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
e f : R
mul : e * f = 0
add : e + f = 1
p : PrimeSpectrum R
⊢ p ∈ (↑(basicOpen e) : Set (PrimeSpectrum R)) ↔ Membership.mem (γ := Set (PrimeSpectrum R)) (zeroLocus {f}) p
have h : e ∉ p.asIdeal ↔ f ∈ p.asIdeal
Tactic: suffices e ∉ p.asIdeal ↔ f ∈ p.asIdeal by simpa
-- 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
p : PrimeSpectrum R
⊢ e ∉ p.asIdeal ↔ f ∈ p.asIdeal
have h : ∀ (h₁ : f ∈ p.asIdeal) (h₂ : e ∈ p.asIdeal), p.asIdeal = ⊤
Tactic: refine ⟨(p.2.mem_or_mem_of_mul_eq_zero mul).resolve_left, fun h₁ h₂ ↦ p.2.1 ?_⟩
-- 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
p : PrimeSpectrum R
h₁ : f ∈ p.asIdeal
h₂ : e ∈ p.asIdeal
⊢ p.asIdeal = ⊤
have h : e + f ∈ p.asIdeal
Tactic: rw [Ideal.eq_top_iff_one, ← add]
-- 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
⊢ zeroLocus {e} = (↑(basicOpen f) : Set (PrimeSpectrum R))
have mul₁ : f * e = 0
Tactic: rw [basicOpen_eq_zeroLocus_of_mul_add f e]
-- 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
mul₁ : f * e = 0
⊢ zeroLocus {e} = (↑(basicOpen f) : Set (PrimeSpectrum R))
have add₁ : f + e = 1
Tactic: rw [basicOpen_eq_zeroLocus_of_mul_add f e]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
x : R
hx : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} x
y : R
hy : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} y
eq : Eq (α := TopologicalSpace.Opens (PrimeSpectrum R)) (basicOpen x) (basicOpen y)
⊢ x = y
have by_contra! : ∀ (ne : x ≠ y), False
Tactic: by_contra! ne
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
x : R
hx : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} x
y : R
hy : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} y
eq : Eq (α := TopologicalSpace.Opens (PrimeSpectrum R)) (basicOpen x) (basicOpen y)
ne : x ≠ y
⊢ False
have inr : ∀
  (this :
    ∀ (x : R) (hx : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} x) (y : R)
      (hy : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} y)
      (eq : Eq (α := TopologicalSpace.Opens (PrimeSpectrum R)) (basicOpen x) (basicOpen y)) (ne : x ≠ y)
      (ne' : x * y ≠ x), False)
  (ne' : ¬x * y ≠ x), False
Tactic: wlog ne' : x * y ≠ x generalizing x y
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
x : R
hx : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} x
y : R
hy : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} y
eq : Eq (α := TopologicalSpace.Opens (PrimeSpectrum R)) (basicOpen x) (basicOpen y)
ne : x ≠ y
inr :
  ∀
    (this :
      ∀ (x : R) (hx : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} x) (y : R)
        (hy : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} y)
        (eq : Eq (α := TopologicalSpace.Opens (PrimeSpectrum R)) (basicOpen x) (basicOpen y)) (ne : x ≠ y)
        (ne' : x * y ≠ x), False)
    (ne' : ¬x * y ≠ x), False
⊢ False
have wlog : ∀ (x : R) (hx : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} x) (y : R)
  (hy : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} y)
  (eq : Eq (α := TopologicalSpace.Opens (PrimeSpectrum R)) (basicOpen x) (basicOpen y)) (ne : x ≠ y) (ne' : x * y ≠ x),
  False
Tactic: wlog ne' : x * y ≠ x generalizing x y
Cleared hypotheses: x, hx, y, hy, eq, ne
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
x : R
hx : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} x
y : R
hy : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} y
eq : Eq (α := TopologicalSpace.Opens (PrimeSpectrum R)) (basicOpen x) (basicOpen y)
ne : x ≠ y
this :
  ∀ (x : R) (hx : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} x) (y : R)
    (hy : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} y)
    (eq : Eq (α := TopologicalSpace.Opens (PrimeSpectrum R)) (basicOpen x) (basicOpen y)) (ne : x ≠ y)
