦃J : Submodule R M⦄, J ∈ t → ¬(t.erase J).inf id ≤ J
⊢ ∃ t ⊆ s, t.inf id = N ∧ ∀ ⦃J : Submodule R M⦄, J ∈ t → ¬(t.erase J).inf id ≤ J

Tactic:
obtain ⟨J, hJ, hJ'⟩ := H
-- Module: Mathlib.RingTheory.Lasker
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
IH :
  ∀ s_1 ∈ s,
    ∀ (hs : (s.erase s_1).inf id = N),
      ∃ t ⊆ s.erase s_1, t.inf id = N ∧ ∀ ⦃J : Submodule R M⦄, J ∈ t → ¬(t.erase J).inf id ≤ J
hs : s.inf id = N
J : Submodule R M
hJ : J ∈ s
hJ' : (s.erase J).inf id ≤ J
⊢ ∃ t ⊆ s, t.inf id = N ∧ ∀ ⦃J : Submodule R M⦄, J ∈ t → ¬(t.erase J).inf id ≤ J

Tactic:
(IH _ hJ ?_).imp
      fun t ↦ And.imp_left (fun ht ↦ ht.trans (Finset.erase_subset _ _))

Premises used: And.imp_left, Exists.imp
-- Module: Mathlib.RingTheory.Lasker
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
IH :
  ∀ s_1 ∈ s,
    ∀ (hs : (s.erase s_1).inf id = N),
      ∃ t ⊆ s.erase s_1, t.inf id = N ∧ ∀ ⦃J : Submodule R M⦄, J ∈ t → ¬(t.erase J).inf id ≤ J
hs : s.inf id = N
J : Submodule R M
hJ : J ∈ s
hJ' : (s.erase J).inf id ≤ J
neg : (s.erase J).inf id = N
⊢ ∃ t ⊆ s, t.inf id = N ∧ ∀ ⦃J : Submodule R M⦄, J ∈ t → ¬(t.erase J).inf id ≤ J

Tactic:
refine (IH _ hJ ?_).imp
      fun t ↦ And.imp_left (fun ht ↦ ht.trans (Finset.erase_subset _ _))

Premises used: And.imp_left, Exists.imp
-- Module: Mathlib.RingTheory.Lasker
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
IH :
  ∀ s_1 ∈ s,
    ∀ (hs : (s.erase s_1).inf id = N),
      ∃ t ⊆ s.erase s_1, t.inf id = N ∧ ∀ ⦃J : Submodule R M⦄, J ∈ t → ¬(t.erase J).inf id ≤ J
hs : s.inf id = N
J : Submodule R M
hJ : J ∈ s
hJ' : (s.erase J).inf id ≤ J
⊢ (insert J (s.erase J)).inf id = N

Tactic:
rw [← Finset.insert_erase hJ] at hs

Premises used: Finset.insert_erase
-- Module: Mathlib.RingTheory.Lasker
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
IH :
  ∀ s_1 ∈ s,
    ∀ (hs : (s.erase s_1).inf id = N),
      ∃ t ⊆ s.erase s_1, t.inf id = N ∧ ∀ ⦃J : Submodule R M⦄, J ∈ t → ¬(t.erase J).inf id ≤ J
hs : s.inf id = N
J : Submodule R M
hJ : J ∈ s
hJ' : (s.erase J).inf id ≤ J
hs₁ : (insert J (s.erase J)).inf id = N
⊢ J ∈ s

Tactic:
rw [← Finset.insert_erase hJ] at hs

Premises used: Finset.insert_erase
-- Module: Mathlib.RingTheory.Lasker
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
IH :
  ∀ s_1 ∈ s,
    ∀ (hs : (s.erase s_1).inf id = N),
      ∃ t ⊆ s.erase s_1, t.inf id = N ∧ ∀ ⦃J : Submodule R M⦄, J ∈ t → ¬(t.erase J).inf id ≤ J
hs : s.inf id = N
J : Submodule R M
hJ : J ∈ s
hJ' : (s.erase J).inf id ≤ J
hs₁ : (insert J (s.erase J)).inf id = N
hJ₁ : J ∈ s
⊢ (s.erase J).inf id ≤ J

Tactic:
rw [← Finset.insert_erase hJ] at hs

Premises used: Finset.insert_erase
-- Module: Mathlib.RingTheory.Lasker
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
IH :
  ∀ s_1 ∈ s,
    ∀ (hs : (s.erase s_1).inf id = N),
      ∃ t ⊆ s.erase s_1, t.inf id = N ∧ ∀ ⦃J : Submodule R M⦄, J ∈ t → ¬(t.erase J).inf id ≤ J
J : Submodule R M
hs : (insert J (s.erase J)).inf id = N
hJ : J ∈ s
hJ' : (s.erase J).inf id ≤ J
⊢ (s.erase J).inf id = N

