mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Abelian.FreydMitchell

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
this : (F C).Faithful
⊢ Ind.yoneda.rightOp.Faithful

Tactic:
]

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
this : (F C).Faithful
⊢ Ind.yoneda.rightOp.Faithful
Module: Mathlib.CategoryTheory.Abelian.FreydMitchell

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
⊢ (F C ⋙ G C ⋙ H C).Full

Tactic:
]

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
⊢ (F C ⋙ G C ⋙ H C).Full
Module: Mathlib.CategoryTheory.Abelian.FreydMitchell

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
⊢ AsSmall.equiv.functor.Full

Tactic:
]

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
⊢ AsSmall.equiv.functor.Full
Module: Mathlib.CategoryTheory.Abelian.Indization

Goal state before:
pp: C : Type v
inst✝¹ : SmallCategory C
inst✝ : Abelian C
X Y : Ind C
f : X ⟶ Y
I : Type v
w✝¹ : SmallCategory I
w✝ : IsFiltered I
F G : I ⥤ C
ϕ : F ⟶ G
i : Arrow.mk f ≅ Arrow.mk ((Ind.lim I).map ϕ)
i' : Arrow.mk (coimageImageComparison f) ≅ Arrow.mk (coimageImageComparison ((Ind.lim I).map ϕ)) :=
  coimageImageComparisonFunctor.mapIso i
this : IsIso i'.hom
⊢ IsIso ((Ind.lim I).map (coimageImageComparison ϕ))

Tactic:
]

Goal state after:
C : Type v
inst✝¹ : SmallCategory C
inst✝ : Abelian C
X Y : Ind C
f : X ⟶ Y
I : Type v
w✝¹ : SmallCategory I
w✝ : IsFiltered I
F G : I ⥤ C
ϕ : F ⟶ G
i : Arrow.mk f ≅ Arrow.mk ((Ind.lim I).map ϕ)
i' : Arrow.mk (coimageImageComparison f) ≅ Arrow.mk (coimageImageComparison ((Ind.lim I).map ϕ)) :=
  coimageImageComparisonFunctor.mapIso i
this : IsIso i'.hom
⊢ IsIso ((Ind.lim I).map (coimageImageComparison ϕ))
Module: Mathlib.CategoryTheory.Abelian.FreydMitchell

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
this : (F C).Full
⊢ Ind.yoneda.rightOp.Full

Tactic:
]

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
this : (F C).Full
⊢ Ind.yoneda.rightOp.Full
Module: Mathlib.CategoryTheory.Abelian.FreydMitchell

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
⊢ PreservesFiniteLimits (F C ⋙ G C ⋙ H C)

Tactic:
]

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
⊢ PreservesFiniteLimits (F C ⋙ G C ⋙ H C)
Module: Mathlib.CategoryTheory.Abelian.FreydMitchell

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
⊢ PreservesFiniteLimits AsSmall.equiv.functor

Tactic:
]

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
⊢ PreservesFiniteLimits AsSmall.equiv.functor
Module: Mathlib.CategoryTheory.Abelian.FreydMitchell

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
this : PreservesFiniteLimits (F C)
⊢ PreservesFiniteLimits Ind.yoneda.rightOp

Tactic:
]

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
this : PreservesFiniteLimits (F C)
⊢ PreservesFiniteLimits Ind.yoneda.rightOp
Module: Mathlib.CategoryTheory.Abelian.FreydMitchell

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
⊢ PreservesFiniteColimits (F C ⋙ G C ⋙ H C)

Tactic:
]

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
⊢ PreservesFiniteColimits (F C ⋙ G C ⋙ H C)
Module: Mathlib.CategoryTheory.Abelian.FreydMitchell

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
⊢ PreservesFiniteColimits AsSmall.equiv.functor

Tactic:
]

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
⊢ PreservesFiniteColimits AsSmall.equiv.functor
Module: Mathlib.CategoryTheory.Abelian.FreydMitchell

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
this : PreservesFiniteColimits (F C)
⊢ PreservesFiniteColimits Ind.yoneda.rightOp

Tactic:
]