    (ne' : x * y ≠ x), False
ne' : ¬x * y ≠ x
⊢ False
have inr : y * x ≠ y
Tactic: apply this y hy x hx eq.symm ne.symm
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
x : R
hx : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} x
y : R
hy : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} y
eq : Eq (α := TopologicalSpace.Opens (PrimeSpectrum R)) (basicOpen x) (basicOpen y)
ne : x ≠ y
this :
  ∀ (x : R) (hx : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} x) (y : R)
    (hy : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} y)
    (eq : Eq (α := TopologicalSpace.Opens (PrimeSpectrum R)) (basicOpen x) (basicOpen y)) (ne : x ≠ y)
    (ne' : x * y ≠ x), False
ne' : ¬x * y ≠ x
⊢ y * x ≠ y
have inr : x * y ≠ y
Tactic: rwa [mul_comm, of_not_not ne']
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
x : R
hx : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} x
y : R
hy : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} y
eq : Eq (α := TopologicalSpace.Opens (PrimeSpectrum R)) (basicOpen x) (basicOpen y)
ne : x ≠ y
ne' : x * y ≠ x
mem : Membership.mem (γ := Ideal R) (Ideal.span {y}) x
⊢ x * y = x
have obtain : ∀ (r : R) (hx : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} (r * y))
  (eq : Eq (α := TopologicalSpace.Opens (PrimeSpectrum R)) (basicOpen (r * y)) (basicOpen y)) (ne : r * y ≠ y)
  (ne' : r * y * y ≠ r * y) (mem : Membership.mem (γ := Ideal R) (Ideal.span {y}) (r * y)), r * y * y = r * y
Tactic: obtain ⟨r, rfl⟩ := Ideal.mem_span_singleton'.mp mem
Cleared hypotheses: x, hx, eq, ne, ne', mem
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
y : R
hy : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} y
r : R
hx : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} (r * y)
eq : Eq (α := TopologicalSpace.Opens (PrimeSpectrum R)) (basicOpen (r * y)) (basicOpen y)
ne : r * y ≠ y
ne' : r * y * y ≠ r * y
mem : Membership.mem (γ := Ideal R) (Ideal.span {y}) (r * y)
⊢ r * y * y = r * y
have rw : r * (y * y) = r * y
Tactic: rw [mul_assoc, hy]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
y : R
hy : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} y
r : R
hx : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} (r * y)
eq : Eq (α := TopologicalSpace.Opens (PrimeSpectrum R)) (basicOpen (r * y)) (basicOpen y)
ne : r * y ≠ y
ne' : r * y * y ≠ r * y
mem : Membership.mem (γ := Ideal R) (Ideal.span {y}) (r * y)
rw : r * (y * y) = r * y
⊢ r * y * y = r * y
have rw₁ : r * y = r * y
Tactic: rw [mul_assoc, hy]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (hypothesis)
Goal:
R : Type u
inst : CommSemiring R
x : R
hx : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} x
y : R
hy : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} y
eq : Eq (α := TopologicalSpace.Opens (PrimeSpectrum R)) (basicOpen x) (basicOpen y)
ne : x ≠ y
ne' : x * y ≠ x
⊢ False
have this : x ∉ Ideal.span {y}
Tactic: have : x ∉ Ideal.span {y} := fun mem ↦ ne' <| by
    obtain ⟨r, rfl⟩ := Ideal.mem_span_singleton'.mp mem
    rw [mul_assoc, hy]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
x : R
hx : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} x
y : R
hy : Membership.mem (γ := Set R) {e : R | IsIdempotentElem e} y
eq : Eq (α := TopologicalSpace.Opens (PrimeSpectrum R)) (basicOpen x) (basicOpen y)
ne : x ≠ y
ne' : x * y ≠ x
this : x ∉ Ideal.span {y}
⊢ False
have have : ∀ (p : Ideal R) (prime : p.IsPrime) (le : Ideal.span {y} ≤ p) (notMem : x ∉ p), False
Tactic: have ⟨p, prime, le, notMem⟩ := Ideal.exists_le_prime_notMem_of_isIdempotentElem _ x hx this
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
⊢ ∃ (e : R) (f : R),