Tactic:
simp [← hs, hJ']
-- Module: Mathlib.RingTheory.Lasker
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
H :
  ∀ (s : Finset (Submodule R M))
    (IH :
      ∀ s_1 ∈ s,
        ∀ (hs : (s.erase s_1).inf id = N),
          ∃ t ⊆ s.erase s_1, t.inf id = N ∧ ∀ ⦃J : Submodule R M⦄, J ∈ t → ¬(t.erase J).inf id ≤ J)
    (hs : s.inf id = N), ∃ t ⊆ s, t.inf id = N ∧ ∀ ⦃J : Submodule R M⦄, J ∈ t → ¬(t.erase J).inf id ≤ J
⊢ ∃ t ⊆ s, t.inf id = N ∧ ∀ ⦃J : Submodule R M⦄, J ∈ t → ¬(t.erase J).inf id ≤ J

Tactic:
induction s using Finset.eraseInduction with
  | H s IH =>
    by_cases! H : ∀ J ∈ s, ¬ (s.erase J).inf id ≤ J
    · exact ⟨s, Finset.Subset.rfl, hs, H⟩
    obtain ⟨J, hJ, hJ'⟩ := H
    refine (IH _ hJ ?_).imp
      fun t ↦ And.imp_left (fun ht ↦ ht.trans (Finset.erase_subset _ _))
    rw [← Finset.insert_erase hJ] at hs
    simp [← hs, hJ']

Premises used: Finset.eraseInduction
-- Module: Mathlib.RingTheory.Lasker
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
⊢ ∃ (t : Finset (Submodule R M)),
    t.inf id = N ∧
      (∀ ⦃J : Submodule R M⦄, J ∈ t → J.IsPrimary) ∧
        (↑t : Set (Submodule R M)).Pairwise
          ((fun (x1 x2 : Ideal R) ↦ x1 ≠ x2) on fun (J : Submodule R M) ↦ (J.colon Set.univ).radical)

Tactic:
⟨(s.image fun J ↦ {I ∈ s | (I.colon .univ).radical = (J.colon .univ).radical}).image
    fun t ↦ t.inf id, ?_, ?_, ?_⟩

Premises used: And.intro, Exists.intro, Finset.filter, Eq, Finset.image
-- Module: Mathlib.RingTheory.Lasker
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
refine_1 :
  Finset.inf
      (Finset.image (fun (t : Finset (Submodule R M)) ↦ t.inf id)
        (Finset.image (fun (J : Submodule R M) ↦ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical}) s))
      id =
    N
refine_2 :
  ∀ ⦃J : Submodule R M⦄,
    Membership.mem (γ := Finset (Submodule R M))
        (Finset.image (fun (t : Finset (Submodule R M)) ↦ t.inf id)
          (Finset.image (fun (J : Submodule R M) ↦ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical})
            s))
        J →
      J.IsPrimary
refine_3 :
  (↑(Finset.image (fun (t : Finset (Submodule R M)) ↦ t.inf id)
            (Finset.image (fun (J : Submodule R M) ↦ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical})
              s)) :
        Set (Submodule R M)).Pairwise
    ((fun (x1 x2 : Ideal R) ↦ x1 ≠ x2) on fun (J : Submodule R M) ↦ (J.colon Set.univ).radical)
⊢ ∃ (t : Finset (Submodule R M)),
    t.inf id = N ∧
      (∀ ⦃J : Submodule R M⦄, J ∈ t → J.IsPrimary) ∧
        (↑t : Set (Submodule R M)).Pairwise
          ((fun (x1 x2 : Ideal R) ↦ x1 ≠ x2) on fun (J : Submodule R M) ↦ (J.colon Set.univ).radical)

Tactic:
refine ⟨(s.image fun J ↦ {I ∈ s | (I.colon .univ).radical = (J.colon .univ).radical}).image
    fun t ↦ t.inf id, ?_, ?_, ?_⟩
-- Module: Mathlib.RingTheory.Lasker
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
refine_1 :
  ∀ (x : M),
    Membership.mem (γ := Submodule R M)
        (Finset.inf
          (Finset.image (fun (t : Finset (Submodule R M)) ↦ t.inf id)
            (Finset.image (fun (J : Submodule R M) ↦ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical})
              s))
          id)
        x ↔
      x ∈ N
⊢ Finset.inf
      (Finset.image (fun (t : Finset (Submodule R M)) ↦ t.inf id)
        (Finset.image (fun (J : Submodule R M) ↦ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical}) s))
      id =
    N

Tactic:
ext
-- Module: Mathlib.RingTheory.Lasker
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
x : M
⊢ Membership.mem (γ := Submodule R M)
      (Finset.inf
        (Finset.image (fun (t : Finset (Submodule R M)) ↦ t.inf id)
          (Finset.image (fun (J : Submodule R M) ↦ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical})
            s))
        id)
      x ↔
    x ∈ N

Tactic:
grind [Finset.inf_image, Submodule.mem_finsetInf]

Premises used: Finset.inf_image, Submodule.mem_finsetInf
-- Module: Mathlib.RingTheory.Lasker
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
refine_2 : ∀ a ∈ s, ({I ∈ s | (I.colon Set.univ).radical = (a.colon Set.univ).radical}.inf id).IsPrimary
⊢ ∀ ⦃J : Submodule R M⦄,
    Membership.mem (γ := Finset (Submodule R M))
        (Finset.image (fun (t : Finset (Submodule R M)) ↦ t.inf id)
          (Finset.image (fun (J : Submodule R M) ↦ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical})
            s))
        J →
      J.IsPrimary

Tactic:
simp only [Finset.mem_image, exists_exists_and_eq_and, forall_exists_index, and_imp,
    forall_apply_eq_imp_iff₂]

Premises used: Finset.mem_image, and_imp, forall_exists_index, exists_exists_and_eq_and, forall_apply_eq_imp_iff₂
-- Module: Mathlib.RingTheory.Lasker
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
J : Submodule R M
hJ : J ∈ s
⊢ ∀ ⦃y : Submodule R M⦄,
    y ∈ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical} →
      ((id y).colon Set.univ).radical = ((id J).colon Set.univ).radical

Tactic:
simp
-- Module: Mathlib.RingTheory.Lasker
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
J : Submodule R M
hJ : J ∈ s
refine_2 : J ∈ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical}
refine_2₁ :
  ∀ ⦃y : Submodule R M⦄, y ∈ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical} → (id y).IsPrimary
⊢ ({I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical}.inf id).IsPrimary

Tactic:
refine isPrimary_finsetInf (i := J) ?_ ?_ (by simp)

Premises used: Submodule.isPrimary_finsetInf
-- Module: Mathlib.RingTheory.Lasker
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
J : Submodule R M
hJ : J ∈ s
⊢ J ∈ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical}

Tactic:
simp [hJ]
-- Module: Mathlib.RingTheory.Lasker
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
J : Submodule R M
hJ : J ∈ s
refine_2 : ∀ ⦃y : Submodule R M⦄, y ∈ s → (y.colon Set.univ).radical = (J.colon Set.univ).radical → y.IsPrimary
⊢ ∀ ⦃y : Submodule R M⦄, y ∈ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical} → (id y).IsPrimary

Tactic:
simp only [Finset.mem_filter, id_eq, and_imp]

Premises used: and_imp, Finset.mem_filter, id_eq
-- Module: Mathlib.RingTheory.Lasker
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
J : Submodule R M
hJ : J ∈ s
y : Submodule R M
hy : y ∈ s
⊢ (y.colon Set.univ).radical = (J.colon Set.univ).radical → y.IsPrimary

Tactic:
simp [hs' hy]
-- Module: Mathlib.RingTheory.Lasker
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
I : Submodule R M
hI :
  I ∈
    (↑(Finset.image (fun (t : Finset (Submodule R M)) ↦ t.inf id)
          (Finset.image (fun (J : Submodule R M) ↦ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical})
            s)) :
      Set (Submodule R M))
J : Submodule R M
hJ :
  J ∈
    (↑(Finset.image (fun (t : Finset (Submodule R M)) ↦ t.inf id)
          (Finset.image (fun (J : Submodule R M) ↦ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical})
            s)) :
      Set (Submodule R M))
hIJ : I ≠ J
⊢ ∃ a ∈ s, {I ∈ s | (I.colon Set.univ).radical = (a.colon Set.univ).radical}.inf id = I

Tactic:
simp only [Finset.coe_image, Set.mem_image, Finset.mem_coe, exists_exists_and_eq_and] at hI hJ

Premises used: Set.mem_image, Finset.coe_image, exists_exists_and_eq_and, Finset.mem_coe
-- Module: Mathlib.RingTheory.Lasker
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
I : Submodule R M
hI :
  I ∈
    (↑(Finset.image (fun (t : Finset (Submodule R M)) ↦ t.inf id)
          (Finset.image (fun (J : Submodule R M) ↦ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical})
            s)) :
      Set (Submodule R M))
J : Submodule R M
hJ :
  J ∈
    (↑(Finset.image (fun (t : Finset (Submodule R M)) ↦ t.inf id)
          (Finset.image (fun (J : Submodule R M) ↦ {I ∈ s | (I.colon Set.univ).radical = (J.colon Set.univ).radical})
            s)) :
      Set (Submodule R M))
hIJ : I ≠ J
hI₁ : ∃ a ∈ s, {I ∈ s | (I.colon Set.univ).radical = (a.colon Set.univ).radical}.inf id = I
⊢ ∃ a ∈ s, {I ∈ s | (I.colon Set.univ).radical = (a.colon Set.univ).radical}.inf id = J

Tactic:
simp only [Finset.coe_image, Set.mem_image, Finset.mem_coe, exists_exists_and_eq_and] at hI hJ

Premises used: Set.mem_image, Finset.coe_image, exists_exists_and_eq_and, Finset.mem_coe
-- Module: Mathlib.RingTheory.Lasker
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
I J : Submodule R M
hIJ : I ≠ J
hI : ∃ a ∈ s, {I ∈ s | (I.colon Set.univ).radical = (a.colon Set.univ).radical}.inf id = I
hJ : ∃ a ∈ s, {I ∈ s | (I.colon Set.univ).radical = (a.colon Set.univ).radical}.inf id = J
refine_3 :
  ∀ (I' : Submodule R M) (hI' : I' ∈ s)
    (hI : {I ∈ s | (I.colon Set.univ).radical = (I'.colon Set.univ).radical}.inf id = I),
    ((fun (x1 x2 : Ideal R) ↦ x1 ≠ x2) on fun (J : Submodule R M) ↦ (J.colon Set.univ).radical) I J
⊢ ((fun (x1 x2 : Ideal R) ↦ x1 ≠ x2) on fun (J : Submodule R M) ↦ (J.colon Set.univ).radical) I J

Tactic:
obtain ⟨I', hI', hI⟩ := hI
-- Module: Mathlib.RingTheory.Lasker
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
I J : Submodule R M
hIJ : I ≠ J
hJ : ∃ a ∈ s, {I ∈ s | (I.colon Set.univ).radical = (a.colon Set.univ).radical}.inf id = J
I' : Submodule R M
hI' : I' ∈ s
hI : {I ∈ s | (I.colon Set.univ).radical = (I'.colon Set.univ).radical}.inf id = I
refine_3 :
  ∀ (J' : Submodule R M) (hJ' : J' ∈ s)
    (hJ : {I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id = J),
    ((fun (x1 x2 : Ideal R) ↦ x1 ≠ x2) on fun (J : Submodule R M) ↦ (J.colon Set.univ).radical) I J
⊢ ((fun (x1 x2 : Ideal R) ↦ x1 ≠ x2) on fun (J : Submodule R M) ↦ (J.colon Set.univ).radical) I J

Tactic:
obtain ⟨J', hJ', hJ⟩ := hJ
-- Module: Mathlib.RingTheory.Lasker
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
I J : Submodule R M
hIJ : I ≠ J
I' : Submodule R M
hI' : I' ∈ s
hI : {I ∈ s | (I.colon Set.univ).radical = (I'.colon Set.univ).radical}.inf id = I
J' : Submodule R M
hJ' : J' ∈ s
hJ : {I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id = J
refine_3 : ∀ (hIJ : (I.colon Set.univ).radical = (J.colon Set.univ).radical), I = J
⊢ ¬(I.colon Set.univ).radical = (J.colon Set.univ).radical

Tactic:
contrapose hIJ
-- Module: Mathlib.RingTheory.Lasker
Goal:
R : Type u_1
M : Type u_2
inst : CommSemiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
N : Submodule R M
s : Finset (Submodule R M)
hs : s.inf id = N
hs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary
I J I' : Submodule R M
hI' : I' ∈ s
hI : {I ∈ s | (I.colon Set.univ).radical = (I'.colon Set.univ).radical}.inf id = I
J' : Submodule R M
hJ' : J' ∈ s
hJ : {I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id = J
hIJ : (I.colon Set.univ).radical = (J.colon Set.univ).radical
this : (I'.colon Set.univ).radical = (J'.colon Set.univ).radical
rw : {I ∈ s | (I.colon Set.univ).radical = (I'.colon Set.univ).radical}.inf id = J
rw₁ :
  {I ∈ s | (I.colon Set.univ).radical = (I'.colon Set.univ).radical}.inf id =
    {I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id
rw₂ rw₃ :
  {I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id =
    {I ∈ s | (I.colon Set.univ).radical = (J'.colon Set.univ).radical}.inf id
⊢ I = J

Tactic:
rw [← hI, ← hJ, this]