awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
x y : PrimeSpectrum R
rw : x.asIdeal ≤ y.asIdeal ↔ Membership.mem (γ := Set (PrimeSpectrum R)) (closure {x}) y
⊢ x ≤ y ↔ Membership.mem (γ := Set (PrimeSpectrum R)) (closure {x}) y
have rw₁ : x.asIdeal ≤ y.asIdeal ↔ Membership.mem (γ := Set (PrimeSpectrum R)) (zeroLocus (↑(vanishingIdeal {x}) : Set R)) y
Tactic: rw [← asIdeal_le_asIdeal, ← zeroLocus_vanishingIdeal_eq_closure, mem_zeroLocus,
    vanishingIdeal_singleton, SetLike.coe_subset_coe]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
x y : PrimeSpectrum R
rw : x.asIdeal ≤ y.asIdeal ↔ Membership.mem (γ := Set (PrimeSpectrum R)) (closure {x}) y
rw₁ : x.asIdeal ≤ y.asIdeal ↔ Membership.mem (γ := Set (PrimeSpectrum R)) (zeroLocus (↑(vanishingIdeal {x}) : Set R)) y
⊢ x ≤ y ↔ Membership.mem (γ := Set (PrimeSpectrum R)) (closure {x}) y
have rw₂ : x.asIdeal ≤ y.asIdeal ↔ (↑(vanishingIdeal {x}) : Set R) ⊆ (↑y.asIdeal : Set R)
Tactic: rw [← asIdeal_le_asIdeal, ← zeroLocus_vanishingIdeal_eq_closure, mem_zeroLocus,
    vanishingIdeal_singleton, SetLike.coe_subset_coe]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
x y : PrimeSpectrum R
rw : x.asIdeal ≤ y.asIdeal ↔ Membership.mem (γ := Set (PrimeSpectrum R)) (closure {x}) y
rw₁ : x.asIdeal ≤ y.asIdeal ↔ Membership.mem (γ := Set (PrimeSpectrum R)) (zeroLocus (↑(vanishingIdeal {x}) : Set R)) y
rw₂ : x.asIdeal ≤ y.asIdeal ↔ (↑(vanishingIdeal {x}) : Set R) ⊆ (↑y.asIdeal : Set R)
⊢ x ≤ y ↔ Membership.mem (γ := Set (PrimeSpectrum R)) (closure {x}) y
have rw₃ : x.asIdeal ≤ y.asIdeal ↔ (↑x.asIdeal : Set R) ⊆ (↑y.asIdeal : Set R)
Tactic: rw [← asIdeal_le_asIdeal, ← zeroLocus_vanishingIdeal_eq_closure, mem_zeroLocus,
    vanishingIdeal_singleton, SetLike.coe_subset_coe]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
x y : PrimeSpectrum R
rw : x.asIdeal ≤ y.asIdeal ↔ Membership.mem (γ := Set (PrimeSpectrum R)) (closure {x}) y
rw₁ : x.asIdeal ≤ y.asIdeal ↔ Membership.mem (γ := Set (PrimeSpectrum R)) (zeroLocus (↑(vanishingIdeal {x}) : Set R)) y
rw₂ : x.asIdeal ≤ y.asIdeal ↔ (↑(vanishingIdeal {x}) : Set R) ⊆ (↑y.asIdeal : Set R)
rw₃ : x.asIdeal ≤ y.asIdeal ↔ (↑x.asIdeal : Set R) ⊆ (↑y.asIdeal : Set R)
⊢ x ≤ y ↔ Membership.mem (γ := Set (PrimeSpectrum R)) (closure {x}) y
have rw₄ : x.asIdeal ≤ y.asIdeal ↔ x.asIdeal ≤ y.asIdeal
Tactic: rw [← asIdeal_le_asIdeal, ← zeroLocus_vanishingIdeal_eq_closure, mem_zeroLocus,
    vanishingIdeal_singleton, SetLike.coe_subset_coe]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
S : Type v
inst : CommSemiring R
inst_1 : CommSemiring S
x y : PrimeSpectrum R
h : x ⤳ y
a : R
ha : a ∈ y.asIdeal.primeCompl
⊢ IsUnit (M := Localization.AtPrime x.asIdeal)
    ((algebraMap R (Localization.AtPrime x.asIdeal) : (a : R) → Localization.AtPrime x.asIdeal)
        (↑(Subtype.mk (p := fun (x : R) ↦ x ∈ y.asIdeal.primeCompl) a ha) : R) :
