mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicGeometry.Modules.Tilde

Goal state before:
pp: R : CommRingCat
M✝ : ModuleCat ↑R
M N : (Spec R).Modules
f : modulesSpecToSheaf.obj M ⟶ modulesSpecToSheaf.obj N
U : (Opens ↥(Spec R))ᵒᵖ
t : ↑((Spec R).ringCatSheaf.obj.obj U)
m : ↑(M.val.1 U)
x : ↥(Spec R)
hxU : x ∈ unop U
r : ↑R
hxr : x ∈ ↑(basicOpen r)
hrU : basicOpen r ≤ unop U
this :
  ∀ (x : ↑((modulesSpecToSheaf.obj M).obj.obj (op (unop U)))),
    (ConcreteCategory.hom (f.hom.app (op (basicOpen r)))) ((ConcreteCategory.hom (M.val.map (homOfLE hrU).op)) x) =
      (ConcreteCategory.hom (N.val.map (homOfLE hrU).op)) ((ConcreteCategory.hom (f.hom.app (op (unop U)))) x)
⊢ (ConcreteCategory.hom (f.hom.app (op (basicOpen r)))) ((ConcreteCategory.hom (M.val.map (homOfLE hrU).op)) (t • m)) =
    (ConcreteCategory.hom ((Spec R).ringCatSheaf.obj.map (homOfLE hrU).op)) t •
      (ConcreteCategory.hom (f.hom.app (op (basicOpen r)))) ((ConcreteCategory.hom (M.val.map (homOfLE hrU).op)) m)

Tactic:
M.val.map_smul

Goal state after:
R : CommRingCat
M✝ : ModuleCat ↑R
M N : (Spec R).Modules
f : modulesSpecToSheaf.obj M ⟶ modulesSpecToSheaf.obj N
U : (Opens ↥(Spec R))ᵒᵖ
t : ↑((Spec R).ringCatSheaf.obj.obj U)
m : ↑(M.val.1 U)
x : ↥(Spec R)
hxU : x ∈ unop U
r : ↑R
hxr : x ∈ ↑(basicOpen r)
hrU : basicOpen r ≤ unop U
this :
  ∀ (x : ↑((modulesSpecToSheaf.obj M).obj.obj (op (unop U)))),
    (ConcreteCategory.hom (f.hom.app (op (basicOpen r)))) ((ConcreteCategory.hom (M.val.map (homOfLE hrU).op)) x) =
      (ConcreteCategory.hom (N.val.map (homOfLE hrU).op)) ((ConcreteCategory.hom (f.hom.app (op (unop U)))) x)
⊢ (ConcreteCategory.hom (f.hom.app (op (basicOpen r))))
      ((ConcreteCategory.hom ((Spec R).ringCatSheaf.obj.map (homOfLE hrU).op)) t •
        (ConcreteCategory.hom (M.val.map (homOfLE hrU).op)) m) =
    (ConcreteCategory.hom ((Spec R).ringCatSheaf.obj.map (homOfLE hrU).op)) t •
      (ConcreteCategory.hom (f.hom.app (op (basicOpen r)))) ((ConcreteCategory.hom (M.val.map (homOfLE hrU).op)) m)
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: case pos.refine_2
ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁶ : CommRing A
inst✝⁵ : SetLike σ A
inst✝⁴ : AddSubgroupClass σ A
inst✝³ : AddCommMonoid ι
inst✝² : DecidableEq ι
𝒜 : ι → σ
inst✝¹ : GradedRing 𝒜
f : A
d : ι
hf : f ∈ 𝒜 d
ι' : Type u_4
inst✝ : Fintype ι'
v : ι' → A
hx : Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤
dv : ι' → ι
hxd : ∀ (i : ι'), v i ∈ 𝒜 (dv i)
HH : ¬Subsingleton (Away 𝒜 f)
n : ι
a : A
j : ℕ
hb' : (fun x ↦ f ^ x) j ∈ 𝒜 n
hfj : ¬f ^ j = 0
ai : ι' → ℕ
H : ∑ i, ai i • dv i = n
⊢ Away.mk 𝒜 hf j (∏ i, v i ^ ai i) ⋯ =
    mk { deg := n, num := ((decompose 𝒜) (∏ i, v i ^ ai i)) n, den := ⟨f ^ j, hb'⟩, den_mem := ⋯ }

