l3lab/ntp-mathlib huggingface dataset:
theorem negMem' (U : (Opens (ProjectiveSpectrum.top 𝒜))ᵒᵖ) (a : ∀ x : U.unop, at x.1)
    (ha : (isLocallyFraction 𝒜).pred a) : (isLocallyFraction 𝒜).pred (-a) := fun x => by
  

Proof state:
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
U : (Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ
a : (x : ↥U.unop) → at ↑x
ha : PrelocalPredicate.pred (isLocallyFraction 𝒜).toPrelocalPredicate a
x : ↥U.unop
⊢ ∃ V,
    ∃ (_ : ↑x ∈ V),
      ∃ i,
        PrelocalPredicate.pred (isFractionPrelocal 𝒜) fun x =>
          (-a) ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)

Next tactic:
rcases ha x with ⟨V, m, i, j, ⟨r, r_mem⟩, ⟨s, s_mem⟩, w⟩
theorem negMem' (U : (Opens (ProjectiveSpectrum.top 𝒜))ᵒᵖ) (a : ∀ x : U.unop, at x.1)
    (ha : (isLocallyFraction 𝒜).pred a) : (isLocallyFraction 𝒜).pred (-a) := fun x => by
  rcases ha x with ⟨V, m, i, j, ⟨r, r_mem⟩, ⟨s, s_mem⟩, w⟩
  

Proof state:
case intro.intro.intro.intro.intro.mk.intro.mk
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
U : (Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ
a : (x : ↥U.unop) → at ↑x
ha : PrelocalPredicate.pred (isLocallyFraction 𝒜).toPrelocalPredicate a
x : ↥U.unop
V : Opens ↑(ProjectiveSpectrum.top 𝒜)
m : ↑x ∈ V
i : V ⟶ U.unop
j : ℕ
r : A
r_mem : r ∈ 𝒜 j
s : A
s_mem : s ∈ 𝒜 j
w :
  ∀ (x : ↥V),
    ∃ (s_nin : ↑{ val := s, property := s_mem } ∉ (↑x).asHomogeneousIdeal),
      (fun x => a ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)) x =
        Quotient.mk''
          { deg := j, num := { val := r, property := r_mem }, den := { val := s, property := s_mem }, den_mem := s_nin }
⊢ ∃ V,
    ∃ (_ : ↑x ∈ V),
      ∃ i,
        PrelocalPredicate.pred (isFractionPrelocal 𝒜) fun x =>
          (-a) ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)

Next tactic:
choose nin hy using w
theorem negMem' (U : (Opens (ProjectiveSpectrum.top 𝒜))ᵒᵖ) (a : ∀ x : U.unop, at x.1)
    (ha : (isLocallyFraction 𝒜).pred a) : (isLocallyFraction 𝒜).pred (-a) := fun x => by
  rcases ha x with ⟨V, m, i, j, ⟨r, r_mem⟩, ⟨s, s_mem⟩, w⟩
  choose nin hy using w
  

Proof state:
case intro.intro.intro.intro.intro.mk.intro.mk
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
U : (Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ
a : (x : ↥U.unop) → at ↑x
ha : PrelocalPredicate.pred (isLocallyFraction 𝒜).toPrelocalPredicate a
x : ↥U.unop
V : Opens ↑(ProjectiveSpectrum.top 𝒜)
m : ↑x ∈ V
i : V ⟶ U.unop
j : ℕ
r : A
r_mem : r ∈ 𝒜 j
s : A
s_mem : s ∈ 𝒜 j
nin : ∀ (x : ↥V), ↑{ val := s, property := s_mem } ∉ (↑x).asHomogeneousIdeal
hy :
  ∀ (x : ↥V),
    (fun x => a ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)) x =
      Quotient.mk''
        { deg := j, num := { val := r, property := r_mem }, den := { val := s, property := s_mem },
          den_mem := (_ : ↑{ val := s, property := s_mem } ∉ (↑x).asHomogeneousIdeal) }
⊢ ∃ V,
    ∃ (_ : ↑x ∈ V),
      ∃ i,
        PrelocalPredicate.pred (isFractionPrelocal 𝒜) fun x =>
          (-a) ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)