    e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R)) ∧ sᶜ = (↑(basicOpen f) : Set (PrimeSpectrum R))
have inl : Subsingleton R →
  ∃ (e : R) (f : R),
    e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R)) ∧ sᶜ = (↑(basicOpen f) : Set (PrimeSpectrum R))
Tactic: cases subsingleton_or_nontrivial R
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
inl :
  Subsingleton R →
    ∃ (e : R) (f : R),
      e * f = 0 ∧
        e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R)) ∧ sᶜ = (↑(basicOpen f) : Set (PrimeSpectrum R))
⊢ ∃ (e : R) (f : R),
    e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R)) ∧ sᶜ = (↑(basicOpen f) : Set (PrimeSpectrum R))
have inr : Nontrivial R →
  ∃ (e : R) (f : R),
    e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R)) ∧ sᶜ = (↑(basicOpen f) : Set (PrimeSpectrum R))
Tactic: cases subsingleton_or_nontrivial R
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Subsingleton R
⊢ ∃ (e : R) (f : R),
    e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R)) ∧ sᶜ = (↑(basicOpen f) : Set (PrimeSpectrum R))
have inl : HMul.hMul (α := R) 0 0 = 0
Tactic: refine ⟨0, 0, ?_, ?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Subsingleton R
inl : HMul.hMul (α := R) 0 0 = 0
⊢ ∃ (e : R) (f : R),
    e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R)) ∧ sᶜ = (↑(basicOpen f) : Set (PrimeSpectrum R))
have inl₁ : HAdd.hAdd (α := R) 0 0 = 1
Tactic: refine ⟨0, 0, ?_, ?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Subsingleton R
inl : HMul.hMul (α := R) 0 0 = 0
inl₁ : HAdd.hAdd (α := R) 0 0 = 1
⊢ ∃ (e : R) (f : R),
    e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R)) ∧ sᶜ = (↑(basicOpen f) : Set (PrimeSpectrum R))
have inl₂ : s = (↑(basicOpen 0) : Set (PrimeSpectrum R))
Tactic: refine ⟨0, 0, ?_, ?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Subsingleton R
inl : HMul.hMul (α := R) 0 0 = 0
inl₁ : HAdd.hAdd (α := R) 0 0 = 1
inl₂ : s = (↑(basicOpen 0) : Set (PrimeSpectrum R))
⊢ ∃ (e : R) (f : R),
    e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R)) ∧ sᶜ = (↑(basicOpen f) : Set (PrimeSpectrum R))
have inl₃ : sᶜ = (↑(basicOpen 0) : Set (PrimeSpectrum R))
Tactic: refine ⟨0, 0, ?_, ?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
⊢ ∃ (e : R) (f : R),
    e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R)) ∧ sᶜ = (↑(basicOpen f) : Set (PrimeSpectrum R))
have inr : ∀ (I : Ideal R) (hI : I.FG) (hI' : (zeroLocus (↑I : Set R))ᶜ = s),
  ∃ (e : R) (f : R),
    e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R)) ∧ sᶜ = (↑(basicOpen f) : Set (PrimeSpectrum R))
Tactic: obtain ⟨I, hI, hI'⟩ := isCompact_isOpen_iff_ideal.mp ⟨hs.1.isCompact, hs.2⟩
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
hI' : (zeroLocus (↑I : Set R))ᶜ = s
⊢ ∃ (e : R) (f : R),
    e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R)) ∧ sᶜ = (↑(basicOpen f) : Set (PrimeSpectrum R))
have inr : ∀ (J : Ideal R) (hJ : J.FG) (hJ' : (zeroLocus (↑J : Set R))ᶜ = sᶜ),
  ∃ (e : R) (f : R),
    e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R)) ∧ sᶜ = (↑(basicOpen f) : Set (PrimeSpectrum R))
Tactic: obtain ⟨J, hJ, hJ'⟩ := isCompact_isOpen_iff_ideal.mp
    ⟨hs.2.isClosed_compl.isCompact, hs.1.isOpen_compl⟩
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (hypothesis)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
hI' : (zeroLocus (↑I : Set R))ᶜ = s
J : Ideal R
hJ : J.FG
hJ' : (zeroLocus (↑J : Set R))ᶜ = sᶜ
⊢ ∃ (e : R) (f : R),
    e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R)) ∧ sᶜ = (↑(basicOpen f) : Set (PrimeSpectrum R))
have hI'₁ : zeroLocus (↑I : Set R) = sᶜ
Tactic: simp only [compl_eq_iff_isCompl, ← eq_compl_iff_isCompl, compl_compl] at hI' hJ'
Cleared hypotheses: hI', hJ'