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
this : PreservesFiniteColimits (F C)
⊢ PreservesFiniteColimits Ind.yoneda.rightOp
Module: Mathlib.CategoryTheory.Limits.FilteredColimitCommutesProduct

Goal state before:
pp: α : Type u
I : α → Type u
inst✝¹ : (i : α) → SmallCategory (I i)
inst✝ : ∀ (i : α), IsFiltered (I i)
F : (i : α) → I i ⥤ Type u
ky : (i : α) → I i
yk₀ : (pointwiseProduct F).obj ky
ky' : (i : α) → I i
yk₀' : (pointwiseProduct F).obj ky'
k : (i : α) → I i := IsFiltered.max ky ky'
yk : ∏ᶜ (Functor.pi F).obj k := (hom ((pointwiseProduct F).map (IsFiltered.leftToMax ky ky'))) yk₀
yk' : ∏ᶜ (Functor.pi F).obj k := (hom ((pointwiseProduct F).map (IsFiltered.rightToMax ky ky'))) yk₀'
hyk₀ : (hom (colimit.ι (pointwiseProduct F) ky)) yk₀ = (hom (colimit.ι (pointwiseProduct F) k)) yk
hyk₀' : (hom (colimit.ι (pointwiseProduct F) ky')) yk₀' = (hom (colimit.ι (pointwiseProduct F) k)) yk'
hy :
  (hom (colimitPointwiseProductToProductColimit F)) ((hom (colimit.ι (pointwiseProduct F) k)) yk) =
    (hom (colimitPointwiseProductToProductColimit F)) ((hom (colimit.ι (pointwiseProduct F) k)) yk')
s : α
hy₁ :
  (hom (Pi.π (fun s ↦ colimit (F s)) s))
      ((hom (colimitPointwiseProductToProductColimit F)) ((hom (colimit.ι (pointwiseProduct F) k)) yk)) =
    (hom (colimit.ι (F s) (k s))) ((hom (Pi.π ((Functor.pi F).obj k) s)) yk)
hy₂ :
  ∃ k_1 f g,
    (hom ((F s).map f)) ((hom (Pi.π ((Functor.pi F).obj k) s)) yk) =
      (hom ((F s).map g)) ((hom (Pi.π ((Functor.pi F).obj k) s)) yk')
⊢ ∃ i' hi',
    (hom ((F s).map hi')) ((hom (Pi.π (fun s ↦ (F s).obj (k s)) s)) yk) =
      (hom ((F s).map hi')) ((hom (Pi.π (fun s ↦ (F s).obj (k s)) s)) yk')

Tactic:
]

Goal state after:
α : Type u
I : α → Type u
inst✝¹ : (i : α) → SmallCategory (I i)
inst✝ : ∀ (i : α), IsFiltered (I i)
F : (i : α) → I i ⥤ Type u
ky : (i : α) → I i
yk₀ : (pointwiseProduct F).obj ky
ky' : (i : α) → I i
yk₀' : (pointwiseProduct F).obj ky'
k : (i : α) → I i := IsFiltered.max ky ky'
yk : ∏ᶜ (Functor.pi F).obj k := (hom ((pointwiseProduct F).map (IsFiltered.leftToMax ky ky'))) yk₀
yk' : ∏ᶜ (Functor.pi F).obj k := (hom ((pointwiseProduct F).map (IsFiltered.rightToMax ky ky'))) yk₀'
hyk₀ : (hom (colimit.ι (pointwiseProduct F) ky)) yk₀ = (hom (colimit.ι (pointwiseProduct F) k)) yk
hyk₀' : (hom (colimit.ι (pointwiseProduct F) ky')) yk₀' = (hom (colimit.ι (pointwiseProduct F) k)) yk'
hy :
  (hom (colimitPointwiseProductToProductColimit F)) ((hom (colimit.ι (pointwiseProduct F) k)) yk) =
    (hom (colimitPointwiseProductToProductColimit F)) ((hom (colimit.ι (pointwiseProduct F) k)) yk')
s : α
hy₁ :
  (hom (Pi.π (fun s ↦ colimit (F s)) s))
      ((hom (colimitPointwiseProductToProductColimit F)) ((hom (colimit.ι (pointwiseProduct F) k)) yk)) =
    (hom (colimit.ι (F s) (k s))) ((hom (Pi.π ((Functor.pi F).obj k) s)) yk)
hy₂ :
  ∃ k_1 f g,
    (hom ((F s).map f)) ((hom (Pi.π ((Functor.pi F).obj k) s)) yk) =
      (hom ((F s).map g)) ((hom (Pi.π ((Functor.pi F).obj k) s)) yk')
⊢ ∃ i' hi',
    (hom ((F s).map hi')) ((hom (Pi.π (fun s ↦ (F s).obj (k s)) s)) yk) =
      (hom ((F s).map hi')) ((hom (Pi.π (fun s ↦ (F s).obj (k s)) s)) yk')
Module: Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Connected