Next tactic:
refine' ⟨V, m, i, j, ⟨-r, Submodule.neg_mem _ r_mem⟩, ⟨s, s_mem⟩, fun y => ⟨nin y, _⟩⟩
theorem negMem' (U : (Opens (ProjectiveSpectrum.top 𝒜))ᵒᵖ) (a : ∀ x : U.unop, at x.1)
    (ha : (isLocallyFraction 𝒜).pred a) : (isLocallyFraction 𝒜).pred (-a) := fun x => by
  rcases ha x with ⟨V, m, i, j, ⟨r, r_mem⟩, ⟨s, s_mem⟩, w⟩
  choose nin hy using w
  refine' ⟨V, m, i, j, ⟨-r, Submodule.neg_mem _ r_mem⟩, ⟨s, s_mem⟩, fun y => ⟨nin y, _⟩⟩
  

Proof state:
case intro.intro.intro.intro.intro.mk.intro.mk
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
U : (Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ
a : (x : ↥U.unop) → at ↑x
ha : PrelocalPredicate.pred (isLocallyFraction 𝒜).toPrelocalPredicate a
x : ↥U.unop
V : Opens ↑(ProjectiveSpectrum.top 𝒜)
m : ↑x ∈ V
i : V ⟶ U.unop
j : ℕ
r : A
r_mem : r ∈ 𝒜 j
s : A
s_mem : s ∈ 𝒜 j
nin : ∀ (x : ↥V), ↑{ val := s, property := s_mem } ∉ (↑x).asHomogeneousIdeal
hy :
  ∀ (x : ↥V),
    (fun x => a ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)) x =
      Quotient.mk''
        { deg := j, num := { val := r, property := r_mem }, den := { val := s, property := s_mem },
          den_mem := (_ : ↑{ val := s, property := s_mem } ∉ (↑x).asHomogeneousIdeal) }
y : ↥V
⊢ (fun x => (-a) ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)) y =
    Quotient.mk''
      { deg := j, num := { val := -r, property := (_ : -r ∈ 𝒜 j) }, den := { val := s, property := s_mem },
        den_mem := (_ : ↑{ val := s, property := s_mem } ∉ (↑y).asHomogeneousIdeal) }

Next tactic:
simp only [ext_iff_val, val_mk''] at hy
theorem negMem' (U : (Opens (ProjectiveSpectrum.top 𝒜))ᵒᵖ) (a : ∀ x : U.unop, at x.1)
    (ha : (isLocallyFraction 𝒜).pred a) : (isLocallyFraction 𝒜).pred (-a) := fun x => by
  rcases ha x with ⟨V, m, i, j, ⟨r, r_mem⟩, ⟨s, s_mem⟩, w⟩
  choose nin hy using w
  refine' ⟨V, m, i, j, ⟨-r, Submodule.neg_mem _ r_mem⟩, ⟨s, s_mem⟩, fun y => ⟨nin y, _⟩⟩
  simp only [ext_iff_val, val_mk''] at hy
  

Proof state:
case intro.intro.intro.intro.intro.mk.intro.mk
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
U : (Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ
a : (x : ↥U.unop) → at ↑x
ha : PrelocalPredicate.pred (isLocallyFraction 𝒜).toPrelocalPredicate a
x : ↥U.unop
V : Opens ↑(ProjectiveSpectrum.top 𝒜)
m : ↑x ∈ V
i : V ⟶ U.unop
j : ℕ
r : A
r_mem : r ∈ 𝒜 j
s : A
s_mem : s ∈ 𝒜 j
nin : ∀ (x : ↥V), ↑{ val := s, property := s_mem } ∉ (↑x).asHomogeneousIdeal
y : ↥V
hy :
  ∀ (x : ↥V),
    val (a { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) =
      Localization.mk r
        { val := s,
          property :=
            (_ :
              ↑{ deg := j, num := { val := r, property := r_mem }, den := { val := s, property := s_mem },
                      den_mem := (_ : ↑{ val := s, property := s_mem } ∉ (↑x).asHomogeneousIdeal) }.den ∈
                Ideal.primeCompl (HomogeneousIdeal.toIdeal (↑x).asHomogeneousIdeal)) }
⊢ (fun x => (-a) ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)) y =
    Quotient.mk''
      { deg := j, num := { val := -r, property := (_ : -r ∈ 𝒜 j) }, den := { val := s, property := s_mem },
        den_mem := (_ : ↑{ val := s, property := s_mem } ∉ (↑y).asHomogeneousIdeal) }

