Module: Mathlib.Algebra.Homology.BifunctorAssociator

Goal state before:
pp: case pos
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝³³ : Category.{v_1, u_1} C₁
inst✝³² : Category.{v_2, u_2} C₂
inst✝³¹ : Category.{v_3, u_5} C₃
inst✝³⁰ : Category.{v_4, u_6} C₄
inst✝²⁹ : Category.{v_5, u_3} C₁₂
inst✝²⁸ : Category.{v_6, u_4} C₂₃
inst✝²⁷ : HasZeroMorphisms C₁
inst✝²⁶ : HasZeroMorphisms C₂
inst✝²⁵ : HasZeroMorphisms C₃
inst✝²⁴ : Preadditive C₁₂
inst✝²³ : Preadditive C₂₃
inst✝²² : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
inst✝²¹ : F₁₂.PreservesZeroMorphisms
inst✝²⁰ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝¹⁹ : G.Additive
inst✝¹⁸ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
inst✝¹⁷ : G₂₃.PreservesZeroMorphisms
inst✝¹⁶ : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst✝¹⁵ : F.PreservesZeroMorphisms
inst✝¹⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
associator : bifunctorComp₁₂ F₁₂ G ≅ bifunctorComp₂₃ F G₂₃
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₂₃ : Type u_11
ι₄ : Type u_12
inst✝¹³ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₂₃ : ComplexShape ι₂₃
c₄ : ComplexShape ι₄
inst✝¹² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹¹ : TotalComplexShape c₁₂ c₃ c₄
inst✝¹⁰ : TotalComplexShape c₂ c₃ c₂₃
inst✝⁹ : TotalComplexShape c₁ c₂₃ c₄
inst✝⁸ : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝⁷ : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst✝⁶ : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst✝⁵ : DecidableEq ι₁₂
inst✝⁴ : DecidableEq ι₂₃
inst✝³ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
inst✝² : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
inst✝¹ : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
inst✝ : HasGoodTrifunctor₂₃Obj F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄
j j' : ι₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
h₁ : c₁.Rel i₁ (c₁.next i₁)
this :
  (G.map ((F₁₂.map (K₁.d i₁ (c₁.next i₁))).app (K₂.X i₂))).app (K₃.X i₃) ≫
      ((associator.hom.app (K₁.X (c₁.next i₁))).app (K₂.X i₂)).app (K₃.X i₃) =
    ((associator.hom.app (K₁.X i₁)).app (K₂.X i₂)).app (K₃.X i₃) ≫
      (F.map (K₁.d i₁ (c₁.next i₁))).app ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃))
⊢ c₁.ε₁ c₂₃ c₄ (i₁, c₂.π c₃ c₂₃ (i₂, i₃)) •
      ((associator.hom.app (K₁.X i₁)).app (K₂.X i₂)).app (K₃.X i₃) ≫
        (F.map (K₁.d i₁ (c₁.next i₁))).app ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃)) ≫
          mapBifunctor₂₃.ιOrZero F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ (c₁.next i₁) i₂ i₃ j' =
    (c₁₂.ε₁ c₃ c₄ (c₁.π c₂ c₁₂ (i₁, i₂), i₃) * c₁.ε₁ c₂ c₁₂ (i₁, i₂)) •
      (G.map ((F₁₂.map (K₁.d i₁ (c₁.next i₁))).app (K₂.X i₂))).app (K₃.X i₃) ≫
        mapBifunctor₁₂.ιOrZero F₁₂ G K₁ K₂ K₃ c₁₂ c₄ (c₁.next i₁) i₂ i₃ j' ≫
          (mapBifunctorAssociatorX associator K₁ K₂ K₃ c₁₂ c₂₃ c₄ j').hom

Tactic:
ComplexShape.associative_ε₁_eq_mul c₁ c₂ c₃ c₁₂ c₂₃ c₄,

Goal state after:
case pos
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝³³ : Category.{v_1, u_1} C₁
inst✝³² : Category.{v_2, u_2} C₂
inst✝³¹ : Category.{v_3, u_5} C₃
inst✝³⁰ : Category.{v_4, u_6} C₄
inst✝²⁹ : Category.{v_5, u_3} C₁₂
inst✝²⁸ : Category.{v_6, u_4} C₂₃
inst✝²⁷ : HasZeroMorphisms C₁
inst✝²⁶ : HasZeroMorphisms C₂
inst✝²⁵ : HasZeroMorphisms C₃
inst✝²⁴ : Preadditive C₁₂
inst✝²³ : Preadditive C₂₃
inst✝²² : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
inst✝²¹ : F₁₂.PreservesZeroMorphisms
inst✝²⁰ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝¹⁹ : G.Additive
inst✝¹⁸ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
inst✝¹⁷ : G₂₃.PreservesZeroMorphisms
inst✝¹⁶ : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst✝¹⁵ : F.PreservesZeroMorphisms
inst✝¹⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
associator : bifunctorComp₁₂ F₁₂ G ≅ bifunctorComp₂₃ F G₂₃
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₂₃ : Type u_11
ι₄ : Type u_12
inst✝¹³ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₂₃ : ComplexShape ι₂₃
c₄ : ComplexShape ι₄
inst✝¹² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹¹ : TotalComplexShape c₁₂ c₃ c₄
inst✝¹⁰ : TotalComplexShape c₂ c₃ c₂₃
inst✝⁹ : TotalComplexShape c₁ c₂₃ c₄
inst✝⁸ : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝⁷ : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst✝⁶ : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst✝⁵ : DecidableEq ι₁₂
inst✝⁴ : DecidableEq ι₂₃
inst✝³ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
inst✝² : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
inst✝¹ : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
inst✝ : HasGoodTrifunctor₂₃Obj F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄
j j' : ι₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
h₁ : c₁.Rel i₁ (c₁.next i₁)
this :
  (G.map ((F₁₂.map (K₁.d i₁ (c₁.next i₁))).app (K₂.X i₂))).app (K₃.X i₃) ≫
      ((associator.hom.app (K₁.X (c₁.next i₁))).app (K₂.X i₂)).app (K₃.X i₃) =
    ((associator.hom.app (K₁.X i₁)).app (K₂.X i₂)).app (K₃.X i₃) ≫
      (F.map (K₁.d i₁ (c₁.next i₁))).app ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃))
⊢ (c₁₂.ε₁ c₃ c₄ (c₁.π c₂ c₁₂ (i₁, i₂), i₃) * c₁.ε₁ c₂ c₁₂ (i₁, i₂)) •
      ((associator.hom.app (K₁.X i₁)).app (K₂.X i₂)).app (K₃.X i₃) ≫
        (F.map (K₁.d i₁ (c₁.next i₁))).app ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃)) ≫
          mapBifunctor₂₃.ιOrZero F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ (c₁.next i₁) i₂ i₃ j' =
    (c₁₂.ε₁ c₃ c₄ (c₁.π c₂ c₁₂ (i₁, i₂), i₃) * c₁.ε₁ c₂ c₁₂ (i₁, i₂)) •
      (G.map ((F₁₂.map (K₁.d i₁ (c₁.next i₁))).app (K₂.X i₂))).app (K₃.X i₃) ≫
        mapBifunctor₁₂.ιOrZero F₁₂ G K₁ K₂ K₃ c₁₂ c₄ (c₁.next i₁) i₂ i₃ j' ≫
          (mapBifunctorAssociatorX associator K₁ K₂ K₃ c₁₂ c₂₃ c₄ j').hom