Goal state before:
pp: J : Type w
inst✝⁵ : Category.{w', w} J
inst✝⁴ : IsConnected J
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasPullbacks C
inst✝¹ : HasColimitsOfShape J C
inst✝ : HasExactColimitsOfShape J C
F : J ⥤ C
c : Cocone F
hc : IsColimit c
X Y : C
f : X ⟶ c.pt
g h : c.pt ⟶ Y
hf : ∀ (j : J), pullback.snd (c.ι.app j) f ≫ f ≫ g = pullback.snd (c.ι.app j) f ≫ f ≫ h
j : J
⊢ (pullbackObjIso c.ι ((Functor.const J).map f) j).inv ≫
      { pt := X, ι := pullback.snd c.ι ((Functor.const J).map f) }.ι.app j ≫ f ≫ g =
    (pullbackObjIso c.ι ((Functor.const J).map f) j).inv ≫
      { pt := X, ι := pullback.snd c.ι ((Functor.const J).map f) }.ι.app j ≫ f ≫ h

Tactic:
]

Goal state after:
J : Type w
inst✝⁵ : Category.{w', w} J
inst✝⁴ : IsConnected J
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasPullbacks C
inst✝¹ : HasColimitsOfShape J C
inst✝ : HasExactColimitsOfShape J C
F : J ⥤ C
c : Cocone F
hc : IsColimit c
X Y : C
f : X ⟶ c.pt
g h : c.pt ⟶ Y
hf : ∀ (j : J), pullback.snd (c.ι.app j) f ≫ f ≫ g = pullback.snd (c.ι.app j) f ≫ f ≫ h
j : J
⊢ (pullbackObjIso c.ι ((Functor.const J).map f) j).inv ≫
      { pt := X, ι := pullback.snd c.ι ((Functor.const J).map f) }.ι.app j ≫ f ≫ g =
    (pullbackObjIso c.ι ((Functor.const J).map f) j).inv ≫
      { pt := X, ι := pullback.snd c.ι ((Functor.const J).map f) }.ι.app j ≫ f ≫ h
Module: Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Connected

Goal state before:
pp: J : Type w
inst✝⁵ : Category.{w', w} J
inst✝⁴ : IsConnected J
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasPullbacks C
inst✝¹ : HasColimitsOfShape J C
inst✝ : HasExactColimitsOfShape J C
F : J ⥤ C
c : Cocone F
hc : IsColimit c
X Y : C
f : X ⟶ c.pt
g h : c.pt ⟶ Y
hf : ∀ (j : J), pullback.snd (c.ι.app j) f ≫ f ≫ g = pullback.snd (c.ι.app j) f ≫ f ≫ h
j : J
⊢ { pt := X, ι := pullback.snd c.ι ((Functor.const J).map f) }.ι.app j ≫ f ≫ g =
    { pt := X, ι := pullback.snd c.ι ((Functor.const J).map f) }.ι.app j ≫ f ≫ h

Tactic:
rw [← cancel_epi (pullbackObjIso _ _ _).inv]

