Module: Mathlib.Algebra.Homology.BifunctorHomotopy

Goal state before:
pp: 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✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
inst✝⁶ : Preadditive D
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K₁ L₁ : HomologicalComplex C₁ c₁
f₁ f₁' : K₁ ⟶ L₁
h₁ : Homotopy f₁ f₁'
K₂ L₂ : HomologicalComplex C₂ c₂
f₂ : K₂ ⟶ L₂
F : C₁ ⥤ C₂ ⥤ D
inst✝⁵ : F.Additive
inst✝⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
c : ComplexShape J
inst✝³ : DecidableEq J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : L₁.HasMapBifunctor L₂ F c
j : J
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c (i₁, i₂) = j
this : ∀ {X Y : D} (a b c d e f : X ⟶ Y), a = c → b = e → f = -d → a + b = c + d + (e + f)
⊢ (F.map (K₁.d i₁ (c₁.next i₁))).app (K₂.X i₂) ≫
        (F.map (h₁.hom (c₁.next i₁) i₁)).app (K₂.X i₂) ≫
          (F.obj (L₁.X i₁)).map (f₂.f i₂) ≫ L₁.ιMapBifunctor L₂ F c i₁ i₂ j h +
      (F.map (h₁.hom i₁ (c₁.prev i₁))).app (K₂.X i₂) ≫
        (F.map (L₁.d (c₁.prev i₁) i₁)).app (K₂.X i₂) ≫
          (F.obj (L₁.X i₁)).map (f₂.f i₂) ≫ L₁.ιMapBifunctor L₂ F c i₁ i₂ j h =
    (((F.mapBifunctorHomologicalComplex c₁ c₂).obj K₁).obj K₂).d₁ c i₁ i₂ (c.next j) ≫ hom₁ h₁ f₂ F c (c.next j) j +
        (((F.mapBifunctorHomologicalComplex c₁ c₂).obj K₁).obj K₂).d₂ c i₁ i₂ (c.next j) ≫ hom₁ h₁ f₂ F c (c.next j) j +
      (K₁.ιMapBifunctor K₂ F c i₁ i₂ j h ≫
          hom₁ h₁ f₂ F c j (c.prev j) ≫ (((F.mapBifunctorHomologicalComplex c₁ c₂).obj L₁).obj L₂).D₁ c (c.prev j) j +
        K₁.ιMapBifunctor K₂ F c i₁ i₂ j h ≫
          hom₁ h₁ f₂ F c j (c.prev j) ≫ (((F.mapBifunctorHomologicalComplex c₁ c₂).obj L₁).obj L₂).D₂ c (c.prev j) j)

Tactic:
apply this

Goal state after:
case a
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✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
inst✝⁶ : Preadditive D
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K₁ L₁ : HomologicalComplex C₁ c₁
f₁ f₁' : K₁ ⟶ L₁
h₁ : Homotopy f₁ f₁'
K₂ L₂ : HomologicalComplex C₂ c₂
f₂ : K₂ ⟶ L₂
F : C₁ ⥤ C₂ ⥤ D
inst✝⁵ : F.Additive
inst✝⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
c : ComplexShape J
inst✝³ : DecidableEq J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : L₁.HasMapBifunctor L₂ F c
j : J
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c (i₁, i₂) = j
this : ∀ {X Y : D} (a b c d e f : X ⟶ Y), a = c → b = e → f = -d → a + b = c + d + (e + f)
⊢ (F.map (K₁.d i₁ (c₁.next i₁))).app (K₂.X i₂) ≫
      (F.map (h₁.hom (c₁.next i₁) i₁)).app (K₂.X i₂) ≫
        (F.obj (L₁.X i₁)).map (f₂.f i₂) ≫ L₁.ιMapBifunctor L₂ F c i₁ i₂ j h =
    (((F.mapBifunctorHomologicalComplex c₁ c₂).obj K₁).obj K₂).d₁ c i₁ i₂ (c.next j) ≫ hom₁ h₁ f₂ F c (c.next j) j

case a
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✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
inst✝⁶ : Preadditive D
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K₁ L₁ : HomologicalComplex C₁ c₁
f₁ f₁' : K₁ ⟶ L₁
h₁ : Homotopy f₁ f₁'
K₂ L₂ : HomologicalComplex C₂ c₂
f₂ : K₂ ⟶ L₂
F : C₁ ⥤ C₂ ⥤ D
inst✝⁵ : F.Additive
inst✝⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
c : ComplexShape J
inst✝³ : DecidableEq J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : L₁.HasMapBifunctor L₂ F c
j : J
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c (i₁, i₂) = j
this : ∀ {X Y : D} (a b c d e f : X ⟶ Y), a = c → b = e → f = -d → a + b = c + d + (e + f)
⊢ (F.map (h₁.hom i₁ (c₁.prev i₁))).app (K₂.X i₂) ≫
      (F.map (L₁.d (c₁.prev i₁) i₁)).app (K₂.X i₂) ≫
        (F.obj (L₁.X i₁)).map (f₂.f i₂) ≫ L₁.ιMapBifunctor L₂ F c i₁ i₂ j h =
    K₁.ιMapBifunctor K₂ F c i₁ i₂ j h ≫
      hom₁ h₁ f₂ F c j (c.prev j) ≫ (((F.mapBifunctorHomologicalComplex c₁ c₂).obj L₁).obj L₂).D₁ c (c.prev j) j

case a
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✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
inst✝⁶ : Preadditive D
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K₁ L₁ : HomologicalComplex C₁ c₁
f₁ f₁' : K₁ ⟶ L₁
h₁ : Homotopy f₁ f₁'
K₂ L₂ : HomologicalComplex C₂ c₂
f₂ : K₂ ⟶ L₂
F : C₁ ⥤ C₂ ⥤ D
inst✝⁵ : F.Additive
inst✝⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
c : ComplexShape J
inst✝³ : DecidableEq J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : L₁.HasMapBifunctor L₂ F c
j : J
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c (i₁, i₂) = j
this : ∀ {X Y : D} (a b c d e f : X ⟶ Y), a = c → b = e → f = -d → a + b = c + d + (e + f)
⊢ K₁.ιMapBifunctor K₂ F c i₁ i₂ j h ≫
      hom₁ h₁ f₂ F c j (c.prev j) ≫ (((F.mapBifunctorHomologicalComplex c₁ c₂).obj L₁).obj L₂).D₂ c (c.prev j) j =
    -(((F.mapBifunctorHomologicalComplex c₁ c₂).obj K₁).obj K₂).d₂ c i₁ i₂ (c.next j) ≫ hom₁ h₁ f₂ F c (c.next j) j