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (hypothesis)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
hI' : (zeroLocus (↑I : Set R))ᶜ = s
J : Ideal R
hJ : J.FG
hJ' : (zeroLocus (↑J : Set R))ᶜ = sᶜ
hI'₁ : zeroLocus (↑I : Set R) = sᶜ
⊢ ∃ (e : R) (f : R),
    e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R)) ∧ sᶜ = (↑(basicOpen f) : Set (PrimeSpectrum R))
have hJ'₁ : zeroLocus (↑J : Set R) = s
Tactic: simp only [compl_eq_iff_isCompl, ← eq_compl_iff_isCompl, compl_compl] at hI' hJ'
Cleared hypotheses: hI', hJ'
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
⊢ I * J ≤ nilradical R
have refine : (I * J).radical = Ideal.radical 0
Tactic: refine Ideal.radical_le_radical_iff.mp (le_of_eq ?_)
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
⊢ (I * J).radical = Ideal.radical 0
have rw : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(I * J) : Set R)) (zeroLocus (↑0 : Set R))
Tactic: rw [← zeroLocus_eq_iff, Ideal.zero_eq_bot, zeroLocus_bot,
      zeroLocus_mul, hI', hJ', Set.compl_union_self]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
rw : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(I * J) : Set R)) (zeroLocus (↑0 : Set R))
⊢ (I * J).radical = Ideal.radical 0
have rw₁ : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(I * J) : Set R)) (zeroLocus (↑⊥ : Set R))
Tactic: rw [← zeroLocus_eq_iff, Ideal.zero_eq_bot, zeroLocus_bot,
      zeroLocus_mul, hI', hJ', Set.compl_union_self]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
rw : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(I * J) : Set R)) (zeroLocus (↑0 : Set R))
rw₁ : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(I * J) : Set R)) (zeroLocus (↑⊥ : Set R))
⊢ (I * J).radical = Ideal.radical 0
have rw₂ : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(I * J) : Set R)) Set.univ
Tactic: rw [← zeroLocus_eq_iff, Ideal.zero_eq_bot, zeroLocus_bot,
      zeroLocus_mul, hI', hJ', Set.compl_union_self]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
rw : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(I * J) : Set R)) (zeroLocus (↑0 : Set R))
rw₁ : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(I * J) : Set R)) (zeroLocus (↑⊥ : Set R))
rw₂ : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(I * J) : Set R)) Set.univ
⊢ (I * J).radical = Ideal.radical 0
have rw₃ : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑I : Set R) ∪ zeroLocus (↑J : Set R)) Set.univ
Tactic: rw [← zeroLocus_eq_iff, Ideal.zero_eq_bot, zeroLocus_bot,
      zeroLocus_mul, hI', hJ', Set.compl_union_self]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
rw : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(I * J) : Set R)) (zeroLocus (↑0 : Set R))
rw₁ : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(I * J) : Set R)) (zeroLocus (↑⊥ : Set R))
rw₂ : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(I * J) : Set R)) Set.univ
rw₃ : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑I : Set R) ∪ zeroLocus (↑J : Set R)) Set.univ
⊢ (I * J).radical = Ideal.radical 0
have rw₄ : sᶜ ∪ zeroLocus (↑J : Set R) = Set.univ
Tactic: rw [← zeroLocus_eq_iff, Ideal.zero_eq_bot, zeroLocus_bot,
      zeroLocus_mul, hI', hJ', Set.compl_union_self]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
rw : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(I * J) : Set R)) (zeroLocus (↑0 : Set R))
rw₁ : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(I * J) : Set R)) (zeroLocus (↑⊥ : Set R))
rw₂ : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(I * J) : Set R)) Set.univ
rw₃ : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑I : Set R) ∪ zeroLocus (↑J : Set R)) Set.univ
rw₄ : sᶜ ∪ zeroLocus (↑J : Set R) = Set.univ
⊢ (I * J).radical = Ideal.radical 0