Next tactic:
simp only [Pi.neg_apply, ext_iff_val, neg_val, hy, val_mk'', neg_mk]
theorem mulMem' (U : (Opens (ProjectiveSpectrum.top 𝒜))ᵒᵖ) (a b : ∀ x : U.unop, at x.1)
    (ha : (isLocallyFraction 𝒜).pred a) (hb : (isLocallyFraction 𝒜).pred b) :
    (isLocallyFraction 𝒜).pred (a * b) := fun x => by
  

Proof state:
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
U : (Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ
a b : (x : ↥U.unop) → at ↑x
ha : PrelocalPredicate.pred (isLocallyFraction 𝒜).toPrelocalPredicate a
hb : PrelocalPredicate.pred (isLocallyFraction 𝒜).toPrelocalPredicate b
x : ↥U.unop
⊢ ∃ V,
    ∃ (_ : ↑x ∈ V),
      ∃ i,
        PrelocalPredicate.pred (isFractionPrelocal 𝒜) fun x =>
          (a * b) ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)

Next tactic:
rcases ha x with ⟨Va, ma, ia, ja, ⟨ra, ra_mem⟩, ⟨sa, sa_mem⟩, wa⟩
theorem mulMem' (U : (Opens (ProjectiveSpectrum.top 𝒜))ᵒᵖ) (a b : ∀ x : U.unop, at x.1)
    (ha : (isLocallyFraction 𝒜).pred a) (hb : (isLocallyFraction 𝒜).pred b) :
    (isLocallyFraction 𝒜).pred (a * b) := fun x => by
  rcases ha x with ⟨Va, ma, ia, ja, ⟨ra, ra_mem⟩, ⟨sa, sa_mem⟩, wa⟩
  

Proof state:
case intro.intro.intro.intro.intro.mk.intro.mk
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
U : (Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ
a b : (x : ↥U.unop) → at ↑x
ha : PrelocalPredicate.pred (isLocallyFraction 𝒜).toPrelocalPredicate a
hb : PrelocalPredicate.pred (isLocallyFraction 𝒜).toPrelocalPredicate b
x : ↥U.unop
Va : Opens ↑(ProjectiveSpectrum.top 𝒜)
ma : ↑x ∈ Va
ia : Va ⟶ U.unop
ja : ℕ
ra : A
ra_mem : ra ∈ 𝒜 ja
sa : A
sa_mem : sa ∈ 𝒜 ja
wa :
  ∀ (x : ↥Va),
    ∃ (s_nin : ↑{ val := sa, property := sa_mem } ∉ (↑x).asHomogeneousIdeal),
      (fun x => a ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)) x =
        Quotient.mk''
          { deg := ja, num := { val := ra, property := ra_mem }, den := { val := sa, property := sa_mem },
            den_mem := s_nin }
⊢ ∃ V,
    ∃ (_ : ↑x ∈ V),
      ∃ i,
        PrelocalPredicate.pred (isFractionPrelocal 𝒜) fun x =>
          (a * b) ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)

Next tactic:
rcases hb x with ⟨Vb, mb, ib, jb, ⟨rb, rb_mem⟩, ⟨sb, sb_mem⟩, wb⟩
theorem mulMem' (U : (Opens (ProjectiveSpectrum.top 𝒜))ᵒᵖ) (a b : ∀ x : U.unop, at x.1)
    (ha : (isLocallyFraction 𝒜).pred a) (hb : (isLocallyFraction 𝒜).pred b) :
    (isLocallyFraction 𝒜).pred (a * b) := fun x => by
  rcases ha x with ⟨Va, ma, ia, ja, ⟨ra, ra_mem⟩, ⟨sa, sa_mem⟩, wa⟩
  rcases hb x with ⟨Vb, mb, ib, jb, ⟨rb, rb_mem⟩, ⟨sb, sb_mem⟩, wb⟩
  

