-- Module: Mathlib.Algebra.Homology.BifunctorHomotopy
Goal:
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
inst_2 : Category.{v_3, u_3} D
inst_3 : Preadditive.{v_1, u_1} C₁
inst_4 : Preadditive.{v_2, u_2} C₂
inst_5 : Preadditive.{v_3, u_3} D
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K₁ L₁ : HomologicalComplex.{v_1, u_1, u_4} C₁ c₁
f₁ f₁' : K₁ ⟶ L₁
h₁ : Homotopy f₁ f₁'
K₂ L₂ : HomologicalComplex.{v_2, u_2, u_5} C₂ c₂
f₂ f₂' : K₂ ⟶ L₂
h₂ : Homotopy f₂ f₂'
F : C₁ ⥤ CategoryTheory.Functor.{v_2, v_3, u_2, u_3} C₂ D
inst_6 : F.Additive
inst_7 : ∀ (X₁ : C₁), (F.obj X₁).Additive
c : ComplexShape J
inst_8 : DecidableEq J
inst_9 : TotalComplexShape c₁ c₂ c
inst_10 : K₁.HasMapBifunctor K₂ F c
inst_11 : L₁.HasMapBifunctor L₂ F c
this_1 : TotalComplexShape c₂ c₁ c := TotalComplexShape.symm c₁ c₂ c
this_2 : TotalComplexShapeSymmetry c₁ c₂ c := TotalComplexShape.symmSymmetry c₁ c₂ c
this_3 : F.flip.Additive
this : ∀ (X₁ : C₂), (F.flip.obj X₁).Additive
H : Homotopy (C := K₁.mapBifunctor K₂ F c) (D := L₁.mapBifunctor L₂ F c) (mapBifunctorMap f₁ f₂ F c)
  (mapBifunctorMap f₁ f₂' F c) :=
  Homotopy.trans (f :=
    CategoryStruct.comp (Y := K₂.mapBifunctor K₁ F.flip c)
      (Iso.inv (K₁.mapBifunctorFlipIso K₂ F c) :
        Quiver.Hom (V := HomologicalComplex.{v_3, u_3, u_6} D c) (K₁.mapBifunctor K₂ F c) (K₂.mapBifunctor K₁ F.flip c))
      (CategoryStruct.comp (Y := L₂.mapBifunctor L₁ F.flip c)
          (mapBifunctorMap f₂ f₁ F.flip c :
            Quiver.Hom (V := HomologicalComplex.{v_3, u_3, u_6} D c) (K₂.mapBifunctor K₁ F.flip c)
              (L₂.mapBifunctor L₁ F.flip c))
          (Iso.hom (L₁.mapBifunctorFlipIso L₂ F c) :
            Quiver.Hom (V := HomologicalComplex.{v_3, u_3, u_6} D c) (L₂.mapBifunctor L₁ F.flip c)
              (L₁.mapBifunctor L₂ F c)) :
        Quiver.Hom (V := HomologicalComplex.{v_3, u_3, u_6} D c) (K₂.mapBifunctor K₁ F.flip c)
          (L₁.mapBifunctor L₂ F c)))
    (Homotopy.ofEq ⋯ :
      Homotopy (C := K₁.mapBifunctor K₂ F c) (D := L₁.mapBifunctor L₂ F c) (mapBifunctorMap f₁ f₂ F c)
        (CategoryStruct.comp (Y := K₂.mapBifunctor K₁ F.flip c)
          (Iso.inv (K₁.mapBifunctorFlipIso K₂ F c) :
            Quiver.Hom (V := HomologicalComplex.{v_3, u_3, u_6} D c) (K₁.mapBifunctor K₂ F c)
              (K₂.mapBifunctor K₁ F.flip c))
          (CategoryStruct.comp (Y := L₂.mapBifunctor L₁ F.flip c)
              (mapBifunctorMap f₂ f₁ F.flip c :
                Quiver.Hom (V := HomologicalComplex.{v_3, u_3, u_6} D c) (K₂.mapBifunctor K₁ F.flip c)
                  (L₂.mapBifunctor L₁ F.flip c))
              (Iso.hom (L₁.mapBifunctorFlipIso L₂ F c) :
                Quiver.Hom (V := HomologicalComplex.{v_3, u_3, u_6} D c) (L₂.mapBifunctor L₁ F.flip c)
                  (L₁.mapBifunctor L₂ F c)) :
