Module: Mathlib.CategoryTheory.Abelian.RightDerived

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{v_1, 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:
← cancel_mono ((HomotopyCategory.homologyFunctor _ _ 0).map
      (I.isoRightDerivedToHomotopyCategoryObj F).hom),

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{v_1, 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 (I.isoRightDerivedToHomotopyCategoryObj F).hom =
    (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) ≫
      (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) 0).map (I.isoRightDerivedToHomotopyCategoryObj F).hom