Theorem: CategoryTheory.InjectiveResolution.toRightDerivedZero_eq
File: Mathlib/CategoryTheory/Abelian/RightDerived.lean

Proof steps:
state_before: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
X : C
I : InjectiveResolution X
F : C ⥤ D
inst✝ : F.Additive
⊢ F.toRightDerivedZero.app X =
    I.toRightDerivedZero' F ≫
      (CochainComplex.isoHomologyπ₀ ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex)).hom ≫
        (I.isoRightDerivedObj F 0).inv
state_after: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
X : C
I : InjectiveResolution X
F : C ⥤ D
inst✝ : F.Additive
⊢ (injectiveResolution X).toRightDerivedZero' F ≫
      ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex).homologyπ 0 ≫
        (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
          ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex) =
    I.toRightDerivedZero' F ≫
      ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex).homologyπ 0 ≫
        (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex) ≫
          (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) 0).map (I.isoRightDerivedToHomotopyCategoryObj F).inv
tactic: dsimp [Functor.toRightDerivedZero, isoRightDerivedObj]
state_before: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
X : C
I : InjectiveResolution X
F : C ⥤ D
inst✝ : F.Additive
⊢ (injectiveResolution X).toRightDerivedZero' F ≫
      ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex).homologyπ 0 ≫
        (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
          ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex) =
    I.toRightDerivedZero' F ≫
      ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex).homologyπ 0 ≫
        (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex) ≫
          (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) 0).map (I.isoRightDerivedToHomotopyCategoryObj F).inv
state_after: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
X : C
I : InjectiveResolution X
F : C ⥤ D
inst✝ : F.Additive
h₁ :
  F.map (𝟙 X) ≫ I.toRightDerivedZero' F =
    (injectiveResolution X).toRightDerivedZero' F ≫
      HomologicalComplex.cyclesMap
        ((F.mapHomologicalComplex (ComplexShape.up ℕ)).map (desc (𝟙 X) I (injectiveResolution X))) 0
⊢ (injectiveResolution X).toRightDerivedZero' F ≫
      ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex).homologyπ 0 ≫
        (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
          ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex) =
    I.toRightDerivedZero' F ≫
      ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex).homologyπ 0 ≫
        (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex) ≫
          (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) 0).map (I.isoRightDerivedToHomotopyCategoryObj F).inv
tactic: have h₁ := InjectiveResolution.toRightDerivedZero'_naturality
    (𝟙 X) (injectiveResolution X) I (desc (𝟙 X) _ _) (by simp) F
state_before: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
X : C
I : InjectiveResolution X
F : C ⥤ D
inst✝ : F.Additive
h₁ :
  F.map (𝟙 X) ≫ I.toRightDerivedZero' F =
    (injectiveResolution X).toRightDerivedZero' F ≫
      HomologicalComplex.cyclesMap
        ((F.mapHomologicalComplex (ComplexShape.up ℕ)).map (desc (𝟙 X) I (injectiveResolution X))) 0
⊢ (injectiveResolution X).toRightDerivedZero' F ≫
      ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex).homologyπ 0 ≫
        (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
          ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex) =
    I.toRightDerivedZero' F ≫
      ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex).homologyπ 0 ≫
        (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex) ≫
          (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) 0).map (I.isoRightDerivedToHomotopyCategoryObj F).inv
state_after: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
X : C
I : InjectiveResolution X
F : C ⥤ D
inst✝ : F.Additive
h₁ :
  I.toRightDerivedZero' F =
    (injectiveResolution X).toRightDerivedZero' F ≫
      HomologicalComplex.cyclesMap
        ((F.mapHomologicalComplex (ComplexShape.up ℕ)).map (desc (𝟙 X) I (injectiveResolution X))) 0
⊢ (injectiveResolution X).toRightDerivedZero' F ≫
      ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex).homologyπ 0 ≫
        (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
          ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex) =
    I.toRightDerivedZero' F ≫
      ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex).homologyπ 0 ≫