have rw₅ : sᶜ ∪ s = Set.univ
Tactic: rw [← zeroLocus_eq_iff, Ideal.zero_eq_bot, zeroLocus_bot,
      zeroLocus_mul, hI', hJ', Set.compl_union_self]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
rw : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(I * J) : Set R)) (zeroLocus (↑0 : Set R))
rw₁ : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(I * J) : Set R)) (zeroLocus (↑⊥ : Set R))
rw₂ : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(I * J) : Set R)) Set.univ
rw₃ : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑I : Set R) ∪ zeroLocus (↑J : Set R)) Set.univ
rw₄ : sᶜ ∪ zeroLocus (↑J : Set R) = Set.univ
rw₅ : sᶜ ∪ s = Set.univ
⊢ (I * J).radical = Ideal.radical 0
have rw₆ : Eq.{u + 1} (α := Set (PrimeSpectrum R)) Set.univ Set.univ
Tactic: rw [← zeroLocus_eq_iff, Ideal.zero_eq_bot, zeroLocus_bot,
      zeroLocus_mul, hI', hJ', Set.compl_union_self]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (hypothesis)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
⊢ ∃ (e : R) (f : R),
    e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R)) ∧ sᶜ = (↑(basicOpen f) : Set (PrimeSpectrum R))
have this : I * J ≤ nilradical R
Tactic: have : I * J ≤ nilradical R := by
    refine Ideal.radical_le_radical_iff.mp (le_of_eq ?_)
    rw [← zeroLocus_eq_iff, Ideal.zero_eq_bot, zeroLocus_bot,
      zeroLocus_mul, hI', hJ', Set.compl_union_self]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
this : I * J ≤ nilradical R
⊢ ∃ (e : R) (f : R),
    e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R)) ∧ sᶜ = (↑(basicOpen f) : Set (PrimeSpectrum R))
have inr : ∀ (n : ℕ) (hn : (I * J) ^ n ≤ 0),
  ∃ (e : R) (f : R),
    e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R)) ∧ sᶜ = (↑(basicOpen f) : Set (PrimeSpectrum R))
Tactic: obtain ⟨n, hn⟩ := Ideal.exists_pow_le_of_le_radical_of_fg this (Submodule.FG.mul hI hJ)
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (hypothesis)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
this : I * J ≤ nilradical R
n : ℕ
hn : (I * J) ^ n ≤ 0
⊢ ∃ (e : R) (f : R),
    e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R)) ∧ sᶜ = (↑(basicOpen f) : Set (PrimeSpectrum R))
have hnz : n ≠ 0
Tactic: have hnz : n ≠ 0 := by rintro rfl; simp at hn
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (hypothesis)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
this : I * J ≤ nilradical R
n : ℕ
hn : (I * J) ^ n ≤ 0
hnz : n ≠ 0
⊢ ∃ (e : R) (f : R),
    e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R)) ∧ sᶜ = (↑(basicOpen f) : Set (PrimeSpectrum R))
have hn₁ : I ^ n * J ^ n ≤ ⊥
Tactic: rw [mul_pow, Ideal.zero_eq_bot] at hn
Cleared hypotheses: hn, hnz
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
⊢ I ^ n ⊔ J ^ n = ⊤
have rw : (fun (x : R) ↦ x ^ n) '' (↑I : Set R) ⊆ (↑(I ^ n ⊔ J ^ n) : Set R) ∧
  (fun (x : R) ↦ x ^ n) '' (↑J : Set R) ⊆ (↑(I ^ n ⊔ J ^ n) : Set R)
Tactic: rw [eq_top_iff, ← Ideal.span_pow_eq_top (I ∪ J : Set R) _ n, Ideal.span_le, Set.image_union,
      Set.union_subset_iff]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
rw :
  (fun (x : R) ↦ x ^ n) '' (↑I : Set R) ⊆ (↑(I ^ n ⊔ J ^ n) : Set R) ∧
    (fun (x : R) ↦ x ^ n) '' (↑J : Set R) ⊆ (↑(I ^ n ⊔ J ^ n) : Set R)
⊢ I ^ n ⊔ J ^ n = ⊤
have rw₁ : Eq (α := Ideal R) (Ideal.span ((↑I : Set R) ∪ (↑J : Set R))) ⊤
Tactic: rw [eq_top_iff, ← Ideal.span_pow_eq_top (I ∪ J : Set R) _ n, Ideal.span_le, Set.image_union,
      Set.union_subset_iff]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
⊢ (fun (x : R) ↦ x ^ n) '' (↑I : Set R) ⊆ (↑(I ^ n ⊔ J ^ n) : Set R) ∧
    (fun (x : R) ↦ x ^ n) '' (↑J : Set R) ⊆ (↑(I ^ n ⊔ J ^ n) : Set R)
have left : (fun (x : R) ↦ x ^ n) '' (↑I : Set R) ⊆ (↑(I ^ n ⊔ J ^ n) : Set R)
Tactic: constructor