            Quiver.Hom (V := HomologicalComplex.{v_3, u_3, u_6} D c) (K₂.mapBifunctor K₁ F.flip c)
              (L₁.mapBifunctor L₂ F c))))
    (Homotopy.trans (f :=
        CategoryStruct.comp (Y := K₂.mapBifunctor K₁ F.flip c)
          (Iso.inv (K₁.mapBifunctorFlipIso K₂ F c) :
            Quiver.Hom (V := HomologicalComplex.{v_3, u_3, u_6} D c) (K₁.mapBifunctor K₂ F c)
              (K₂.mapBifunctor K₁ F.flip c))
          (CategoryStruct.comp (Y := L₂.mapBifunctor L₁ F.flip c)
              (mapBifunctorMap f₂' f₁ F.flip c :
                Quiver.Hom (V := HomologicalComplex.{v_3, u_3, u_6} D c) (K₂.mapBifunctor K₁ F.flip c)
                  (L₂.mapBifunctor L₁ F.flip c))
              (Iso.hom (L₁.mapBifunctorFlipIso L₂ F c) :
                Quiver.Hom (V := HomologicalComplex.{v_3, u_3, u_6} D c) (L₂.mapBifunctor L₁ F.flip c)
                  (L₁.mapBifunctor L₂ F c)) :
            Quiver.Hom (V := HomologicalComplex.{v_3, u_3, u_6} D c) (K₂.mapBifunctor K₁ F.flip c)
              (L₁.mapBifunctor L₂ F c)))
        (Homotopy.compLeft
            (Homotopy.compRight (mapBifunctorMapHomotopy₁ h₂ f₁ F.flip c) (Iso.hom (L₁.mapBifunctorFlipIso L₂ F c)))
            (Iso.inv (K₁.mapBifunctorFlipIso K₂ F c)) :
          Homotopy (C := K₁.mapBifunctor K₂ F c) (D := L₁.mapBifunctor L₂ F c)
            (CategoryStruct.comp (Y := K₂.mapBifunctor K₁ F.flip c)
              (Iso.inv (K₁.mapBifunctorFlipIso K₂ F c) :
                Quiver.Hom (V := HomologicalComplex.{v_3, u_3, u_6} D c) (K₁.mapBifunctor K₂ F c)
                  (K₂.mapBifunctor K₁ F.flip c))
              (CategoryStruct.comp (Y := L₂.mapBifunctor L₁ F.flip c)
                  (mapBifunctorMap f₂ f₁ F.flip c :
                    Quiver.Hom (V := HomologicalComplex.{v_3, u_3, u_6} D c) (K₂.mapBifunctor K₁ F.flip c)
                      (L₂.mapBifunctor L₁ F.flip c))
                  (Iso.hom (L₁.mapBifunctorFlipIso L₂ F c) :
                    Quiver.Hom (V := HomologicalComplex.{v_3, u_3, u_6} D c) (L₂.mapBifunctor L₁ F.flip c)
                      (L₁.mapBifunctor L₂ F c)) :
                Quiver.Hom (V := HomologicalComplex.{v_3, u_3, u_6} D c) (K₂.mapBifunctor K₁ F.flip c)
                  (L₁.mapBifunctor L₂ F c)))
            (CategoryStruct.comp (Y := K₂.mapBifunctor K₁ F.flip c)
              (Iso.inv (K₁.mapBifunctorFlipIso K₂ F c) :
                Quiver.Hom (V := HomologicalComplex.{v_3, u_3, u_6} D c) (K₁.mapBifunctor K₂ F c)
                  (K₂.mapBifunctor K₁ F.flip c))
              (CategoryStruct.comp (Y := L₂.mapBifunctor L₁ F.flip c)
                  (mapBifunctorMap f₂' f₁ F.flip c :
                    Quiver.Hom (V := HomologicalComplex.{v_3, u_3, u_6} D c) (K₂.mapBifunctor K₁ F.flip c)
                      (L₂.mapBifunctor L₁ F.flip c))