        (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex) ≫
          (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) 0).map (I.isoRightDerivedToHomotopyCategoryObj F).inv
tactic: simp only [Functor.map_id, id_comp] at h₁
state_before: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
X : C
I : InjectiveResolution X
F : C ⥤ D
inst✝ : F.Additive
h₁ :
  I.toRightDerivedZero' F =
    (injectiveResolution X).toRightDerivedZero' F ≫
      HomologicalComplex.cyclesMap
        ((F.mapHomologicalComplex (ComplexShape.up ℕ)).map (desc (𝟙 X) I (injectiveResolution X))) 0
⊢ (injectiveResolution X).toRightDerivedZero' F ≫
      ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex).homologyπ 0 ≫
        (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
          ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex) =
    I.toRightDerivedZero' F ≫
      ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex).homologyπ 0 ≫
        (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex) ≫
          (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) 0).map (I.isoRightDerivedToHomotopyCategoryObj F).inv
state_after: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
X : C
I : InjectiveResolution X
F : C ⥤ D
inst✝ : F.Additive
h₁ :
  I.toRightDerivedZero' F =
    (injectiveResolution X).toRightDerivedZero' F ≫
      HomologicalComplex.cyclesMap
        ((F.mapHomologicalComplex (ComplexShape.up ℕ)).map (desc (𝟙 X) I (injectiveResolution X))) 0
h₂ :
  (I.isoRightDerivedToHomotopyCategoryObj F).hom =
    (F.mapHomologicalComplex (ComplexShape.up ℕ) ⋙ HomotopyCategory.quotient D (ComplexShape.up ℕ)).map
      (desc (𝟙 X) I (injectiveResolution X))
⊢ (injectiveResolution X).toRightDerivedZero' F ≫
      ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex).homologyπ 0 ≫
        (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
          ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex) =
    I.toRightDerivedZero' F ≫
      ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex).homologyπ 0 ≫
        (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex) ≫
          (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) 0).map (I.isoRightDerivedToHomotopyCategoryObj F).inv
tactic: have h₂ : (I.isoRightDerivedToHomotopyCategoryObj F).hom =
    (F.mapHomologicalComplex _ ⋙ HomotopyCategory.quotient _ _).map (desc (𝟙 X) _ _) :=
    comp_id _
state_before: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
X : C
I : InjectiveResolution X
F : C ⥤ D
inst✝ : F.Additive
h₁ :
  I.toRightDerivedZero' F =
    (injectiveResolution X).toRightDerivedZero' F ≫
      HomologicalComplex.cyclesMap
        ((F.mapHomologicalComplex (ComplexShape.up ℕ)).map (desc (𝟙 X) I (injectiveResolution X))) 0
h₂ :
  (I.isoRightDerivedToHomotopyCategoryObj F).hom =
    (F.mapHomologicalComplex (ComplexShape.up ℕ) ⋙ HomotopyCategory.quotient D (ComplexShape.up ℕ)).map
      (desc (𝟙 X) I (injectiveResolution X))
⊢ (injectiveResolution X).toRightDerivedZero' F ≫
      ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex).homologyπ 0 ≫
        (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
          ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex) =
    I.toRightDerivedZero' F ≫
      ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex).homologyπ 0 ≫
        (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex) ≫
          (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) 0).map (I.isoRightDerivedToHomotopyCategoryObj F).inv
state_after: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
X : C
I : InjectiveResolution X
F : C ⥤ D
inst✝ : F.Additive
h₁ :
  I.toRightDerivedZero' F =
    (injectiveResolution X).toRightDerivedZero' F ≫
      HomologicalComplex.cyclesMap
        ((F.mapHomologicalComplex (ComplexShape.up ℕ)).map (desc (𝟙 X) I (injectiveResolution X))) 0
h₂ :
  (I.isoRightDerivedToHomotopyCategoryObj F).hom =
    (F.mapHomologicalComplex (ComplexShape.up ℕ) ⋙ HomotopyCategory.quotient D (ComplexShape.up ℕ)).map
      (desc (𝟙 X) I (injectiveResolution X))
⊢ (injectiveResolution X).toRightDerivedZero' F ≫
      ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex).homologyπ 0 ≫
        (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex) ≫
          (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) 0).map
            ((F.mapHomologicalComplex (ComplexShape.up ℕ) ⋙ HomotopyCategory.quotient D (ComplexShape.up ℕ)).map
              (desc (𝟙 X) I (injectiveResolution X))) =
    (injectiveResolution X).toRightDerivedZero' F ≫
      ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex).homologyπ 0 ≫
        HomologicalComplex.homologyMap
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).map (desc (𝟙 X) I (injectiveResolution X))) 0 ≫
          (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex)
tactic: rw [← cancel_mono ((HomotopyCategory.homologyFunctor _ _ 0).map
      (I.isoRightDerivedToHomotopyCategoryObj F).hom),
    assoc, assoc, assoc, assoc, assoc, ← Functor.map_comp,
    Iso.inv_hom_id, Functor.map_id, comp_id,
    reassoc_of% h₁, h₂, ← HomologicalComplex.homologyπ_naturality_assoc]
state_before: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
X : C
I : InjectiveResolution X
F : C ⥤ D
inst✝ : F.Additive
h₁ :
  I.toRightDerivedZero' F =
    (injectiveResolution X).toRightDerivedZero' F ≫
      HomologicalComplex.cyclesMap
        ((F.mapHomologicalComplex (ComplexShape.up ℕ)).map (desc (𝟙 X) I (injectiveResolution X))) 0
h₂ :
  (I.isoRightDerivedToHomotopyCategoryObj F).hom =
    (F.mapHomologicalComplex (ComplexShape.up ℕ) ⋙ HomotopyCategory.quotient D (ComplexShape.up ℕ)).map
      (desc (𝟙 X) I (injectiveResolution X))
⊢ (injectiveResolution X).toRightDerivedZero' F ≫
      ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex).homologyπ 0 ≫
        (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex) ≫
          (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) 0).map
            ((F.mapHomologicalComplex (ComplexShape.up ℕ) ⋙ HomotopyCategory.quotient D (ComplexShape.up ℕ)).map
              (desc (𝟙 X) I (injectiveResolution X))) =
    (injectiveResolution X).toRightDerivedZero' F ≫
      ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex).homologyπ 0 ≫
        HomologicalComplex.homologyMap
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).map (desc (𝟙 X) I (injectiveResolution X))) 0 ≫
          (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex)
state_after: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
X : C
I : InjectiveResolution X
F : C ⥤ D
inst✝ : F.Additive
h₁ :
  I.toRightDerivedZero' F =
    (injectiveResolution X).toRightDerivedZero' F ≫
      HomologicalComplex.cyclesMap
        ((F.mapHomologicalComplex (ComplexShape.up ℕ)).map (desc (𝟙 X) I (injectiveResolution X))) 0
h₂ :
  (I.isoRightDerivedToHomotopyCategoryObj F).hom =
    (F.mapHomologicalComplex (ComplexShape.up ℕ) ⋙ HomotopyCategory.quotient D (ComplexShape.up ℕ)).map
      (desc (𝟙 X) I (injectiveResolution X))
⊢ (injectiveResolution X).toRightDerivedZero' F ≫
      ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex).homologyπ 0 ≫
        (HomologicalComplex.homologyFunctor D (ComplexShape.up ℕ) 0).map
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).map (desc (𝟙 X) I (injectiveResolution X))) ≫
          (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex) =
    (injectiveResolution X).toRightDerivedZero' F ≫
      ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex).homologyπ 0 ≫
        HomologicalComplex.homologyMap
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).map (desc (𝟙 X) I (injectiveResolution X))) 0 ≫
          (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex)
tactic: erw [← NatTrans.naturality]
state_before: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
X : C
I : InjectiveResolution X
F : C ⥤ D
inst✝ : F.Additive
h₁ :
  I.toRightDerivedZero' F =
    (injectiveResolution X).toRightDerivedZero' F ≫
      HomologicalComplex.cyclesMap
        ((F.mapHomologicalComplex (ComplexShape.up ℕ)).map (desc (𝟙 X) I (injectiveResolution X))) 0
h₂ :
  (I.isoRightDerivedToHomotopyCategoryObj F).hom =
    (F.mapHomologicalComplex (ComplexShape.up ℕ) ⋙ HomotopyCategory.quotient D (ComplexShape.up ℕ)).map
      (desc (𝟙 X) I (injectiveResolution X))
⊢ (injectiveResolution X).toRightDerivedZero' F ≫
      ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex).homologyπ 0 ≫
        (HomologicalComplex.homologyFunctor D (ComplexShape.up ℕ) 0).map
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).map (desc (𝟙 X) I (injectiveResolution X))) ≫
          (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex) =
    (injectiveResolution X).toRightDerivedZero' F ≫
      ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex).homologyπ 0 ≫
        HomologicalComplex.homologyMap
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).map (desc (𝟙 X) I (injectiveResolution X))) 0 ≫
          (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app
            ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex)
state_after: no goals
tactic: rfl