Tactic:
ext

Goal state after:
case pos.refine_2
ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁶ : CommRing A
inst✝⁵ : SetLike σ A
inst✝⁴ : AddSubgroupClass σ A
inst✝³ : AddCommMonoid ι
inst✝² : DecidableEq ι
𝒜 : ι → σ
inst✝¹ : GradedRing 𝒜
f : A
d : ι
hf : f ∈ 𝒜 d
ι' : Type u_4
inst✝ : Fintype ι'
v : ι' → A
hx : Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤
dv : ι' → ι
hxd : ∀ (i : ι'), v i ∈ 𝒜 (dv i)
HH : ¬Subsingleton (Away 𝒜 f)
n : ι
a : A
j : ℕ
hb' : (fun x ↦ f ^ x) j ∈ 𝒜 n
hfj : ¬f ^ j = 0
ai : ι' → ℕ
H : ∑ i, ai i • dv i = n
⊢ val (Away.mk 𝒜 hf j (∏ i, v i ^ ai i) ⋯) =
    (mk { deg := n, num := ((decompose 𝒜) (∏ i, v i ^ ai i)) n, den := ⟨f ^ j, hb'⟩, den_mem := ⋯ }).val
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: case e'_5
ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁶ : CommRing A
inst✝⁵ : SetLike σ A
inst✝⁴ : AddSubgroupClass σ A
inst✝³ : AddCommMonoid ι
inst✝² : DecidableEq ι
𝒜 : ι → σ
inst✝¹ : GradedRing 𝒜
f : A
d : ι
hf : f ∈ 𝒜 d
ι' : Type u_4
inst✝ : Fintype ι'
v : ι' → A
hx : Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤
dv : ι' → ι
hxd : ∀ (i : ι'), v i ∈ 𝒜 (dv i)
HH : ¬Subsingleton (Away 𝒜 f)
n : ι
a : A
j : ℕ
hb' : (fun x ↦ f ^ x) j ∈ 𝒜 n
hfj : ¬f ^ j = 0
ai : ι' → ℕ
H : ¬∑ i, ai i • dv i = n
⊢ mk { deg := n, num := ((decompose 𝒜) (∏ i, v i ^ ai i)) n, den := ⟨f ^ j, hb'⟩, den_mem := ⋯ } = 0

Tactic:
ext

Goal state after:
case e'_5
ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁶ : CommRing A
inst✝⁵ : SetLike σ A
inst✝⁴ : AddSubgroupClass σ A
inst✝³ : AddCommMonoid ι
inst✝² : DecidableEq ι
𝒜 : ι → σ
inst✝¹ : GradedRing 𝒜
f : A
d : ι
hf : f ∈ 𝒜 d
ι' : Type u_4
inst✝ : Fintype ι'
v : ι' → A
hx : Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤
dv : ι' → ι
hxd : ∀ (i : ι'), v i ∈ 𝒜 (dv i)
HH : ¬Subsingleton (Away 𝒜 f)
n : ι
a : A
j : ℕ
hb' : (fun x ↦ f ^ x) j ∈ 𝒜 n
hfj : ¬f ^ j = 0
ai : ι' → ℕ
H : ¬∑ i, ai i • dv i = n
⊢ (mk { deg := n, num := ((decompose 𝒜) (∏ i, v i ^ ai i)) n, den := ⟨f ^ j, hb'⟩, den_mem := ⋯ }).val = val 0
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: case e'_5
ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁶ : CommRing A
inst✝⁵ : SetLike σ A
inst✝⁴ : AddSubgroupClass σ A
inst✝³ : AddCommMonoid ι
inst✝² : DecidableEq ι
𝒜 : ι → σ
inst✝¹ : GradedRing 𝒜
f : A
d : ι
hf : f ∈ 𝒜 d
ι' : Type u_4
inst✝ : Fintype ι'
v : ι' → A
hx : Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤
dv : ι' → ι
hxd : ∀ (i : ι'), v i ∈ 𝒜 (dv i)
HH : ¬Subsingleton (Away 𝒜 f)
n : ι
a : A
j : ℕ
hb' : (fun x ↦ f ^ x) j ∈ 𝒜 n
hfj : ¬f ^ j = 0
⊢ mk { deg := n, num := ((decompose 𝒜) 0) n, den := ⟨f ^ j, hb'⟩, den_mem := ⋯ } = 0