Goal state after:
J : Type w
inst✝⁵ : Category.{w', w} J
inst✝⁴ : IsConnected J
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasPullbacks C
inst✝¹ : HasColimitsOfShape J C
inst✝ : HasExactColimitsOfShape J C
F : J ⥤ C
c : Cocone F
hc : IsColimit c
X Y : C
f : X ⟶ c.pt
g h : c.pt ⟶ Y
hf : ∀ (j : J), pullback.snd (c.ι.app j) f ≫ f ≫ g = pullback.snd (c.ι.app j) f ≫ f ≫ h
j : J
⊢ (pullbackObjIso c.ι ((Functor.const J).map f) j).inv ≫
      { pt := X, ι := pullback.snd c.ι ((Functor.const J).map f) }.ι.app j ≫ f ≫ g =
    (pullbackObjIso c.ι ((Functor.const J).map f) j).inv ≫
      { pt := X, ι := pullback.snd c.ι ((Functor.const J).map f) }.ι.app j ≫ f ≫ h
Module: Mathlib.CategoryTheory.Preadditive.LiftToFinset

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
inst✝² : HasFiniteCoproducts C
α : Type w
inst✝¹ : DecidableEq α
f : α → C
inst✝ : HasCoproduct f
S : Finset (Discrete α)
v : ↥S
⊢ Sigma.ι f (↑v).as = Sigma.ι (fun x ↦ f (↑x).as) v ≫ Sigma.π (fun a ↦ f (↑a).as) v ≫ Sigma.ι f (↑v).as

pp: case h₀
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
inst✝² : HasFiniteCoproducts C
α : Type w
inst✝¹ : DecidableEq α
f : α → C
inst✝ : HasCoproduct f
S : Finset (Discrete α)
v : ↥S
⊢ ∀ b ∈ S.attach, b ≠ v → Sigma.ι (fun x ↦ f (↑x).as) v ≫ Sigma.π (fun a ↦ f (↑a).as) b ≫ Sigma.ι f (↑b).as = 0

pp: case h₁
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
inst✝² : HasFiniteCoproducts C
α : Type w
inst✝¹ : DecidableEq α
f : α → C
inst✝ : HasCoproduct f
S : Finset (Discrete α)
v : ↥S
⊢ v ∉ S.attach → Sigma.ι (fun x ↦ f (↑x).as) v ≫ Sigma.π (fun a ↦ f (↑a).as) v ≫ Sigma.ι f (↑v).as = 0

Tactic:
]

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
inst✝² : HasFiniteCoproducts C
α : Type w
inst✝¹ : DecidableEq α
f : α → C
inst✝ : HasCoproduct f
S : Finset (Discrete α)
v : ↥S
⊢ Sigma.ι f (↑v).as = Sigma.ι (fun x ↦ f (↑x).as) v ≫ Sigma.π (fun a ↦ f (↑a).as) v ≫ Sigma.ι f (↑v).as

case h₀
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
inst✝² : HasFiniteCoproducts C
α : Type w
inst✝¹ : DecidableEq α
f : α → C
inst✝ : HasCoproduct f
S : Finset (Discrete α)
v : ↥S
⊢ ∀ b ∈ S.attach, b ≠ v → Sigma.ι (fun x ↦ f (↑x).as) v ≫ Sigma.π (fun a ↦ f (↑a).as) b ≫ Sigma.ι f (↑b).as = 0

case h₁
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
inst✝² : HasFiniteCoproducts C
α : Type w
inst✝¹ : DecidableEq α
f : α → C
inst✝ : HasCoproduct f
S : Finset (Discrete α)
v : ↥S
⊢ v ∉ S.attach → Sigma.ι (fun x ↦ f (↑x).as) v ≫ Sigma.π (fun a ↦ f (↑a).as) v ≫ Sigma.ι f (↑v).as = 0
Module: Mathlib.CategoryTheory.Preadditive.LiftToFinset

Goal state before:
pp: case h₀
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
inst✝² : HasFiniteCoproducts C
α : Type w
inst✝¹ : DecidableEq α
f : α → C
inst✝ : HasCoproduct f
S : Finset (Discrete α)
v b : ↥S
hb : b ∈ S.attach
hb₁ : b ≠ v
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Connected

