-- Module: Mathlib.Algebra.Homology.BifunctorAssociator
Goal:
C₁ : Type u_1
C₂ : Type u_2
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
inst_2 : Category.{v_3, u_5} C₃
inst_3 : Category.{v_4, u_6} C₄
inst_4 : Category.{v_6, u_4} C₂₃
inst_5 : HasZeroMorphisms.{v_1, u_1} C₁
inst_6 : HasZeroMorphisms.{v_2, u_2} C₂
inst_7 : HasZeroMorphisms.{v_3, u_5} C₃
inst_8 : Preadditive.{v_6, u_4} C₂₃
inst_9 : Preadditive.{v_4, u_6} C₄
F : C₁ ⥤ CategoryTheory.Functor.{v_6, v_4, u_4, u_6} C₂₃ C₄
G₂₃ : C₂ ⥤ CategoryTheory.Functor.{v_3, v_6, u_5, u_4} C₃ C₂₃
inst_10 : G₂₃.PreservesZeroMorphisms
inst_11 : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst_12 : F.PreservesZeroMorphisms
inst_13 : ∀ (X₁ : C₁), (F.obj X₁).Additive
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_14 : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex.{v_1, u_1, u_7} C₁ c₁
K₂ : HomologicalComplex.{v_2, u_2, u_8} C₂ c₂
K₃ : HomologicalComplex.{v_3, u_5, u_9} C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₂₃ : ComplexShape ι₂₃
c₄ : ComplexShape ι₄
inst_15 : TotalComplexShape c₁ c₂ c₁₂
inst_16 : TotalComplexShape c₁₂ c₃ c₄
inst_17 : TotalComplexShape c₂ c₃ c₂₃
inst_18 : TotalComplexShape c₁ c₂₃ c₄
inst_19 : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst_20 : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst_21 : DecidableEq ι₂₃
inst_22 : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
inst_23 : 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
i₂₃ : ι₂₃ := c₂.π c₃ c₂₃ (i₂, i₃)
h₁ : c₂₃.Rel i₂₃ (c₂₃.next i₂₃)
h₂ : c₁.π c₂₃ c₄ (i₁, c₂₃.next i₂₃) = j'
h₃ : c₃.Rel i₃ (c₃.next i₃)
pos :
  Eq (α :=
    (F.obj (K₁.X i₁)).obj ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃)) ⟶
      X (K₁.mapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄) j')
    ((F.obj (K₁.X i₁)).map ((G₂₃.obj (K₂.X i₂)).map (K₃.d i₃ (c₃.next i₃))) ≫
        (F.obj (K₁.X i₁)).map (K₂.ιMapBifunctor K₃ G₂₃ c₂₃ i₂ (c₃.next i₃) (c₂₃.next (c₂.π c₃ c₂₃ (i₂, i₃))) ⋯) ≫
          K₁.ιMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄ i₁ (c₂₃.next i₂₃) j' h₂ :
      (F.obj (K₁.X i₁)).obj ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃)) ⟶
        X (K₁.mapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄) j')
    (CategoryStruct.comp (Y := (F.obj (K₁.X i₁)).obj ((G₂₃.obj (K₂.X i₂)).obj (K₃.X (c₃.next i₃))))
        ((F.obj (K₁.X i₁)).map ((G₂₃.obj (K₂.X i₂)).map (K₃.d i₃ (c₃.next i₃))))
        (ι F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ (c₃.next i₃) j' ⋯) :
      (F.obj (K₁.X i₁)).obj ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃)) ⟶
        X (K₁.mapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄) j')
pos₁ :
  Eq (α :=
    (F.obj (K₁.X i₁)).obj ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃)) ⟶
      X (K₁.mapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄) j')
    ((F.obj (K₁.X i₁)).map ((G₂₃.obj (K₂.X i₂)).map (K₃.d i₃ (c₃.next i₃))) ≫
      (F.obj (K₁.X i₁)).map (K₂.ιMapBifunctor K₃ G₂₃ c₂₃ i₂ (c₃.next i₃) (c₂₃.next (c₂.π c₃ c₂₃ (i₂, i₃))) ⋯) ≫
        K₁.ιMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄ i₁ (c₂₃.next i₂₃) j' h₂)
    ((F.obj (K₁.X i₁)).map ((G₂₃.obj (K₂.X i₂)).map (K₃.d i₃ (c₃.next i₃))) ≫
      (F.obj (K₁.X i₁)).map (K₂.ιMapBifunctor K₃ G₂₃ c₂₃ i₂ (c₃.next i₃) (c₂₃.next (c₂.π c₃ c₂₃ (i₂, i₃))) ⋯) ≫
        K₁.ιMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄ i₁ (c₂₃.next (c₂.π c₃ c₂₃ (i₂, i₃))) j' h₂)
pos₂ : ι₂₃
pos₃ : c₂.π c₃ c₂₃ (i₂, c₃.next i₃) = c₂₃.next (c₂.π c₃ c₂₃ (i₂, i₃))
pos₄ : c₁.π c₂₃ c₄ (i₁, c₂₃.next (c₂.π c₃ c₂₃ (i₂, i₃))) = j'
pos₅ :
  Eq (α :=
    (F.obj (K₁.X i₁)).obj ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃)) ⟶
      X (K₁.mapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄) j')
    ((F.obj (K₁.X i₁)).map ((G₂₃.obj (K₂.X i₂)).map (K₃.d i₃ (c₃.next i₃))) ≫
      (F.obj (K₁.X i₁)).map (K₂.ιMapBifunctor K₃ G₂₃ c₂₃ i₂ (c₃.next i₃) (c₂₃.next (c₂.π c₃ c₂₃ (i₂, i₃))) ⋯) ≫
        K₁.ιMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄ i₁ (c₂₃.next i₂₃) j' h₂)
    ((F.obj (K₁.X i₁)).map ((G₂₃.obj (K₂.X i₂)).map (K₃.d i₃ (c₃.next i₃))) ≫
      (F.obj (K₁.X i₁)).map (K₂.ιMapBifunctor K₃ G₂₃ c₂₃ i₂ (c₃.next i₃) (c₂₃.next (c₂.π c₃ c₂₃ (i₂, i₃))) ⋯) ≫
        K₁.ιMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄ i₁ (c₂₃.next (c₂.π c₃ c₂₃ (i₂, i₃))) j' h₂)
pos₆ : ι₂₃
pos₇ : c₂.π c₃ c₂₃ (i₂, c₃.next i₃) = c₂₃.next (c₂.π c₃ c₂₃ (i₂, i₃))
pos₈ : c₁.π c₂₃ c₄ (i₁, c₂₃.next (c₂.π c₃ c₂₃ (i₂, i₃))) = j'
⊢ Eq (α :=
    (F.obj (K₁.X i₁)).obj ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃)) ⟶
      X (K₁.mapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄) j')
    ((F.obj (K₁.X i₁)).map ((G₂₃.obj (K₂.X i₂)).map (K₃.d i₃ (c₃.next i₃))) ≫
        (F.obj (K₁.X i₁)).map (K₂.ιMapBifunctor K₃ G₂₃ c₂₃ i₂ (c₃.next i₃) (c₂₃.next (c₂.π c₃ c₂₃ (i₂, i₃))) ⋯) ≫
          K₁.ιMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄ i₁ (c₂₃.next i₂₃) j' h₂ :
      (F.obj (K₁.X i₁)).obj ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃)) ⟶
        X (K₁.mapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄) j')
    (CategoryStruct.comp (Y := (F.obj (K₁.X i₁)).obj ((G₂₃.obj (K₂.X i₂)).obj (K₃.X (c₃.next i₃))))
        ((F.obj (K₁.X i₁)).map ((G₂₃.obj (K₂.X i₂)).map (K₃.d i₃ (c₃.next i₃))))
        (ιOrZero F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ (c₃.next i₃) j') :
      (F.obj (K₁.X i₁)).obj ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃)) ⟶
        X (K₁.mapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄) j')

Tactic:
rw [ιOrZero_eq _ _ _ _ _ _ _ _ _ _ _ _ (by
            dsimp [ComplexShape.r]
            rw [← h₂, ComplexShape.assoc c₁ c₂ c₃ c₁₂ c₂₃ c₄, ComplexShape.next_π₂ c₂ c₂₃ i₂ h₃]),
            ι_eq]

Premises used: HomologicalComplex.mapBifunctor₂₃.ιOrZero_eq, HomologicalComplex.mapBifunctor₂₃.ι_eq