                  (Iso.hom (L₁.mapBifunctorFlipIso L₂ F c) :
                    Quiver.Hom (V := HomologicalComplex.{v_3, u_3, u_6} D c) (L₂.mapBifunctor L₁ F.flip c)
                      (L₁.mapBifunctor L₂ F c)) :
                Quiver.Hom (V := HomologicalComplex.{v_3, u_3, u_6} D c) (K₂.mapBifunctor K₁ F.flip c)
                  (L₁.mapBifunctor L₂ F c))))
        (Homotopy.ofEq ⋯ :
          Homotopy (C := K₁.mapBifunctor K₂ F c) (D := L₁.mapBifunctor L₂ F c)
            (CategoryStruct.comp (Y := K₂.mapBifunctor K₁ F.flip c)
              (Iso.inv (K₁.mapBifunctorFlipIso K₂ F c) :
                Quiver.Hom (V := HomologicalComplex.{v_3, u_3, u_6} D c) (K₁.mapBifunctor K₂ F c)
                  (K₂.mapBifunctor K₁ F.flip c))
              (CategoryStruct.comp (Y := L₂.mapBifunctor L₁ F.flip c)
                  (mapBifunctorMap f₂' f₁ F.flip c :
                    Quiver.Hom (V := HomologicalComplex.{v_3, u_3, u_6} D c) (K₂.mapBifunctor K₁ F.flip c)
                      (L₂.mapBifunctor L₁ F.flip c))
                  (Iso.hom (L₁.mapBifunctorFlipIso L₂ F c) :
                    Quiver.Hom (V := HomologicalComplex.{v_3, u_3, u_6} D c) (L₂.mapBifunctor L₁ F.flip c)
                      (L₁.mapBifunctor L₂ F c)) :
                Quiver.Hom (V := HomologicalComplex.{v_3, u_3, u_6} D c) (K₂.mapBifunctor K₁ F.flip c)
                  (L₁.mapBifunctor L₂ F c)))
            (mapBifunctorMap f₁ f₂' F c)) :
      Homotopy (C := K₁.mapBifunctor K₂ F c) (D := L₁.mapBifunctor L₂ F c)
        (CategoryStruct.comp (Y := K₂.mapBifunctor K₁ F.flip c)
          (Iso.inv (K₁.mapBifunctorFlipIso K₂ F c) :
            Quiver.Hom (V := HomologicalComplex.{v_3, u_3, u_6} D c) (K₁.mapBifunctor K₂ F c)
              (K₂.mapBifunctor K₁ F.flip c))
          (CategoryStruct.comp (Y := L₂.mapBifunctor L₁ F.flip c)
              (mapBifunctorMap f₂ f₁ F.flip c :
                Quiver.Hom (V := HomologicalComplex.{v_3, u_3, u_6} D c) (K₂.mapBifunctor K₁ F.flip c)
                  (L₂.mapBifunctor L₁ F.flip c))
              (Iso.hom (L₁.mapBifunctorFlipIso L₂ F c) :
                Quiver.Hom (V := HomologicalComplex.{v_3, u_3, u_6} D c) (L₂.mapBifunctor L₁ F.flip c)
                  (L₁.mapBifunctor L₂ F c)) :
            Quiver.Hom (V := HomologicalComplex.{v_3, u_3, u_6} D c) (K₂.mapBifunctor K₁ F.flip c)
              (L₁.mapBifunctor L₂ F c)))
        (mapBifunctorMap f₁ f₂' F c))
j j' : J
i₁ : I₁
i₂ : I₂
hj : c₁.π c₂ c (i₁, i₂) = j
hj' : c₁.π c₂ c (i₁, c₂.prev i₂) = j'
hi₂ : c₂.Rel (c₂.prev i₂) i₂
pos : c₁.ε₂ c₂ c (i₁, c₂.prev i₂) = c₂.ε₁ c₁ c (c₂.prev i₂, i₁) * c₁.σ c₂ c i₁ i₂ * c₁.σ c₂ c i₁ (c₂.prev i₂)
⊢ c₁.ε₂ c₂ c (i₁, c₂.prev i₂) = c₁.σ c₂ c i₁ i₂ * c₂.ε₁ c₁ c (c₂.prev i₂, i₁) * c₁.σ c₂ c i₁ (c₂.prev i₂)

Tactic:
nth_rw 2 [mul_comm]

Premises used: mul_comm