Tactic:
ext

Goal state after:
case e'_5
ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁶ : CommRing A
inst✝⁵ : SetLike σ A
inst✝⁴ : AddSubgroupClass σ A
inst✝³ : AddCommMonoid ι
inst✝² : DecidableEq ι
𝒜 : ι → σ
inst✝¹ : GradedRing 𝒜
f : A
d : ι
hf : f ∈ 𝒜 d
ι' : Type u_4
inst✝ : Fintype ι'
v : ι' → A
hx : Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤
dv : ι' → ι
hxd : ∀ (i : ι'), v i ∈ 𝒜 (dv i)
HH : ¬Subsingleton (Away 𝒜 f)
n : ι
a : A
j : ℕ
hb' : (fun x ↦ f ^ x) j ∈ 𝒜 n
hfj : ¬f ^ j = 0
⊢ (mk { deg := n, num := ((decompose 𝒜) 0) n, den := ⟨f ^ j, hb'⟩, den_mem := ⋯ }).val = val 0
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁶ : CommRing A
inst✝⁵ : SetLike σ A
inst✝⁴ : AddSubgroupClass σ A
inst✝³ : AddCommMonoid ι
inst✝² : DecidableEq ι
𝒜 : ι → σ
inst✝¹ : GradedRing 𝒜
f : A
d : ι
hf : f ∈ 𝒜 d
ι' : Type u_4
inst✝ : Fintype ι'
v : ι' → A
hx : Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤
dv : ι' → ι
hxd : ∀ (i : ι'), v i ∈ 𝒜 (dv i)
HH : ¬Subsingleton (Away 𝒜 f)
n : ι
a : A
j : ℕ
hb' : (fun x ↦ f ^ x) j ∈ 𝒜 n
hfj : ¬f ^ j = 0
s t : A
hs : s ∈ Submodule.span ↥(𝒜 0) ↑(Submonoid.closure (Set.range v))
ht : t ∈ Submodule.span ↥(𝒜 0) ↑(Submonoid.closure (Set.range v))
hs' :
  mk { deg := n, num := ((decompose 𝒜) s) n, den := ⟨f ^ j, hb'⟩, den_mem := ⋯ } ∈
    Submodule.span ↥(𝒜 0) {x | ∃ a ai, ∃ (hai : ∑ i, ai i • dv i = a • d), Away.mk 𝒜 hf a (∏ i, v i ^ ai i) ⋯ = x}
ht' :
  mk { deg := n, num := ((decompose 𝒜) t) n, den := ⟨f ^ j, hb'⟩, den_mem := ⋯ } ∈
    Submodule.span ↥(𝒜 0) {x | ∃ a ai, ∃ (hai : ∑ i, ai i • dv i = a • d), Away.mk 𝒜 hf a (∏ i, v i ^ ai i) ⋯ = x}
⊢ mk { deg := n, num := ((decompose 𝒜) (s + t)) n, den := ⟨f ^ j, hb'⟩, den_mem := ⋯ } =
    mk { deg := n, num := ((decompose 𝒜) s) n, den := ⟨f ^ j, hb'⟩, den_mem := ⋯ } +
      mk { deg := n, num := ((decompose 𝒜) t) n, den := ⟨f ^ j, hb'⟩, den_mem := ⋯ }

Tactic:
ext

Goal state after:
ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁶ : CommRing A
inst✝⁵ : SetLike σ A
inst✝⁴ : AddSubgroupClass σ A
inst✝³ : AddCommMonoid ι
inst✝² : DecidableEq ι
𝒜 : ι → σ
inst✝¹ : GradedRing 𝒜
f : A
d : ι
hf : f ∈ 𝒜 d
ι' : Type u_4
inst✝ : Fintype ι'
v : ι' → A
hx : Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤
dv : ι' → ι
hxd : ∀ (i : ι'), v i ∈ 𝒜 (dv i)
HH : ¬Subsingleton (Away 𝒜 f)
n : ι
a : A
j : ℕ
hb' : (fun x ↦ f ^ x) j ∈ 𝒜 n
hfj : ¬f ^ j = 0
s t : A
hs : s ∈ Submodule.span ↥(𝒜 0) ↑(Submonoid.closure (Set.range v))
ht : t ∈ Submodule.span ↥(𝒜 0) ↑(Submonoid.closure (Set.range v))
hs' :
  mk { deg := n, num := ((decompose 𝒜) s) n, den := ⟨f ^ j, hb'⟩, den_mem := ⋯ } ∈
    Submodule.span ↥(𝒜 0) {x | ∃ a ai, ∃ (hai : ∑ i, ai i • dv i = a • d), Away.mk 𝒜 hf a (∏ i, v i ^ ai i) ⋯ = x}
ht' :
  mk { deg := n, num := ((decompose 𝒜) t) n, den := ⟨f ^ j, hb'⟩, den_mem := ⋯ } ∈
    Submodule.span ↥(𝒜 0) {x | ∃ a ai, ∃ (hai : ∑ i, ai i • dv i = a • d), Away.mk 𝒜 hf a (∏ i, v i ^ ai i) ⋯ = x}
⊢ (mk { deg := n, num := ((decompose 𝒜) (s + t)) n, den := ⟨f ^ j, hb'⟩, den_mem := ⋯ }).val =
    val
      (mk { deg := n, num := ((decompose 𝒜) s) n, den := ⟨f ^ j, hb'⟩, den_mem := ⋯ } +
        mk { deg := n, num := ((decompose 𝒜) t) n, den := ⟨f ^ j, hb'⟩, den_mem := ⋯ })
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁶ : CommRing A
inst✝⁵ : SetLike σ A
inst✝⁴ : AddSubgroupClass σ A
inst✝³ : AddCommMonoid ι
inst✝² : DecidableEq ι
𝒜 : ι → σ
inst✝¹ : GradedRing 𝒜
f : A
d : ι
hf : f ∈ 𝒜 d
ι' : Type u_4
inst✝ : Fintype ι'
v : ι' → A
hx✝ : Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤
dv : ι' → ι
hxd : ∀ (i : ι'), v i ∈ 𝒜 (dv i)
HH : ¬Subsingleton (Away 𝒜 f)
n : ι
a : A
j : ℕ
hb' : (fun x ↦ f ^ x) j ∈ 𝒜 n
hfj : ¬f ^ j = 0
r : ↥(𝒜 0)
x : A
hx : x ∈ Submodule.span ↥(𝒜 0) ↑(Submonoid.closure (Set.range v))
hx' :
  mk { deg := n, num := ((decompose 𝒜) x) n, den := ⟨f ^ j, hb'⟩, den_mem := ⋯ } ∈
    Submodule.span ↥(𝒜 0) {x | ∃ a ai, ∃ (hai : ∑ i, ai i • dv i = a • d), Away.mk 𝒜 hf a (∏ i, v i ^ ai i) ⋯ = x}
⊢ mk { deg := n, num := ((decompose 𝒜) (r • x)) n, den := ⟨f ^ j, hb'⟩, den_mem := ⋯ } =
    r • mk { deg := n, num := ((decompose 𝒜) x) n, den := ⟨f ^ j, hb'⟩, den_mem := ⋯ }