Goal state before:
pp: J : Type w
inst✝⁵ : Category.{w', w} J
inst✝⁴ : IsConnected J
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasPushouts C
inst✝¹ : HasLimitsOfShape J C
inst✝ : HasExactLimitsOfShape J C
F : J ⥤ C
c : Cone F
hc : IsLimit c
X Y : C
g h : Y ⟶ c.pt
f : c.pt ⟶ X
hf : ∀ (j : J), g ≫ f ≫ pushout.inr (c.π.app j) f = h ≫ f ≫ pushout.inr (c.π.app j) f
j : J
⊢ ((g ≫ f) ≫ { pt := X, π := pushout.inr c.π ((Functor.const J).map f) }.π.app j) ≫
      (pushoutObjIso c.π ((Functor.const J).map f) j).hom =
    ((h ≫ f) ≫ { pt := X, π := pushout.inr c.π ((Functor.const J).map f) }.π.app j) ≫
      (pushoutObjIso c.π ((Functor.const J).map f) j).hom

Tactic:
]

Goal state after:
J : Type w
inst✝⁵ : Category.{w', w} J
inst✝⁴ : IsConnected J
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasPushouts C
inst✝¹ : HasLimitsOfShape J C
inst✝ : HasExactLimitsOfShape J C
F : J ⥤ C
c : Cone F
hc : IsLimit c
X Y : C
g h : Y ⟶ c.pt
f : c.pt ⟶ X
hf : ∀ (j : J), g ≫ f ≫ pushout.inr (c.π.app j) f = h ≫ f ≫ pushout.inr (c.π.app j) f
j : J
⊢ ((g ≫ f) ≫ { pt := X, π := pushout.inr c.π ((Functor.const J).map f) }.π.app j) ≫
      (pushoutObjIso c.π ((Functor.const J).map f) j).hom =
    ((h ≫ f) ≫ { pt := X, π := pushout.inr c.π ((Functor.const J).map f) }.π.app j) ≫
      (pushoutObjIso c.π ((Functor.const J).map f) j).hom
Module: Mathlib.CategoryTheory.Preadditive.LiftToFinset

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
inst✝² : HasFiniteProducts C
α : Type w
inst✝¹ : DecidableEq α
f : α → C
inst✝ : HasProduct f
S : (Finset (Discrete α))ᵒᵖ
v : ↥(Opposite.unop S)
⊢ Pi.π f (↑v).as = Pi.π f (↑v).as ≫ Pi.ι (fun a ↦ f (↑a).as) v ≫ Pi.π (fun x ↦ f (↑x).as) v

pp: case h₀
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
inst✝² : HasFiniteProducts C
α : Type w
inst✝¹ : DecidableEq α
f : α → C
inst✝ : HasProduct f
S : (Finset (Discrete α))ᵒᵖ
v : ↥(Opposite.unop S)
⊢ ∀ b ∈ (Opposite.unop S).attach, b ≠ v → Pi.π f (↑b).as ≫ Pi.ι (fun a ↦ f (↑a).as) b ≫ Pi.π (fun x ↦ f (↑x).as) v = 0

pp: case h₁
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
inst✝² : HasFiniteProducts C
α : Type w
inst✝¹ : DecidableEq α
f : α → C
inst✝ : HasProduct f
S : (Finset (Discrete α))ᵒᵖ
v : ↥(Opposite.unop S)
⊢ v ∉ (Opposite.unop S).attach → Pi.π f (↑v).as ≫ Pi.ι (fun a ↦ f (↑a).as) v ≫ Pi.π (fun x ↦ f (↑x).as) v = 0

Tactic:
]

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
inst✝² : HasFiniteProducts C
α : Type w
inst✝¹ : DecidableEq α
f : α → C
inst✝ : HasProduct f
S : (Finset (Discrete α))ᵒᵖ
v : ↥(Opposite.unop S)
⊢ Pi.π f (↑v).as = Pi.π f (↑v).as ≫ Pi.ι (fun a ↦ f (↑a).as) v ≫ Pi.π (fun x ↦ f (↑x).as) v