Proof state:
case intro.intro.intro.intro.intro.mk.intro.mk.intro.intro.intro.intro.intro.mk.intro.mk
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
U : (Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ
a b : (x : ↥U.unop) → at ↑x
ha : PrelocalPredicate.pred (isLocallyFraction 𝒜).toPrelocalPredicate a
hb : PrelocalPredicate.pred (isLocallyFraction 𝒜).toPrelocalPredicate b
x : ↥U.unop
Va : Opens ↑(ProjectiveSpectrum.top 𝒜)
ma : ↑x ∈ Va
ia : Va ⟶ U.unop
ja : ℕ
ra : A
ra_mem : ra ∈ 𝒜 ja
sa : A
sa_mem : sa ∈ 𝒜 ja
wa :
  ∀ (x : ↥Va),
    ∃ (s_nin : ↑{ val := sa, property := sa_mem } ∉ (↑x).asHomogeneousIdeal),
      (fun x => a ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)) x =
        Quotient.mk''
          { deg := ja, num := { val := ra, property := ra_mem }, den := { val := sa, property := sa_mem },
            den_mem := s_nin }
Vb : Opens ↑(ProjectiveSpectrum.top 𝒜)
mb : ↑x ∈ Vb
ib : Vb ⟶ U.unop
jb : ℕ
rb : A
rb_mem : rb ∈ 𝒜 jb
sb : A
sb_mem : sb ∈ 𝒜 jb
wb :
  ∀ (x : ↥Vb),
    ∃ (s_nin : ↑{ val := sb, property := sb_mem } ∉ (↑x).asHomogeneousIdeal),
      (fun x => b ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)) x =
        Quotient.mk''
          { deg := jb, num := { val := rb, property := rb_mem }, den := { val := sb, property := sb_mem },
            den_mem := s_nin }
⊢ ∃ V,
    ∃ (_ : ↑x ∈ V),
      ∃ i,
        PrelocalPredicate.pred (isFractionPrelocal 𝒜) fun x =>
          (a * b) ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)

Next tactic:
refine'
    ⟨Va ⊓ Vb, ⟨ma, mb⟩, Opens.infLELeft _ _ ≫ ia, ja + jb,
      ⟨ra * rb, SetLike.mul_mem_graded ra_mem rb_mem⟩,
      ⟨sa * sb, SetLike.mul_mem_graded sa_mem sb_mem⟩, fun y => ⟨fun h => _, _⟩⟩
theorem mulMem' (U : (Opens (ProjectiveSpectrum.top 𝒜))ᵒᵖ) (a b : ∀ x : U.unop, at x.1)
    (ha : (isLocallyFraction 𝒜).pred a) (hb : (isLocallyFraction 𝒜).pred b) :
    (isLocallyFraction 𝒜).pred (a * b) := fun x => by
  rcases ha x with ⟨Va, ma, ia, ja, ⟨ra, ra_mem⟩, ⟨sa, sa_mem⟩, wa⟩
  rcases hb x with ⟨Vb, mb, ib, jb, ⟨rb, rb_mem⟩, ⟨sb, sb_mem⟩, wb⟩
  refine'
    ⟨Va ⊓ Vb, ⟨ma, mb⟩, Opens.infLELeft _ _ ≫ ia, ja + jb,
      ⟨ra * rb, SetLike.mul_mem_graded ra_mem rb_mem⟩,
      ⟨sa * sb, SetLike.mul_mem_graded sa_mem sb_mem⟩, fun y => ⟨fun h => _, _⟩⟩
  · 