Tactic:
ext

Goal state after:
ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁶ : CommRing A
inst✝⁵ : SetLike σ A
inst✝⁴ : AddSubgroupClass σ A
inst✝³ : AddCommMonoid ι
inst✝² : DecidableEq ι
𝒜 : ι → σ
inst✝¹ : GradedRing 𝒜
f : A
d : ι
hf : f ∈ 𝒜 d
ι' : Type u_4
inst✝ : Fintype ι'
v : ι' → A
hx✝ : Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤
dv : ι' → ι
hxd : ∀ (i : ι'), v i ∈ 𝒜 (dv i)
HH : ¬Subsingleton (Away 𝒜 f)
n : ι
a : A
j : ℕ
hb' : (fun x ↦ f ^ x) j ∈ 𝒜 n
hfj : ¬f ^ j = 0
r : ↥(𝒜 0)
x : A
hx : x ∈ Submodule.span ↥(𝒜 0) ↑(Submonoid.closure (Set.range v))
hx' :
  mk { deg := n, num := ((decompose 𝒜) x) n, den := ⟨f ^ j, hb'⟩, den_mem := ⋯ } ∈
    Submodule.span ↥(𝒜 0) {x | ∃ a ai, ∃ (hai : ∑ i, ai i • dv i = a • d), Away.mk 𝒜 hf a (∏ i, v i ^ ai i) ⋯ = x}
⊢ (mk { deg := n, num := ((decompose 𝒜) (r • x)) n, den := ⟨f ^ j, hb'⟩, den_mem := ⋯ }).val =
    val (r • mk { deg := n, num := ((decompose 𝒜) x) n, den := ⟨f ^ j, hb'⟩, den_mem := ⋯ })
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
X : Scheme
f : A →+* ↑Γ(X, ⊤)
hf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤
r : A
n : ℕ
hn : 0 < n
hr : r ∈ 𝒜 n
⊢ ((X.isoOfEq ⋯).inv ≫ fromOfGlobalSections 𝒜 f hf ∣_ basicOpen 𝒜 r) ≫ (basicOpen 𝒜 r).ι =
    toBasicOpenOfGlobalSections 𝒜 f ⋯ hn hr ≫ (basicOpen 𝒜 r).ι

Tactic:
simp only [Scheme.isoOfEq_inv, Category.assoc, morphismRestrict_ι, Scheme.homOfLE_ι_assoc,
    fromOfGlobalSections]

Goal state after:
σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
X : Scheme
f : A →+* ↑Γ(X, ⊤)
hf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤
r : A
n : ℕ
hn : 0 < n
hr : r ∈ 𝒜 n
⊢ (X.basicOpen (f r)).ι ≫
      Scheme.Cover.glueMorphisms (openCoverOfMapIrrelevantEqTop 𝒜 f hf)
        (fun ri ↦ toBasicOpenOfGlobalSections 𝒜 f ⋯ ⋯ ⋯ ≫ (basicOpen 𝒜 ri.snd.fst).ι) ⋯ =
    toBasicOpenOfGlobalSections 𝒜 f ⋯ hn hr ≫ (basicOpen 𝒜 r).ι
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Functor

Goal state before:
pp: A : Type u_1
B : Type u_2
C : Type u_3
σ : Type u_4
τ : Type u_5
ψ : Type u_6
inst✝¹¹ : CommRing A
inst✝¹⁰ : SetLike σ A
inst✝⁹ : AddSubgroupClass σ A
inst✝⁸ : CommRing B
inst✝⁷ : SetLike τ B
inst✝⁶ : AddSubgroupClass τ B
inst✝⁵ : CommRing C
inst✝⁴ : SetLike ψ C
inst✝³ : AddSubgroupClass ψ C
𝒜 : ℕ → σ
ℬ : ℕ → τ
𝒞 : ℕ → ψ
inst✝² : GradedRing 𝒜
inst✝¹ : GradedRing ℬ
inst✝ : GradedRing 𝒞
f : 𝒜 →+*ᵍ ℬ
g : ℬ →+*ᵍ 𝒞
hf : ℬ₊ ≤ HomogeneousIdeal.map f (𝒜₊)
hg : 𝒞₊ ≤ HomogeneousIdeal.map g (ℬ₊)
s : Set A
⊢ comapFun f hf ⁻¹' zeroLocus 𝒜 s = zeroLocus ℬ (⇑f '' s)