case h₀
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
inst✝² : HasFiniteProducts C
α : Type w
inst✝¹ : DecidableEq α
f : α → C
inst✝ : HasProduct f
S : (Finset (Discrete α))ᵒᵖ
v : ↥(Opposite.unop S)
⊢ ∀ b ∈ (Opposite.unop S).attach, b ≠ v → Pi.π f (↑b).as ≫ Pi.ι (fun a ↦ f (↑a).as) b ≫ Pi.π (fun x ↦ f (↑x).as) v = 0

case h₁
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
inst✝² : HasFiniteProducts C
α : Type w
inst✝¹ : DecidableEq α
f : α → C
inst✝ : HasProduct f
S : (Finset (Discrete α))ᵒᵖ
v : ↥(Opposite.unop S)
⊢ v ∉ (Opposite.unop S).attach → Pi.π f (↑v).as ≫ Pi.ι (fun a ↦ f (↑a).as) v ≫ Pi.π (fun x ↦ f (↑x).as) v = 0
Module: Mathlib.CategoryTheory.Preadditive.LiftToFinset

Goal state before:
pp: case h₀
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
inst✝² : HasFiniteProducts C
α : Type w
inst✝¹ : DecidableEq α
f : α → C
inst✝ : HasProduct f
S : (Finset (Discrete α))ᵒᵖ
v b : ↥(Opposite.unop S)
hb : b ∈ (Opposite.unop S).attach
hb₁ : b ≠ v
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.Yoneda.Injective

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
J : C
⊢ (yoneda.obj J).PreservesEpimorphisms ↔ (preadditiveYoneda.obj J).PreservesEpimorphisms

Tactic:
]

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
J : C
⊢ (yoneda.obj J).PreservesEpimorphisms ↔ (preadditiveYoneda.obj J).PreservesEpimorphisms
Module: Mathlib.CategoryTheory.Preadditive.Yoneda.Injective

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
J : C
⊢ (yoneda.obj J).PreservesEpimorphisms ↔ (preadditiveYonedaObj J).PreservesEpimorphisms

Tactic:
]

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
J : C
⊢ (yoneda.obj J).PreservesEpimorphisms ↔ (preadditiveYonedaObj J).PreservesEpimorphisms
Module: Mathlib.CategoryTheory.Abelian.Injective.Basic

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
J : C
hP : PreservesFiniteColimits (preadditiveYonedaObj J)
⊢ (preadditiveYonedaObj J).PreservesEpimorphisms

Tactic:
]

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
J : C
hP : PreservesFiniteColimits (preadditiveYonedaObj J)
⊢ (preadditiveYonedaObj J).PreservesEpimorphisms
Module: Mathlib.CategoryTheory.Abelian.Injective.Extend

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroObject C
inst✝ : Preadditive C
X : C
R : InjectiveResolution X
⊢ ((HomologicalComplex.singleObjXSelf (ComplexShape.up ℤ) 0 X).hom ≫
        (HomologicalComplex.singleObjXSelf (ComplexShape.up ℕ) 0 X).inv ≫
          (((HomologicalComplex.single C (ComplexShape.up ℕ) 0).obj X).extendXIso ComplexShape.embeddingUpNat
              ι'._proof_1).inv) ≫
      (((HomologicalComplex.single C (ComplexShape.up ℕ) 0).obj X).extendXIso ComplexShape.embeddingUpNat ⋯).hom ≫
        R.ι.f 0 ≫ (HomologicalComplex.extendXIso R.cocomplex ComplexShape.embeddingUpNat ⋯).inv =
    (HomologicalComplex.singleObjXSelf (ComplexShape.up ℤ) 0 X).hom ≫
      R.ι.f 0 ≫ (R.cochainComplexXIso 0 0 ι'_f_zero._proof_1).inv

Tactic:
]