Proof state:
case intro.intro.intro.intro.intro.mk.intro.mk.intro.intro.intro.intro.intro.mk.intro.mk.refine'_1
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
U : (Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ
a b : (x : ↥U.unop) → at ↑x
ha : PrelocalPredicate.pred (isLocallyFraction 𝒜).toPrelocalPredicate a
hb : PrelocalPredicate.pred (isLocallyFraction 𝒜).toPrelocalPredicate b
x : ↥U.unop
Va : Opens ↑(ProjectiveSpectrum.top 𝒜)
ma : ↑x ∈ Va
ia : Va ⟶ U.unop
ja : ℕ
ra : A
ra_mem : ra ∈ 𝒜 ja
sa : A
sa_mem : sa ∈ 𝒜 ja
wa :
  ∀ (x : ↥Va),
    ∃ (s_nin : ↑{ val := sa, property := sa_mem } ∉ (↑x).asHomogeneousIdeal),
      (fun x => a ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)) x =
        Quotient.mk''
          { deg := ja, num := { val := ra, property := ra_mem }, den := { val := sa, property := sa_mem },
            den_mem := s_nin }
Vb : Opens ↑(ProjectiveSpectrum.top 𝒜)
mb : ↑x ∈ Vb
ib : Vb ⟶ U.unop
jb : ℕ
rb : A
rb_mem : rb ∈ 𝒜 jb
sb : A
sb_mem : sb ∈ 𝒜 jb
wb :
  ∀ (x : ↥Vb),
    ∃ (s_nin : ↑{ val := sb, property := sb_mem } ∉ (↑x).asHomogeneousIdeal),
      (fun x => b ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)) x =
        Quotient.mk''
          { deg := jb, num := { val := rb, property := rb_mem }, den := { val := sb, property := sb_mem },
            den_mem := s_nin }
y : ↥(Va ⊓ Vb)
h : ↑{ val := sa * sb, property := (_ : sa * sb ∈ 𝒜 (ja + jb)) } ∈ (↑y).asHomogeneousIdeal
⊢ False

Next tactic:
cases' (y : ProjectiveSpectrum.top 𝒜).isPrime.mem_or_mem h with h h
theorem mulMem' (U : (Opens (ProjectiveSpectrum.top 𝒜))ᵒᵖ) (a b : ∀ x : U.unop, at x.1)
    (ha : (isLocallyFraction 𝒜).pred a) (hb : (isLocallyFraction 𝒜).pred b) :
    (isLocallyFraction 𝒜).pred (a * b) := fun x => by
  rcases ha x with ⟨Va, ma, ia, ja, ⟨ra, ra_mem⟩, ⟨sa, sa_mem⟩, wa⟩
  rcases hb x with ⟨Vb, mb, ib, jb, ⟨rb, rb_mem⟩, ⟨sb, sb_mem⟩, wb⟩
  refine'
    ⟨Va ⊓ Vb, ⟨ma, mb⟩, Opens.infLELeft _ _ ≫ ia, ja + jb,
      ⟨ra * rb, SetLike.mul_mem_graded ra_mem rb_mem⟩,
      ⟨sa * sb, SetLike.mul_mem_graded sa_mem sb_mem⟩, fun y => ⟨fun h => _, _⟩⟩
  · cases' (y : ProjectiveSpectrum.top 𝒜).isPrime.mem_or_mem h with h h
    · 

Proof state:
case intro.intro.intro.intro.intro.mk.intro.mk.intro.intro.intro.intro.intro.mk.intro.mk.refine'_1.inl
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
U : (Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ
a b : (x : ↥U.unop) → at ↑x
ha : PrelocalPredicate.pred (isLocallyFraction 𝒜).toPrelocalPredicate a
hb : PrelocalPredicate.pred (isLocallyFraction 𝒜).toPrelocalPredicate b
x : ↥U.unop
Va : Opens ↑(ProjectiveSpectrum.top 𝒜)
ma : ↑x ∈ Va
ia : Va ⟶ U.unop
ja : ℕ
ra : A
ra_mem : ra ∈ 𝒜 ja
sa : A
sa_mem : sa ∈ 𝒜 ja
wa :
  ∀ (x : ↥Va),
    ∃ (s_nin : ↑{ val := sa, property := sa_mem } ∉ (↑x).asHomogeneousIdeal),
      (fun x => a ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)) x =
        Quotient.mk''
          { deg := ja, num := { val := ra, property := ra_mem }, den := { val := sa, property := sa_mem },
            den_mem := s_nin }
Vb : Opens ↑(ProjectiveSpectrum.top 𝒜)
mb : ↑x ∈ Vb
ib : Vb ⟶ U.unop
jb : ℕ
rb : A
rb_mem : rb ∈ 𝒜 jb
sb : A
sb_mem : sb ∈ 𝒜 jb
wb :
  ∀ (x : ↥Vb),
    ∃ (s_nin : ↑{ val := sb, property := sb_mem } ∉ (↑x).asHomogeneousIdeal),
      (fun x => b ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)) x =
        Quotient.mk''
          { deg := jb, num := { val := rb, property := rb_mem }, den := { val := sb, property := sb_mem },
            den_mem := s_nin }
y : ↥(Va ⊓ Vb)
h✝ : ↑{ val := sa * sb, property := (_ : sa * sb ∈ 𝒜 (ja + jb)) } ∈ (↑y).asHomogeneousIdeal
h : sa ∈ HomogeneousIdeal.toIdeal (↑y).asHomogeneousIdeal
⊢ False