Tactic:
ext

Goal state after:
A : Type u_1
B : Type u_2
C : Type u_3
σ : Type u_4
τ : Type u_5
ψ : Type u_6
inst✝¹¹ : CommRing A
inst✝¹⁰ : SetLike σ A
inst✝⁹ : AddSubgroupClass σ A
inst✝⁸ : CommRing B
inst✝⁷ : SetLike τ B
inst✝⁶ : AddSubgroupClass τ B
inst✝⁵ : CommRing C
inst✝⁴ : SetLike ψ C
inst✝³ : AddSubgroupClass ψ C
𝒜 : ℕ → σ
ℬ : ℕ → τ
𝒞 : ℕ → ψ
inst✝² : GradedRing 𝒜
inst✝¹ : GradedRing ℬ
inst✝ : GradedRing 𝒞
f : 𝒜 →+*ᵍ ℬ
g : ℬ →+*ᵍ 𝒞
hf : ℬ₊ ≤ HomogeneousIdeal.map f (𝒜₊)
hg : 𝒞₊ ≤ HomogeneousIdeal.map g (ℬ₊)
s : Set A
x✝ : ProjectiveSpectrum ℬ
⊢ x✝ ∈ comapFun f hf ⁻¹' zeroLocus 𝒜 s ↔ x✝ ∈ zeroLocus ℬ (⇑f '' s)
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: A : Type u_2
σ : Type u_3
inst✝⁴ : CommRing A
inst✝³ : SetLike σ A
inst✝² : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝¹ : GradedRing 𝒜
f : A
d : ℕ
hf : f ∈ 𝒜 d
ι' : Type u_4
inst✝ : Fintype ι'
v : ι' → A
hx : Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤
dv : ι' → ℕ
hxd : ∀ (i : ι'), v i ∈ 𝒜 (dv i)
hxd' : ∀ (i : ι'), 0 < dv i
a : ℕ
ai : ι' → ℕ
hai : ∑ i, ai i • dv i = a • d
⊢ ∑ x, ai x % d * dv x + (∑ i, ai i / d * dv i) * d - (∑ i, dv i * (ai i / d)) * d = ∑ x, ai x % d * dv x

Tactic:
mul_comm (_ / _)

Goal state after:
A : Type u_2
σ : Type u_3
inst✝⁴ : CommRing A
inst✝³ : SetLike σ A
inst✝² : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝¹ : GradedRing 𝒜
f : A
d : ℕ
hf : f ∈ 𝒜 d
ι' : Type u_4
inst✝ : Fintype ι'
v : ι' → A
hx : Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤
dv : ι' → ℕ
hxd : ∀ (i : ι'), v i ∈ 𝒜 (dv i)
hxd' : ∀ (i : ι'), 0 < dv i
a : ℕ
ai : ι' → ℕ
hai : ∑ i, ai i • dv i = a • d
⊢ ∑ x, ai x % d * dv x + (∑ x, dv x * (ai x / d)) * d - (∑ i, dv i * (ai i / d)) * d = ∑ x, ai x % d * dv x
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: A : Type u_2
σ : Type u_3
inst✝⁴ : CommRing A
inst✝³ : SetLike σ A
inst✝² : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝¹ : GradedRing 𝒜
f : A
d : ℕ
hf : f ∈ 𝒜 d
ι' : Type u_4
inst✝ : Fintype ι'
v : ι' → A
hx : Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤
dv : ι' → ℕ
hxd : ∀ (i : ι'), v i ∈ 𝒜 (dv i)
hxd' : ∀ (i : ι'), 0 < dv i
a : ℕ
ai : ι' → ℕ
hai : ∑ i, ai i • dv i = a • d
H₀ : (a - ∑ i, dv i * (ai i / d)) • d = ∑ k, (ai k % d) • dv k
⊢ Away.mk 𝒜 hf a (∏ i, v i ^ ai i) ⋯ =
    Away.mk 𝒜 hf (a - ∑ i, dv i * (ai i / d)) (∏ i, v i ^ (ai i % d)) ⋯ * ∏ i, isLocalizationElem hf ⋯ ^ (ai i / d)

Tactic:
apply (show Function.Injective (algebraMap (Away 𝒜 f) (Localization.Away f))
      from val_injective _)

Goal state after:
A : Type u_2
σ : Type u_3
inst✝⁴ : CommRing A
inst✝³ : SetLike σ A
inst✝² : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝¹ : GradedRing 𝒜
f : A
d : ℕ
hf : f ∈ 𝒜 d
ι' : Type u_4
inst✝ : Fintype ι'
v : ι' → A
hx : Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤
dv : ι' → ℕ
hxd : ∀ (i : ι'), v i ∈ 𝒜 (dv i)
hxd' : ∀ (i : ι'), 0 < dv i
a : ℕ
ai : ι' → ℕ
hai : ∑ i, ai i • dv i = a • d
H₀ : (a - ∑ i, dv i * (ai i / d)) • d = ∑ k, (ai k % d) • dv k
⊢ (algebraMap (Away 𝒜 f) (Localization.Away f)) (Away.mk 𝒜 hf a (∏ i, v i ^ ai i) ⋯) =
    (algebraMap (Away 𝒜 f) (Localization.Away f))
      (Away.mk 𝒜 hf (a - ∑ i, dv i * (ai i / d)) (∏ i, v i ^ (ai i % d)) ⋯ * ∏ i, isLocalizationElem hf ⋯ ^ (ai i / d))
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Functor