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroObject C
inst✝ : Preadditive C
X : C
R : InjectiveResolution X
⊢ ((HomologicalComplex.singleObjXSelf (ComplexShape.up ℤ) 0 X).hom ≫
        (HomologicalComplex.singleObjXSelf (ComplexShape.up ℕ) 0 X).inv ≫
          (((HomologicalComplex.single C (ComplexShape.up ℕ) 0).obj X).extendXIso ComplexShape.embeddingUpNat
              ι'._proof_1).inv) ≫
      (((HomologicalComplex.single C (ComplexShape.up ℕ) 0).obj X).extendXIso ComplexShape.embeddingUpNat ⋯).hom ≫
        R.ι.f 0 ≫ (HomologicalComplex.extendXIso R.cocomplex ComplexShape.embeddingUpNat ⋯).inv =
    (HomologicalComplex.singleObjXSelf (ComplexShape.up ℤ) 0 X).hom ≫
      R.ι.f 0 ≫ (R.cochainComplexXIso 0 0 ι'_f_zero._proof_1).inv
Module: Mathlib.CategoryTheory.Preadditive.Injective.Resolution

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
Z : C
I : InjectiveResolution Z
n : ℕ
⊢ QuasiIsoAt I.ι (n + 1)

Tactic:
]

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
Z : C
I : InjectiveResolution Z
n : ℕ
⊢ QuasiIsoAt I.ι (n + 1)
Module: Mathlib.CategoryTheory.Preadditive.Injective.Resolution

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
Z : C
I : InjectiveResolution Z
⊢ (singleObjCyclesSelfIso (ComplexShape.up ℕ) 0 I.kernelFork.pt).inv ≫
      (isoHomologyπ₀ ((single C (ComplexShape.up ℕ) 0).obj I.kernelFork.pt)).hom ≫
        (singleObjHomologySelfIso (ComplexShape.up ℕ) 0 I.kernelFork.pt).hom ≫
          ((singleObjHomologySelfIso (ComplexShape.up ℕ) 0 I.kernelFork.pt).symm ≪≫
                isoOfQuasiIsoAt I.ι 0 ≪≫ I.cocomplex.isoHomologyπ₀.symm).hom ≫
            Fork.ι (KernelFork.ofι (iCycles I.cocomplex 0) ⋯) =
    (singleObjCyclesSelfIso (ComplexShape.up ℕ) 0 I.kernelFork.pt).inv ≫
      (isoHomologyπ₀ ((single C (ComplexShape.up ℕ) 0).obj I.kernelFork.pt)).hom ≫
        (singleObjHomologySelfIso (ComplexShape.up ℕ) 0 I.kernelFork.pt).hom ≫ Fork.ι I.kernelFork

Tactic:
]

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
Z : C
I : InjectiveResolution Z
⊢ (singleObjCyclesSelfIso (ComplexShape.up ℕ) 0 I.kernelFork.pt).inv ≫
      (isoHomologyπ₀ ((single C (ComplexShape.up ℕ) 0).obj I.kernelFork.pt)).hom ≫
        (singleObjHomologySelfIso (ComplexShape.up ℕ) 0 I.kernelFork.pt).hom ≫
          ((singleObjHomologySelfIso (ComplexShape.up ℕ) 0 I.kernelFork.pt).symm ≪≫
                isoOfQuasiIsoAt I.ι 0 ≪≫ I.cocomplex.isoHomologyπ₀.symm).hom ≫
            Fork.ι (KernelFork.ofι (iCycles I.cocomplex 0) ⋯) =
    (singleObjCyclesSelfIso (ComplexShape.up ℕ) 0 I.kernelFork.pt).inv ≫
      (isoHomologyπ₀ ((single C (ComplexShape.up ℕ) 0).obj I.kernelFork.pt)).hom ≫
        (singleObjHomologySelfIso (ComplexShape.up ℕ) 0 I.kernelFork.pt).hom ≫ Fork.ι I.kernelFork
Module: Mathlib.CategoryTheory.Preadditive.Injective.Resolution

Goal state before:
pp: case succ
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
Z : C
I : InjectiveResolution Z
n✝ : ℕ
⊢ Mono 0

Tactic:
]

Goal state after:
case succ
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
Z : C
I : InjectiveResolution Z
n✝ : ℕ
⊢ Mono 0
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.ModuleEmbedding.GabrielPopescu