Next tactic:
choose nin _ using wa ⟨y, (Opens.infLELeft Va Vb y).2⟩
theorem mulMem' (U : (Opens (ProjectiveSpectrum.top 𝒜))ᵒᵖ) (a b : ∀ x : U.unop, at x.1)
    (ha : (isLocallyFraction 𝒜).pred a) (hb : (isLocallyFraction 𝒜).pred b) :
    (isLocallyFraction 𝒜).pred (a * b) := fun x => by
  rcases ha x with ⟨Va, ma, ia, ja, ⟨ra, ra_mem⟩, ⟨sa, sa_mem⟩, wa⟩
  rcases hb x with ⟨Vb, mb, ib, jb, ⟨rb, rb_mem⟩, ⟨sb, sb_mem⟩, wb⟩
  refine'
    ⟨Va ⊓ Vb, ⟨ma, mb⟩, Opens.infLELeft _ _ ≫ ia, ja + jb,
      ⟨ra * rb, SetLike.mul_mem_graded ra_mem rb_mem⟩,
      ⟨sa * sb, SetLike.mul_mem_graded sa_mem sb_mem⟩, fun y => ⟨fun h => _, _⟩⟩
  · cases' (y : ProjectiveSpectrum.top 𝒜).isPrime.mem_or_mem h with h h
    · choose nin _ using wa ⟨y, (Opens.infLELeft Va Vb y).2⟩; 

Proof state:
case intro.intro.intro.intro.intro.mk.intro.mk.intro.intro.intro.intro.intro.mk.intro.mk.refine'_1.inl
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
U : (Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ
a b : (x : ↥U.unop) → at ↑x
ha : PrelocalPredicate.pred (isLocallyFraction 𝒜).toPrelocalPredicate a
hb : PrelocalPredicate.pred (isLocallyFraction 𝒜).toPrelocalPredicate b
x : ↥U.unop
Va : Opens ↑(ProjectiveSpectrum.top 𝒜)
ma : ↑x ∈ Va
ia : Va ⟶ U.unop
ja : ℕ
ra : A
ra_mem : ra ∈ 𝒜 ja
sa : A
sa_mem : sa ∈ 𝒜 ja
wa :
  ∀ (x : ↥Va),
    ∃ (s_nin : ↑{ val := sa, property := sa_mem } ∉ (↑x).asHomogeneousIdeal),
      (fun x => a ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)) x =
        Quotient.mk''
          { deg := ja, num := { val := ra, property := ra_mem }, den := { val := sa, property := sa_mem },
            den_mem := s_nin }
Vb : Opens ↑(ProjectiveSpectrum.top 𝒜)
mb : ↑x ∈ Vb
ib : Vb ⟶ U.unop
jb : ℕ
rb : A
rb_mem : rb ∈ 𝒜 jb
sb : A
sb_mem : sb ∈ 𝒜 jb
wb :
  ∀ (x : ↥Vb),
    ∃ (s_nin : ↑{ val := sb, property := sb_mem } ∉ (↑x).asHomogeneousIdeal),
      (fun x => b ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)) x =
        Quotient.mk''
          { deg := jb, num := { val := rb, property := rb_mem }, den := { val := sb, property := sb_mem },
            den_mem := s_nin }
y : ↥(Va ⊓ Vb)
h✝ : ↑{ val := sa * sb, property := (_ : sa * sb ∈ 𝒜 (ja + jb)) } ∈ (↑y).asHomogeneousIdeal
h : sa ∈ HomogeneousIdeal.toIdeal (↑y).asHomogeneousIdeal
nin :
  ↑{ val := sa, property := sa_mem } ∉
    (↑{ val := ↑y,
          property := (_ : ↑((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑Va) }) y) ∈ Va) }).asHomogeneousIdeal
x✝ :
  (fun x => a ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x))
      { val := ↑y, property := (_ : ↑((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑Va) }) y) ∈ Va) } =
    Quotient.mk''
      { deg := ja, num := { val := ra, property := ra_mem }, den := { val := sa, property := sa_mem }, den_mem := nin }
⊢ False

Next tactic:
exact nin h
theorem mulMem' (U : (Opens (ProjectiveSpectrum.top 𝒜))ᵒᵖ) (a b : ∀ x : U.unop, at x.1)
    (ha : (isLocallyFraction 𝒜).pred a) (hb : (isLocallyFraction 𝒜).pred b) :
    (isLocallyFraction 𝒜).pred (a * b) := fun x => by
  rcases ha x with ⟨Va, ma, ia, ja, ⟨ra, ra_mem⟩, ⟨sa, sa_mem⟩, wa⟩
  rcases hb x with ⟨Vb, mb, ib, jb, ⟨rb, rb_mem⟩, ⟨sb, sb_mem⟩, wb⟩
  refine'
    ⟨Va ⊓ Vb, ⟨ma, mb⟩, Opens.infLELeft _ _ ≫ ia, ja + jb,
      ⟨ra * rb, SetLike.mul_mem_graded ra_mem rb_mem⟩,
      ⟨sa * sb, SetLike.mul_mem_graded sa_mem sb_mem⟩, fun y => ⟨fun h => _, _⟩⟩
  · cases' (y : ProjectiveSpectrum.top 𝒜).isPrime.mem_or_mem h with h h
    · choose nin _ using wa ⟨y, (Opens.infLELeft Va Vb y).2⟩; exact nin h
    · 

Proof state:
case intro.intro.intro.intro.intro.mk.intro.mk.intro.intro.intro.intro.intro.mk.intro.mk.refine'_1.inr
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
U : (Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ
a b : (x : ↥U.unop) → at ↑x
ha : PrelocalPredicate.pred (isLocallyFraction 𝒜).toPrelocalPredicate a
hb : PrelocalPredicate.pred (isLocallyFraction 𝒜).toPrelocalPredicate b
x : ↥U.unop
Va : Opens ↑(ProjectiveSpectrum.top 𝒜)
ma : ↑x ∈ Va
ia : Va ⟶ U.unop
ja : ℕ
ra : A
ra_mem : ra ∈ 𝒜 ja
sa : A
sa_mem : sa ∈ 𝒜 ja
wa :
  ∀ (x : ↥Va),
    ∃ (s_nin : ↑{ val := sa, property := sa_mem } ∉ (↑x).asHomogeneousIdeal),
      (fun x => a ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)) x =
        Quotient.mk''
          { deg := ja, num := { val := ra, property := ra_mem }, den := { val := sa, property := sa_mem },
            den_mem := s_nin }
Vb : Opens ↑(ProjectiveSpectrum.top 𝒜)
mb : ↑x ∈ Vb
ib : Vb ⟶ U.unop
jb : ℕ
rb : A
rb_mem : rb ∈ 𝒜 jb
sb : A
sb_mem : sb ∈ 𝒜 jb
wb :
  ∀ (x : ↥Vb),
    ∃ (s_nin : ↑{ val := sb, property := sb_mem } ∉ (↑x).asHomogeneousIdeal),
      (fun x => b ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)) x =
        Quotient.mk''
          { deg := jb, num := { val := rb, property := rb_mem }, den := { val := sb, property := sb_mem },
            den_mem := s_nin }
y : ↥(Va ⊓ Vb)
h✝ : ↑{ val := sa * sb, property := (_ : sa * sb ∈ 𝒜 (ja + jb)) } ∈ (↑y).asHomogeneousIdeal
h : sb ∈ HomogeneousIdeal.toIdeal (↑y).asHomogeneousIdeal
⊢ False

Next tactic:
choose nin _ using wb ⟨y, (Opens.infLERight Va Vb y).2⟩
theorem mulMem' (U : (Opens (ProjectiveSpectrum.top 𝒜))ᵒᵖ) (a b : ∀ x : U.unop, at x.1)
    (ha : (isLocallyFraction 𝒜).pred a) (hb : (isLocallyFraction 𝒜).pred b) :
    (isLocallyFraction 𝒜).pred (a * b) := fun x => by
  rcases ha x with ⟨Va, ma, ia, ja, ⟨ra, ra_mem⟩, ⟨sa, sa_mem⟩, wa⟩
  rcases hb x with ⟨Vb, mb, ib, jb, ⟨rb, rb_mem⟩, ⟨sb, sb_mem⟩, wb⟩
  refine'
    ⟨Va ⊓ Vb, ⟨ma, mb⟩, Opens.infLELeft _ _ ≫ ia, ja + jb,
      ⟨ra * rb, SetLike.mul_mem_graded ra_mem rb_mem⟩,
      ⟨sa * sb, SetLike.mul_mem_graded sa_mem sb_mem⟩, fun y => ⟨fun h => _, _⟩⟩
  · cases' (y : ProjectiveSpectrum.top 𝒜).isPrime.mem_or_mem h with h h
    · choose nin _ using wa ⟨y, (Opens.infLELeft Va Vb y).2⟩; exact nin h
    · choose nin _ using wb ⟨y, (Opens.infLERight Va Vb y).2⟩; 

Proof state:
case intro.intro.intro.intro.intro.mk.intro.mk.intro.intro.intro.intro.intro.mk.intro.mk.refine'_1.inr
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
U : (Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ
a b : (x : ↥U.unop) → at ↑x
ha : PrelocalPredicate.pred (isLocallyFraction 𝒜).toPrelocalPredicate a
hb : PrelocalPredicate.pred (isLocallyFraction 𝒜).toPrelocalPredicate b
x : ↥U.unop
Va : Opens ↑(ProjectiveSpectrum.top 𝒜)
ma : ↑x ∈ Va
ia : Va ⟶ U.unop
ja : ℕ
ra : A
ra_mem : ra ∈ 𝒜 ja
sa : A
sa_mem : sa ∈ 𝒜 ja
wa :
  ∀ (x : ↥Va),
    ∃ (s_nin : ↑{ val := sa, property := sa_mem } ∉ (↑x).asHomogeneousIdeal),
      (fun x => a ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)) x =
        Quotient.mk''
          { deg := ja, num := { val := ra, property := ra_mem }, den := { val := sa, property := sa_mem },
            den_mem := s_nin }
Vb : Opens ↑(ProjectiveSpectrum.top 𝒜)
mb : ↑x ∈ Vb
ib : Vb ⟶ U.unop
jb : ℕ
rb : A
rb_mem : rb ∈ 𝒜 jb
sb : A
sb_mem : sb ∈ 𝒜 jb
wb :
  ∀ (x : ↥Vb),
    ∃ (s_nin : ↑{ val := sb, property := sb_mem } ∉ (↑x).asHomogeneousIdeal),
      (fun x => b ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x)) x =
        Quotient.mk''
          { deg := jb, num := { val := rb, property := rb_mem }, den := { val := sb, property := sb_mem },
            den_mem := s_nin }
y : ↥(Va ⊓ Vb)
h✝ : ↑{ val := sa * sb, property := (_ : sa * sb ∈ 𝒜 (ja + jb)) } ∈ (↑y).asHomogeneousIdeal
h : sb ∈ HomogeneousIdeal.toIdeal (↑y).asHomogeneousIdeal
nin :
  ↑{ val := sb, property := sb_mem } ∉
    (↑{ val := ↑y,
          property := (_ : ↑((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑Vb) }) y) ∈ Vb) }).asHomogeneousIdeal
x✝ :
  (fun x => b ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑U.unop) }) x))
      { val := ↑y, property := (_ : ↑((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑Vb) }) y) ∈ Vb) } =
    Quotient.mk''
      { deg := jb, num := { val := rb, property := rb_mem }, den := { val := sb, property := sb_mem }, den_mem := nin }
⊢ False

Next tactic:
exact nin h