Goal state before:
pp: A : Type u_1
B : Type u_2
σ : Type u_4
τ : Type u_5
inst✝⁷ : CommRing A
inst✝⁶ : SetLike σ A
inst✝⁵ : AddSubgroupClass σ A
inst✝⁴ : CommRing B
inst✝³ : SetLike τ B
inst✝² : AddSubgroupClass τ B
𝒜 : ℕ → σ
ℬ : ℕ → τ
inst✝¹ : GradedRing 𝒜
inst✝ : GradedRing ℬ
f : 𝒜 →+*ᵍ ℬ
hf : ℬ₊ ≤ HomogeneousIdeal.map f (𝒜₊)
U : Opens (ProjectiveSpectrum 𝒜)
V : Opens (ProjectiveSpectrum ℬ)
hUV : V.carrier ⊆ ⇑(ProjectiveSpectrum.comap f hf) ⁻¹' U.carrier
s : (x : ↥U) → AtPrime 𝒜 (↑x).asHomogeneousIdeal.toSubmodule
hs : (ProjectiveSpectrum.StructureSheaf.isLocallyFraction 𝒜).pred s
p : ↑(top ℬ)
hpV : p ∈ V
W : Opens ↑(top 𝒜)
m : ↑⟨(ProjectiveSpectrum.comap f hf) p, ⋯⟩ ∈ W
iWU : W ⟶ U
i : ℕ
a b : ↥(𝒜 i)
hb : ∀ (x : ↥W), ↑b ∉ (↑x).asHomogeneousIdeal
h_frac : ∀ (x : ↥W), (fun x ↦ s (iWU x)) x = HomogeneousLocalization.mk { deg := i, num := a, den := b, den_mem := ⋯ }
x✝ : ↥((Opens.comap (ProjectiveSpectrum.comap f hf)) W ⊓ V)
q : ↑(top ℬ)
hqW : q ∈ ↑((Opens.comap (ProjectiveSpectrum.comap f hf)) W)
hqV : q ∈ ↑V
⊢ (fun x ↦ comapStructureSheafFun f hf U V hUV s ((((Opens.comap (ProjectiveSpectrum.comap f hf)) W).infLERight V) x))
      ⟨q, ⋯⟩ =
    HomogeneousLocalization.mk { deg := i, num := (f.gradedAddHom i) a, den := (f.gradedAddHom i) b, den_mem := ⋯ }

Tactic:
ext

Goal state after:
A : Type u_1
B : Type u_2
σ : Type u_4
τ : Type u_5
inst✝⁷ : CommRing A
inst✝⁶ : SetLike σ A
inst✝⁵ : AddSubgroupClass σ A
inst✝⁴ : CommRing B
inst✝³ : SetLike τ B
inst✝² : AddSubgroupClass τ B
𝒜 : ℕ → σ
ℬ : ℕ → τ
inst✝¹ : GradedRing 𝒜
inst✝ : GradedRing ℬ
f : 𝒜 →+*ᵍ ℬ
hf : ℬ₊ ≤ HomogeneousIdeal.map f (𝒜₊)
U : Opens (ProjectiveSpectrum 𝒜)
V : Opens (ProjectiveSpectrum ℬ)
hUV : V.carrier ⊆ ⇑(ProjectiveSpectrum.comap f hf) ⁻¹' U.carrier
s : (x : ↥U) → AtPrime 𝒜 (↑x).asHomogeneousIdeal.toSubmodule
hs : (ProjectiveSpectrum.StructureSheaf.isLocallyFraction 𝒜).pred s
p : ↑(top ℬ)
hpV : p ∈ V
W : Opens ↑(top 𝒜)
m : ↑⟨(ProjectiveSpectrum.comap f hf) p, ⋯⟩ ∈ W
iWU : W ⟶ U
i : ℕ
a b : ↥(𝒜 i)
hb : ∀ (x : ↥W), ↑b ∉ (↑x).asHomogeneousIdeal
h_frac : ∀ (x : ↥W), (fun x ↦ s (iWU x)) x = HomogeneousLocalization.mk { deg := i, num := a, den := b, den_mem := ⋯ }
x✝ : ↥((Opens.comap (ProjectiveSpectrum.comap f hf)) W ⊓ V)
q : ↑(top ℬ)
hqW : q ∈ ↑((Opens.comap (ProjectiveSpectrum.comap f hf)) W)
hqV : q ∈ ↑V
⊢ val
      ((fun x ↦
          comapStructureSheafFun f hf U V hUV s ((((Opens.comap (ProjectiveSpectrum.comap f hf)) W).infLERight V) x))
        ⟨q, ⋯⟩) =
    (HomogeneousLocalization.mk
        { deg := i, num := (f.gradedAddHom i) a, den := (f.gradedAddHom i) b, den_mem := ⋯ }).val
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Functor

Goal state before:
pp: A : Type u_1
B : Type u_2
C : Type u_3
σ : Type u_4
τ : Type u_5
ψ : Type u_6
inst✝¹¹ : CommRing A
inst✝¹⁰ : SetLike σ A
inst✝⁹ : AddSubgroupClass σ A
inst✝⁸ : CommRing B
inst✝⁷ : SetLike τ B
inst✝⁶ : AddSubgroupClass τ B
inst✝⁵ : CommRing C
inst✝⁴ : SetLike ψ C
inst✝³ : AddSubgroupClass ψ C
𝒜 : ℕ → σ
ℬ : ℕ → τ
𝒞 : ℕ → ψ
inst✝² : GradedRing 𝒜
inst✝¹ : GradedRing ℬ
inst✝ : GradedRing 𝒞
f : 𝒜 →+*ᵍ ℬ
g : ℬ →+*ᵍ 𝒞
hf : ℬ₊ ≤ HomogeneousIdeal.map f (𝒜₊)
hg : 𝒞₊ ≤ HomogeneousIdeal.map g (ℬ₊)
U : Opens (ProjectiveSpectrum 𝒜)
V : Opens (ProjectiveSpectrum ℬ)
hUV : V.carrier ⊆ ⇑(ProjectiveSpectrum.comap f hf) ⁻¹' U.carrier
⊢ ⟨comapStructureSheafFun f hf U (Opposite.unop (Opposite.op V)) hUV ↑1, ⋯⟩ = 1

Tactic:
ext

Goal state after:
A : Type u_1
B : Type u_2
C : Type u_3
σ : Type u_4
τ : Type u_5
ψ : Type u_6
inst✝¹¹ : CommRing A
inst✝¹⁰ : SetLike σ A
inst✝⁹ : AddSubgroupClass σ A
inst✝⁸ : CommRing B
inst✝⁷ : SetLike τ B
inst✝⁶ : AddSubgroupClass τ B
inst✝⁵ : CommRing C
inst✝⁴ : SetLike ψ C
inst✝³ : AddSubgroupClass ψ C
𝒜 : ℕ → σ
ℬ : ℕ → τ
𝒞 : ℕ → ψ
inst✝² : GradedRing 𝒜
inst✝¹ : GradedRing ℬ
inst✝ : GradedRing 𝒞
f : 𝒜 →+*ᵍ ℬ
g : ℬ →+*ᵍ 𝒞
hf : ℬ₊ ≤ HomogeneousIdeal.map f (𝒜₊)
hg : 𝒞₊ ≤ HomogeneousIdeal.map g (ℬ₊)
U : Opens (ProjectiveSpectrum 𝒜)
V : Opens (ProjectiveSpectrum ℬ)
hUV : V.carrier ⊆ ⇑(ProjectiveSpectrum.comap f hf) ⁻¹' U.carrier
x✝ : ↥(Opposite.unop (Opposite.op V))
⊢ val (↑⟨comapStructureSheafFun f hf U (Opposite.unop (Opposite.op V)) hUV ↑1, ⋯⟩ x✝) = val (↑1 x✝)
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Functor

Goal state before:
pp: A : Type u_1
B : Type u_2
C : Type u_3
σ : Type u_4
τ : Type u_5
ψ : Type u_6
inst✝¹¹ : CommRing A
inst✝¹⁰ : SetLike σ A
inst✝⁹ : AddSubgroupClass σ A
inst✝⁸ : CommRing B
inst✝⁷ : SetLike τ B
inst✝⁶ : AddSubgroupClass τ B
inst✝⁵ : CommRing C
inst✝⁴ : SetLike ψ C
inst✝³ : AddSubgroupClass ψ C
𝒜 : ℕ → σ
ℬ : ℕ → τ
𝒞 : ℕ → ψ
inst✝² : GradedRing 𝒜
inst✝¹ : GradedRing ℬ
inst✝ : GradedRing 𝒞
f : 𝒜 →+*ᵍ ℬ
g : ℬ →+*ᵍ 𝒞
hf : ℬ₊ ≤ HomogeneousIdeal.map f (𝒜₊)
hg : 𝒞₊ ≤ HomogeneousIdeal.map g (ℬ₊)
U : Opens (ProjectiveSpectrum 𝒜)
V : Opens (ProjectiveSpectrum ℬ)
hUV : V.carrier ⊆ ⇑(ProjectiveSpectrum.comap f hf) ⁻¹' U.carrier
x y : ↑((structureSheaf 𝒜).obj.obj (Opposite.op U))
⊢ ⟨comapStructureSheafFun f hf U (Opposite.unop (Opposite.op V)) hUV ↑(x * y), ⋯⟩ =
    ⟨comapStructureSheafFun f hf U (Opposite.unop (Opposite.op V)) hUV ↑x, ⋯⟩ *
      ⟨comapStructureSheafFun f hf U (Opposite.unop (Opposite.op V)) hUV ↑y, ⋯⟩

Tactic:
ext

Goal state after:
A : Type u_1
B : Type u_2
C : Type u_3
σ : Type u_4
τ : Type u_5
ψ : Type u_6
inst✝¹¹ : CommRing A
inst✝¹⁰ : SetLike σ A
inst✝⁹ : AddSubgroupClass σ A
inst✝⁸ : CommRing B
inst✝⁷ : SetLike τ B
inst✝⁶ : AddSubgroupClass τ B
inst✝⁵ : CommRing C
inst✝⁴ : SetLike ψ C
inst✝³ : AddSubgroupClass ψ C
𝒜 : ℕ → σ
ℬ : ℕ → τ
𝒞 : ℕ → ψ
inst✝² : GradedRing 𝒜
inst✝¹ : GradedRing ℬ
inst✝ : GradedRing 𝒞
f : 𝒜 →+*ᵍ ℬ
g : ℬ →+*ᵍ 𝒞
hf : ℬ₊ ≤ HomogeneousIdeal.map f (𝒜₊)
hg : 𝒞₊ ≤ HomogeneousIdeal.map g (ℬ₊)
U : Opens (ProjectiveSpectrum 𝒜)
V : Opens (ProjectiveSpectrum ℬ)
hUV : V.carrier ⊆ ⇑(ProjectiveSpectrum.comap f hf) ⁻¹' U.carrier
x y : ↑((structureSheaf 𝒜).obj.obj (Opposite.op U))
x✝ : ↥(Opposite.unop (Opposite.op V))
⊢ val (↑⟨comapStructureSheafFun f hf U (Opposite.unop (Opposite.op V)) hUV ↑(x * y), ⋯⟩ x✝) =
    val
      (↑(⟨comapStructureSheafFun f hf U (Opposite.unop (Opposite.op V)) hUV ↑x, ⋯⟩ *
            ⟨comapStructureSheafFun f hf U (Opposite.unop (Opposite.op V)) hUV ↑y, ⋯⟩)
        x✝)
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Functor

Goal state before:
pp: A : Type u_1
B : Type u_2
C : Type u_3
σ : Type u_4
τ : Type u_5
ψ : Type u_6
inst✝¹¹ : CommRing A
inst✝¹⁰ : SetLike σ A
inst✝⁹ : AddSubgroupClass σ A
inst✝⁸ : CommRing B
inst✝⁷ : SetLike τ B
inst✝⁶ : AddSubgroupClass τ B
inst✝⁵ : CommRing C
inst✝⁴ : SetLike ψ C
inst✝³ : AddSubgroupClass ψ C
𝒜 : ℕ → σ
ℬ : ℕ → τ
𝒞 : ℕ → ψ
inst✝² : GradedRing 𝒜
inst✝¹ : GradedRing ℬ
inst✝ : GradedRing 𝒞
f : 𝒜 →+*ᵍ ℬ
g : ℬ →+*ᵍ 𝒞
hf : ℬ₊ ≤ HomogeneousIdeal.map f (𝒜₊)
hg : 𝒞₊ ≤ HomogeneousIdeal.map g (ℬ₊)
U : Opens (ProjectiveSpectrum 𝒜)
V : Opens (ProjectiveSpectrum ℬ)
hUV : V.carrier ⊆ ⇑(ProjectiveSpectrum.comap f hf) ⁻¹' U.carrier
⊢ ⟨comapStructureSheafFun f hf U (Opposite.unop (Opposite.op V)) hUV ↑0, ⋯⟩ = 0

Tactic:
ext

Goal state after:
A : Type u_1
B : Type u_2
C : Type u_3
σ : Type u_4
τ : Type u_5
ψ : Type u_6
inst✝¹¹ : CommRing A
inst✝¹⁰ : SetLike σ A
inst✝⁹ : AddSubgroupClass σ A
inst✝⁸ : CommRing B
inst✝⁷ : SetLike τ B
inst✝⁶ : AddSubgroupClass τ B
inst✝⁵ : CommRing C
inst✝⁴ : SetLike ψ C
inst✝³ : AddSubgroupClass ψ C
𝒜 : ℕ → σ
ℬ : ℕ → τ
𝒞 : ℕ → ψ
inst✝² : GradedRing 𝒜
inst✝¹ : GradedRing ℬ
inst✝ : GradedRing 𝒞
f : 𝒜 →+*ᵍ ℬ
g : ℬ →+*ᵍ 𝒞
hf : ℬ₊ ≤ HomogeneousIdeal.map f (𝒜₊)
hg : 𝒞₊ ≤ HomogeneousIdeal.map g (ℬ₊)
U : Opens (ProjectiveSpectrum 𝒜)
V : Opens (ProjectiveSpectrum ℬ)
hUV : V.carrier ⊆ ⇑(ProjectiveSpectrum.comap f hf) ⁻¹' U.carrier
x✝ : ↥(Opposite.unop (Opposite.op V))
⊢ val (↑⟨comapStructureSheafFun f hf U (Opposite.unop (Opposite.op V)) hUV ↑0, ⋯⟩ x✝) = val (↑0 x✝)