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : IsGrothendieckAbelian.{v, v, u} C
G : C
hG : IsSeparator G
A B : C
M : ModuleCat (End G)ᵐᵒᵖ
g : M ⟶ ModuleCat.of (End G)ᵐᵒᵖ (G ⟶ A)
hg : Mono g
f : M ⟶ ModuleCat.of (End G)ᵐᵒᵖ (G ⟶ B)
F : Finset (Discrete ↑M)
⊢ pullback.fst (∑ a ∈ F.attach, Sigma.π (fun a ↦ G) a ≫ Sigma.ι (fun x ↦ G) (↑a).as) (kernel.ι (d g)) ≫
      (∑ a ∈ F.attach, Sigma.π (fun a ↦ G) a ≫ Sigma.ι (fun x ↦ G) (↑a).as) ≫ d f =
    0

Tactic:
]

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : IsGrothendieckAbelian.{v, v, u} C
G : C
hG : IsSeparator G
A B : C
M : ModuleCat (End G)ᵐᵒᵖ
g : M ⟶ ModuleCat.of (End G)ᵐᵒᵖ (G ⟶ A)
hg : Mono g
f : M ⟶ ModuleCat.of (End G)ᵐᵒᵖ (G ⟶ B)
F : Finset (Discrete ↑M)
⊢ pullback.fst (∑ a ∈ F.attach, Sigma.π (fun a ↦ G) a ≫ Sigma.ι (fun x ↦ G) (↑a).as) (kernel.ι (d g)) ≫
      (∑ a ∈ F.attach, Sigma.π (fun a ↦ G) a ≫ Sigma.ι (fun x ↦ G) (↑a).as) ≫ d f =
    0
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.ModuleEmbedding.GabrielPopescu

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : IsGrothendieckAbelian.{v, v, u} C
G : C
hG : IsSeparator G
A B : C
M : ModuleCat (End G)ᵐᵒᵖ
g : M ⟶ ModuleCat.of (End G)ᵐᵒᵖ (G ⟶ A)
hg : Mono g
f : M ⟶ ModuleCat.of (End G)ᵐᵒᵖ (G ⟶ B)
F : Finset (Discrete ↑M)
h : G ⟶ pullback (∑ a ∈ F.attach, Sigma.π (fun a ↦ G) a ≫ Sigma.ι (fun x ↦ G) (↑a).as) (kernel.ι (d g))
⊢ ∑ j ∈ F.attach,
      (h ≫
          pullback.fst (∑ a ∈ F.attach, Sigma.π (fun a ↦ G) a ≫ Sigma.ι (fun x ↦ G) (↑a).as) (kernel.ι (d g)) ≫
            Sigma.π (fun a ↦ G) j ≫ Sigma.ι (fun x ↦ G) (↑j).as) ≫
        d f =
    0

Tactic:
]

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : IsGrothendieckAbelian.{v, v, u} C
G : C
hG : IsSeparator G
A B : C
M : ModuleCat (End G)ᵐᵒᵖ
g : M ⟶ ModuleCat.of (End G)ᵐᵒᵖ (G ⟶ A)
hg : Mono g
f : M ⟶ ModuleCat.of (End G)ᵐᵒᵖ (G ⟶ B)
F : Finset (Discrete ↑M)
h : G ⟶ pullback (∑ a ∈ F.attach, Sigma.π (fun a ↦ G) a ≫ Sigma.ι (fun x ↦ G) (↑a).as) (kernel.ι (d g))
⊢ ∑ j ∈ F.attach,
      (h ≫
          pullback.fst (∑ a ∈ F.attach, Sigma.π (fun a ↦ G) a ≫ Sigma.ι (fun x ↦ G) (↑a).as) (kernel.ι (d g)) ≫
            Sigma.π (fun a ↦ G) j ≫ Sigma.ι (fun x ↦ G) (↑j).as) ≫
        d f =
    0
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.ModuleEmbedding.GabrielPopescu

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : IsGrothendieckAbelian.{v, v, u} C
G : C
hG : IsSeparator G
A B : C
inst✝ : Injective B
M : ModuleCat (End G)ᵐᵒᵖ
g : M ⟶ ModuleCat.of (End G)ᵐᵒᵖ (G ⟶ A)
hg : Mono g
f : M ⟶ ModuleCat.of (End G)ᵐᵒᵖ (G ⟶ B)
⊢ 0 = 0

Tactic:
]

Goal state after: