-- Module: Mathlib.Algebra.Homology.BifunctorAssociator
Goal:
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
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_5, u_3} C₁₂
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_5, u_3} C₁₂
inst_9 : Preadditive.{v_4, u_6} C₄
F₁₂ : C₁ ⥤ CategoryTheory.Functor.{v_2, v_5, u_2, u_3} C₂ C₁₂
G : C₁₂ ⥤ CategoryTheory.Functor.{v_3, v_4, u_5, u_6} C₃ C₄
inst_10 : F₁₂.PreservesZeroMorphisms
inst_11 : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst_12 : G.Additive
inst_13 : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
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
c₄ : ComplexShape ι₄
inst_15 : TotalComplexShape c₁ c₂ c₁₂
inst_16 : TotalComplexShape c₁₂ c₃ c₄
inst_17 : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst_18 : DecidableEq ι₁₂
inst_19 : HasMapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄
j j' : ι₄
inst_20 : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
e_a :
  CategoryStruct.comp (Y := X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j)
      (ι F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j h :
        (G.obj ((F₁₂.obj (K₁.X i₁)).obj (K₂.X i₂))).obj (K₃.X i₃) ⟶
          X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j)
      (mapBifunctor.D₁ (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄ j j' :
        X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j ⟶
          X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j') =
    HAdd.hAdd (α :=
      (G.obj ((F₁₂.obj (K₁.X i₁)).obj (K₂.X i₂))).obj (K₃.X i₃) ⟶
        X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j')
      (β :=
      (G.obj ((F₁₂.obj (K₁.X i₁)).obj (K₂.X i₂))).obj (K₃.X i₃) ⟶
        X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j')
      (d₁ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j' :
        (G.obj ((F₁₂.obj (K₁.X i₁)).obj (K₂.X i₂))).obj (K₃.X i₃) ⟶
          X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j')
      (d₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j' :
        (G.obj ((F₁₂.obj (K₁.X i₁)).obj (K₂.X i₂))).obj (K₃.X i₃) ⟶
          X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j')
⊢ CategoryStruct.comp (Y := X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j)
      (ι F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j h :
        (G.obj ((F₁₂.obj (K₁.X i₁)).obj (K₂.X i₂))).obj (K₃.X i₃) ⟶
          X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j)
      (mapBifunctor.D₁ (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄ j j' :
        X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j ⟶
          X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j') =
    ι F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j h ≫
      HAdd.hAdd (α :=
        X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j ⟶
          X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j')
        (β :=
        X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j ⟶
          X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j')
        (D₁ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ j j' :
          X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j ⟶
            X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j')
        (D₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ j j' :
          X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j ⟶
            X (mapBifunctor (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄) j')

Tactic:
simp only [Preadditive.comp_add, ι_D₁, ι_D₂]

Premises used: HomologicalComplex.mapBifunctor₁₂.ι_D₁, CategoryTheory.Preadditive.comp_add, HomologicalComplex.mapBifunctor₁₂.ι_D₂