      Localization.AtPrime x.asIdeal)
have rw : ∀ (h : (↑y.asIdeal : Set R)ᶜ ⊆ (↑x.asIdeal : Set R)ᶜ) (a : R) (ha : a ∈ y.asIdeal.primeCompl),
  IsUnit (M := Localization.AtPrime x.asIdeal)
    ((algebraMap R (Localization.AtPrime x.asIdeal) : (a : R) → Localization.AtPrime x.asIdeal)
        (↑(Subtype.mk (p := fun (x : R) ↦ x ∈ y.asIdeal.primeCompl) a ha) : R) :
      Localization.AtPrime x.asIdeal)
Tactic: rw [← PrimeSpectrum.le_iff_specializes, ← asIdeal_le_asIdeal, ← SetLike.coe_subset_coe, ←
        Set.compl_subset_compl] at h
Cleared hypotheses: h, a, ha
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
Z : Set (PrimeSpectrum S)
hZ : IsClosed Z
hf : StableUnderSpecialization (comap f '' Z)
⊢ IsClosed (comap f '' Z)
have obtain : ∀ (I : Ideal S) (hZ : IsClosed (X := PrimeSpectrum S) (zeroLocus (↑I : Set S)))
  (hf : StableUnderSpecialization (comap f '' zeroLocus (↑I : Set S))), IsClosed (comap f '' zeroLocus (↑I : Set S))
Tactic: obtain ⟨I, rfl⟩ := (PrimeSpectrum.isClosed_iff_zeroLocus_ideal Z).mp hZ
Cleared hypotheses: Z, hZ, hf
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
I : Ideal S
hZ : IsClosed (X := PrimeSpectrum S) (zeroLocus (↑I : Set S))
hf : StableUnderSpecialization (comap f '' zeroLocus (↑I : Set S))
⊢ IsClosed (comap f '' zeroLocus (↑I : Set S))
have refine_1 : comap f '' zeroLocus (↑I : Set S) ≤ zeroLocus (↑(Ideal.comap f I) : Set R)
Tactic: refine (isClosed_iff_zeroLocus _).mpr ⟨I.comap f, le_antisymm ?_ fun p hp ↦ ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
I : Ideal S
hZ : IsClosed (X := PrimeSpectrum S) (zeroLocus (↑I : Set S))
hf : StableUnderSpecialization (comap f '' zeroLocus (↑I : Set S))
refine_1 : comap f '' zeroLocus (↑I : Set S) ≤ zeroLocus (↑(Ideal.comap f I) : Set R)
⊢ IsClosed (comap f '' zeroLocus (↑I : Set S))
have refine_2 : ∀ (p : PrimeSpectrum R) (hp : Membership.mem (γ := Set (PrimeSpectrum R)) (zeroLocus (↑(Ideal.comap f I) : Set R)) p),
  p ∈ comap f '' zeroLocus (↑I : Set S)
Tactic: refine (isClosed_iff_zeroLocus _).mpr ⟨I.comap f, le_antisymm ?_ fun p hp ↦ ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
I : Ideal S
hZ : IsClosed (X := PrimeSpectrum S) (zeroLocus (↑I : Set S))
hf : StableUnderSpecialization (comap f '' zeroLocus (↑I : Set S))
p : PrimeSpectrum R
hp : Membership.mem (γ := Set (PrimeSpectrum R)) (zeroLocus (↑(Ideal.comap f I) : Set R)) p
⊢ p ∈ comap f '' zeroLocus (↑I : Set S)
have refine_2 : ∀ (q : Ideal S) (hqI : q ≥ I) (hq : q.IsPrime) (hqle : Ideal.comap f q ≤ p.asIdeal),
  p ∈ comap f '' zeroLocus (↑I : Set S)
Tactic: obtain ⟨q, hqI, hq, hqle⟩ := p.asIdeal.exists_ideal_comap_le_prime I hp
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
hf : StableUnderSpecialization (Set.range (comap f))
⊢ IsClosed (Set.range (comap f))
have rw : ∀ (hf : StableUnderSpecialization (comap f '' Set.univ)), IsClosed (comap f '' Set.univ)
Tactic: rw [← Set.image_univ] at hf ⊢
Cleared hypotheses: hf
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
h₁ : Function.Surjective (comap f)
h₂ : SpecializingMap (comap f)
⊢ IsQuotientMap (comap f)
have rw : Function.Surjective (comap f) ∧ ∀ (s : Set (PrimeSpectrum R)), IsClosed s ↔ IsClosed (comap f ⁻¹' s)
Tactic: rw [Topology.isQuotientMap_iff_isClosed]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
h₁ : Function.Surjective (comap f)
h₂ : GeneralizingMap (comap f)
⊢ IsQuotientMap (comap f)
have rw : Function.Surjective (comap f) ∧ ∀ (s : Set (PrimeSpectrum R)), IsClosed s ↔ IsClosed (comap f ⁻¹' s)
Tactic: rw [Topology.isQuotientMap_iff_isClosed]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
h₁ : Function.Surjective (comap f)
h₂ : GeneralizingMap (comap f)
⊢ Function.Surjective (comap f) ∧ ∀ (s : Set (PrimeSpectrum R)), IsClosed s ↔ IsClosed (comap f ⁻¹' s)
have refine : ∀ (s : Set (PrimeSpectrum R)) (hsc : IsClosed (comap f ⁻¹' s)), IsClosed (comap f '' comap f ⁻¹' s)
Tactic: refine ⟨h₁, fun s ↦ ⟨fun hs ↦ hs.preimage (continuous_comap f),
    fun hsc ↦ Set.image_preimage_eq s h₁ ▸ ?_⟩⟩
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
h₁ : Function.Surjective (comap f)
h₂ : GeneralizingMap (comap f)
s : Set (PrimeSpectrum R)
hsc : IsClosed (comap f ⁻¹' s)
⊢ IsClosed (comap f '' comap f ⁻¹' s)
have apply : StableUnderSpecialization (comap f '' comap f ⁻¹' s)
Tactic: apply isClosed_image_of_stableUnderSpecialization _ _ hsc
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
h₁ : Function.Surjective (comap f)
h₂ : GeneralizingMap (comap f)
s : Set (PrimeSpectrum R)
hsc : IsClosed (comap f ⁻¹' s)
⊢ StableUnderSpecialization (comap f '' comap f ⁻¹' s)
have rw : StableUnderGeneralization sᶜ
Tactic: rw [Set.image_preimage_eq s h₁, ← stableUnderGeneralization_compl_iff]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
h₁ : Function.Surjective (comap f)
h₂ : GeneralizingMap (comap f)
s : Set (PrimeSpectrum R)
hsc : IsClosed (comap f ⁻¹' s)
⊢ StableUnderGeneralization sᶜ
have h : sᶜ = comap f '' (comap f ⁻¹' s)ᶜ
Tactic: convert! h₂.stableUnderGeneralization_image hsc.isOpen_compl.stableUnderGeneralization
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
h₁ : Function.Surjective (comap f)
h₂ : GeneralizingMap (comap f)
s : Set (PrimeSpectrum R)
hsc : IsClosed (comap f ⁻¹' s)
⊢ sᶜ = comap f '' (comap f ⁻¹' s)ᶜ
have h : sᶜ = comap f '' comap f ⁻¹' sᶜ
Tactic: rw [← Set.preimage_compl, Set.image_preimage_eq _ h₁]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
h₁ : Function.Surjective (comap f)
h₂ : GeneralizingMap (comap f)
s : Set (PrimeSpectrum R)
hsc : IsClosed (comap f ⁻¹' s)
h : sᶜ = comap f '' comap f ⁻¹' sᶜ
⊢ sᶜ = comap f '' (comap f ⁻¹' s)ᶜ
have h₃ : sᶜ = sᶜ
Tactic: rw [← Set.preimage_compl, Set.image_preimage_eq _ h₁]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
⊢ vanishingIdeal (Set.range (comap f)) = Ideal.radical (RingHom.ker f)
have h : ∀ (x : R), x ∈ vanishingIdeal (Set.range (comap f)) ↔ Membership.mem (γ := Ideal R) (Ideal.radical (RingHom.ker f)) x
Tactic: ext x
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
x : R
⊢ x ∈ vanishingIdeal (Set.range (comap f)) ↔ Membership.mem (γ := Ideal R) (Ideal.radical (RingHom.ker f)) x
have h : x ∈ vanishingIdeal (Set.range (comap f)) ↔
  Membership.mem (γ := Ideal R) (Ideal.comap f (sInf {J : Ideal S | ⊥ ≤ J ∧ J.IsPrime})) x
Tactic: rw [RingHom.ker_eq_comap_bot, ← Ideal.comap_radical, Ideal.radical_eq_sInf]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
x : R
⊢ x ∈ vanishingIdeal (Set.range (comap f)) ↔
    Membership.mem (γ := Ideal R) (Ideal.comap f (sInf {J : Ideal S | ⊥ ≤ J ∧ J.IsPrime})) x
have h : (∀ (a : PrimeSpectrum S), Membership.mem (α := S) a.asIdeal ((f : (a : R) → S) x : S)) ↔
  ∀ (p : Submodule S S), Ideal.IsPrime p → Membership.mem (α := S) p ((f : (a : R) → S) x : S)
Tactic: simp only [mem_vanishingIdeal, Set.mem_range, forall_exists_index, forall_apply_eq_imp_iff,
    comap_asIdeal, Ideal.mem_comap, bot_le, true_and, Submodule.mem_sInf, Set.mem_setOf_eq]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
⊢ closure (Set.range (comap f)) = zeroLocus (↑(RingHom.ker f) : Set R)
have rw : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(vanishingIdeal (Set.range (comap f))) : Set R))
  (zeroLocus (↑(RingHom.ker f) : Set R))
Tactic: rw [← zeroLocus_vanishingIdeal_eq_closure, vanishingIdeal_range_comap, zeroLocus_radical]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
rw :
  Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(vanishingIdeal (Set.range (comap f))) : Set R))
    (zeroLocus (↑(RingHom.ker f) : Set R))
⊢ closure (Set.range (comap f)) = zeroLocus (↑(RingHom.ker f) : Set R)
have rw₁ : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(Ideal.radical (RingHom.ker f)) : Set R))
  (zeroLocus (↑(RingHom.ker f) : Set R))
Tactic: rw [← zeroLocus_vanishingIdeal_eq_closure, vanishingIdeal_range_comap, zeroLocus_radical]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
rw :
  Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(vanishingIdeal (Set.range (comap f))) : Set R))
    (zeroLocus (↑(RingHom.ker f) : Set R))
rw₁ :
  Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(Ideal.radical (RingHom.ker f)) : Set R))
    (zeroLocus (↑(RingHom.ker f) : Set R))
⊢ closure (Set.range (comap f)) = zeroLocus (↑(RingHom.ker f) : Set R)
have rw₂ : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(RingHom.ker f) : Set R)) (zeroLocus (↑(RingHom.ker f) : Set R))
Tactic: rw [← zeroLocus_vanishingIdeal_eq_closure, vanishingIdeal_range_comap, zeroLocus_radical]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
⊢ DenseRange (comap f) ↔ LE.le (α := Ideal R) (RingHom.ker f) (nilradical R)
have rw : closure (Set.range (comap f)) = Set.univ ↔ LE.le (α := Ideal R) (RingHom.ker f) (nilradical R)
Tactic: rw [denseRange_iff_closure_range, closure_range_comap, zeroLocus_eq_univ_iff,
    SetLike.coe_subset_coe]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
rw : closure (Set.range (comap f)) = Set.univ ↔ LE.le (α := Ideal R) (RingHom.ker f) (nilradical R)
⊢ DenseRange (comap f) ↔ LE.le (α := Ideal R) (RingHom.ker f) (nilradical R)
have rw₁ : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(RingHom.ker f) : Set R)) Set.univ ↔
  LE.le (α := Ideal R) (RingHom.ker f) (nilradical R)
Tactic: rw [denseRange_iff_closure_range, closure_range_comap, zeroLocus_eq_univ_iff,
    SetLike.coe_subset_coe]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
rw : closure (Set.range (comap f)) = Set.univ ↔ LE.le (α := Ideal R) (RingHom.ker f) (nilradical R)
rw₁ :
  Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(RingHom.ker f) : Set R)) Set.univ ↔
    LE.le (α := Ideal R) (RingHom.ker f) (nilradical R)
⊢ DenseRange (comap f) ↔ LE.le (α := Ideal R) (RingHom.ker f) (nilradical R)
have rw₂ : (↑(RingHom.ker f) : Set R) ⊆ (↑(nilradical R) : Set R) ↔ LE.le (α := Ideal R) (RingHom.ker f) (nilradical R)
Tactic: rw [denseRange_iff_closure_range, closure_range_comap, zeroLocus_eq_univ_iff,
    SetLike.coe_subset_coe]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
rw : closure (Set.range (comap f)) = Set.univ ↔ LE.le (α := Ideal R) (RingHom.ker f) (nilradical R)
rw₁ :
  Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(RingHom.ker f) : Set R)) Set.univ ↔
    LE.le (α := Ideal R) (RingHom.ker f) (nilradical R)
rw₂ : (↑(RingHom.ker f) : Set R) ⊆ (↑(nilradical R) : Set R) ↔ LE.le (α := Ideal R) (RingHom.ker f) (nilradical R)
⊢ DenseRange (comap f) ↔ LE.le (α := Ideal R) (RingHom.ker f) (nilradical R)
have rw₃ : LE.le (α := Ideal R) (RingHom.ker f) (nilradical R) ↔ LE.le (α := Ideal R) (RingHom.ker f) (nilradical R)
Tactic: rw [denseRange_iff_closure_range, closure_range_comap, zeroLocus_eq_univ_iff,
    SetLike.coe_subset_coe]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
⊢ DenseRange (comap f) ↔
    ∀ (I : Ideal R) (h : Membership.mem (γ := Set (Ideal R)) (minimalPrimes R) I),
      { asIdeal := I, isPrime := ⋯ : PrimeSpectrum R } ∈ Set.range (comap f)
have mp : DenseRange (comap f) →
  ∀ (I : Ideal R) (h : Membership.mem (γ := Set (Ideal R)) (minimalPrimes R) I),
    { asIdeal := I, isPrime := sorry : PrimeSpectrum R } ∈ Set.range (comap f)
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
mp :
  DenseRange (comap f) →
    ∀ (I : Ideal R) (h : Membership.mem (γ := Set (Ideal R)) (minimalPrimes R) I),
      { asIdeal := I, isPrime := ⋯ : PrimeSpectrum R } ∈ Set.range (comap f)
⊢ DenseRange (comap f) ↔
    ∀ (I : Ideal R) (h : Membership.mem (γ := Set (Ideal R)) (minimalPrimes R) I),
      { asIdeal := I, isPrime := ⋯ : PrimeSpectrum R } ∈ Set.range (comap f)
have mpr : (∀ (I : Ideal R) (h : Membership.mem (γ := Set (Ideal R)) (minimalPrimes R) I),
    { asIdeal := I, isPrime := sorry : PrimeSpectrum R } ∈ Set.range (comap f)) →
  DenseRange (comap f)
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
H : DenseRange (comap f)
I : Ideal R
hI : Membership.mem (γ := Set (Ideal R)) (minimalPrimes R) I
⊢ Membership.mem (γ := Set (Ideal R)) (Ideal.minimalPrimes (RingHom.ker f)) I
have rw : ∀ (H : LE.le (α := Ideal R) (RingHom.ker f) (nilradical R)) (I : Ideal R)
  (hI : Membership.mem (γ := Set (Ideal R)) (minimalPrimes R) I),
  Membership.mem (γ := Set (Ideal R)) (Ideal.minimalPrimes (RingHom.ker f)) I
Tactic: rw [denseRange_comap_iff_ker_le_nilRadical] at H
Cleared hypotheses: H, I, hI
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (hypothesis)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
H : LE.le (α := Ideal R) (RingHom.ker f) (nilradical R)
I : Ideal R
hI : Membership.mem (γ := Set (Ideal R)) (minimalPrimes R) I
⊢ Membership.mem (γ := Set (Ideal R)) (Ideal.minimalPrimes (RingHom.ker f)) I
have hI₁ : Minimal (fun (q : Ideal R) ↦ q.IsPrime ∧ ⊥ ≤ q) I
Tactic: simp only [Set.mem_setOf, Ideal.IsMinimalPrime] at hI ⊢
Cleared hypotheses: hI
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
H : LE.le (α := Ideal R) (RingHom.ker f) (nilradical R)
I : Ideal R
hI : Minimal (fun (q : Ideal R) ↦ q.IsPrime ∧ ⊥ ≤ q) I
⊢ Minimal (fun (q : Ideal R) ↦ q.IsPrime ∧ RingHom.ker f ≤ q) I
have h : ∀ (p : Ideal R), p.IsPrime ∧ RingHom.ker f ≤ p ↔ p.IsPrime ∧ ⊥ ≤ p
Tactic: convert! hI using 2 with p
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (hypothesis)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
H : DenseRange (comap f)
I : Ideal R
hI : Membership.mem (γ := Set (Ideal R)) (minimalPrimes R) I
⊢ { asIdeal := I, isPrime := ⋯ : PrimeSpectrum R } ∈ Set.range (comap f)
have this : Membership.mem (γ := Set (Ideal R)) (Ideal.minimalPrimes (RingHom.ker f)) I
Tactic: 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 ⊢
      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)⟩⟩
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
H : DenseRange (comap f)
I : Ideal R
hI : Membership.mem (γ := Set (Ideal R)) (minimalPrimes R) I
this : Membership.mem (γ := Set (Ideal R)) (Ideal.minimalPrimes (RingHom.ker f)) I
⊢ { asIdeal := I, isPrime := ⋯ : PrimeSpectrum R } ∈ Set.range (comap f)
have mp : ∀ (p : Ideal S) (hp : p.IsPrime),
  ⊥ ≤ p →
    ∀ (hI : Membership.mem (γ := Set (Ideal R)) (minimalPrimes R) (Ideal.comap f p))
      (this : Membership.mem (γ := Set (Ideal R)) (Ideal.minimalPrimes (RingHom.ker f)) (Ideal.comap f p)),
      { asIdeal := Ideal.comap f p, isPrime := sorry : PrimeSpectrum R } ∈ Set.range (comap f)
Tactic: obtain ⟨p, hp, _, rfl⟩ := Ideal.exists_comap_eq_of_mem_minimalPrimes f (I := ⊥) I this
Cleared hypotheses: I, hI, this
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommSemiring R
inst_1 : CommSemiring S
f : R →+* S
H :
  ∀ (I : Ideal R) (h : Membership.mem (γ := Set (Ideal R)) (minimalPrimes R) I),
    { asIdeal := I, isPrime := ⋯ : PrimeSpectrum R } ∈ Set.range (comap f)
p : PrimeSpectrum R
⊢ p ∈ closure (Set.range (comap f))
have mpr : ∀ (q : Ideal R) (hq : Membership.mem (γ := Set (Ideal R)) (Ideal.minimalPrimes ⊥) q) (hq' : q ≤ p.asIdeal),
  p ∈ closure (Set.range (comap f))
Tactic: obtain ⟨q, hq, hq'⟩ := Ideal.exists_minimalPrimes_le (J := p.asIdeal) bot_le
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
x : PrimeSpectrum R
⊢ StableUnderSpecialization (X := PrimeSpectrum R) {x} ↔ x.asIdeal.IsMaximal
have simp_rw : (∀ (a : PrimeSpectrum R) (a_1 : x ≤ a), a = x) ↔ IsMax x
Tactic: simp_rw [← isMax_iff, StableUnderSpecialization, ← le_iff_specializes, Set.mem_singleton_iff,
    @forall_comm _ (_ = _), forall_eq]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
x : PrimeSpectrum R
⊢ StableUnderGeneralization (X := PrimeSpectrum R) {x} ↔ Membership.mem (γ := Set (Ideal R)) (minimalPrimes R) x.asIdeal
have simp_rw : (∀ (a : PrimeSpectrum R) (a_1 : a ≤ x), a = x) ↔ IsMin x
Tactic: simp_rw [← isMin_iff, StableUnderGeneralization, ← le_iff_specializes, Set.mem_singleton_iff,
    @forall_comm _ (_ = _), forall_eq]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
⊢ IsCompact s ∧ IsOpen s ↔ ∃ (t : Finset R), (zeroLocus (↑t : Set R))ᶜ = s
have rw : (∃ (s_1 : Set R), s_1.Finite ∧ s = ⋃ i ∈ s_1, (↑(basicOpen i) : Set (PrimeSpectrum R))) ↔
  ∃ (t : Finset R), (zeroLocus (↑t : Set R))ᶜ = s
Tactic: rw [isCompact_open_iff_eq_finite_iUnion_of_isTopologicalBasis _
    isTopologicalBasis_basic_opens isCompact_basicOpen]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
⊢ (∃ (s_1 : Set R), s_1.Finite ∧ s = ⋃ i ∈ s_1, (↑(basicOpen i) : Set (PrimeSpectrum R))) ↔
    ∃ (t : Finset R), (zeroLocus (↑t : Set R))ᶜ = s
have simp : (∃ (s_1 : Set R), s_1.Finite ∧ s = (zeroLocus s_1)ᶜ) ↔ ∃ (t : Finset R), (zeroLocus (↑t : Set R))ᶜ = s
Tactic: simp only [basicOpen_eq_zeroLocus_compl, ← Set.compl_iInter₂, ← zeroLocus_iUnion₂,
    Set.biUnion_of_singleton]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
⊢ IsCompact s ∧ IsOpen s ↔ ∃ (I : Ideal R), I.FG ∧ (zeroLocus (↑I : Set R))ᶜ = s
have rw : (∃ (t : Finset R), (zeroLocus (↑t : Set R))ᶜ = s) ↔ ∃ (I : Ideal R), I.FG ∧ (zeroLocus (↑I : Set R))ᶜ = s
Tactic: rw [isCompact_isOpen_iff]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
e f : R
mul : e * f = 0
add : e + f = 1
⊢ (↑(basicOpen e) : Set (PrimeSpectrum R)) = zeroLocus {f}
have h : ∀ (p : PrimeSpectrum R),
  p ∈ (↑(basicOpen e) : Set (PrimeSpectrum R)) ↔ Membership.mem (γ := Set (PrimeSpectrum R)) (zeroLocus {f}) p
Tactic: ext p