Theorem: CategoryTheory.ProjectiveResolution.leftDerivedToHomotopyCategory_app_eq
File: Mathlib/CategoryTheory/Abelian/LeftDerived.lean

Proof steps:
state_before: C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : ProjectiveResolution X
⊢ (NatTrans.leftDerivedToHomotopyCategory α).app X =
    (P.isoLeftDerivedToHomotopyCategoryObj F).hom ≫
      (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
          ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app P.complex) ≫
        (P.isoLeftDerivedToHomotopyCategoryObj G).inv
state_after: C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : ProjectiveResolution X
⊢ (NatTrans.leftDerivedToHomotopyCategory α).app X ≫ (P.isoLeftDerivedToHomotopyCategoryObj G).hom =
    (P.isoLeftDerivedToHomotopyCategoryObj F).hom ≫
      (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
        ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app P.complex)
tactic: rw [← cancel_mono (P.isoLeftDerivedToHomotopyCategoryObj G).hom, assoc, assoc,
      Iso.inv_hom_id, comp_id]
state_before: C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : ProjectiveResolution X
⊢ (NatTrans.leftDerivedToHomotopyCategory α).app X ≫ (P.isoLeftDerivedToHomotopyCategoryObj G).hom =
    (P.isoLeftDerivedToHomotopyCategoryObj F).hom ≫
      (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
        ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app P.complex)
state_after: C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : ProjectiveResolution X
⊢ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
        ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app ((projectiveResolutions C).obj X).as) ≫
      (G.mapHomotopyCategory (ComplexShape.down ℕ)).map P.iso.hom ≫
        𝟙
          ((HomotopyCategory.quotient D (ComplexShape.down ℕ)).obj
            ((G.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex)) =
    ((F.mapHomotopyCategory (ComplexShape.down ℕ)).map P.iso.hom ≫
        𝟙
          ((HomotopyCategory.quotient D (ComplexShape.down ℕ)).obj
            ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex))) ≫
      (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
        ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app P.complex)
tactic: dsimp [isoLeftDerivedToHomotopyCategoryObj, Functor.mapHomotopyCategoryFactors,
    NatTrans.leftDerivedToHomotopyCategory]
state_before: C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : ProjectiveResolution X
⊢ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
        ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app ((projectiveResolutions C).obj X).as) ≫
      (G.mapHomotopyCategory (ComplexShape.down ℕ)).map P.iso.hom ≫
        𝟙
          ((HomotopyCategory.quotient D (ComplexShape.down ℕ)).obj
            ((G.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex)) =
    ((F.mapHomotopyCategory (ComplexShape.down ℕ)).map P.iso.hom ≫
        𝟙
          ((HomotopyCategory.quotient D (ComplexShape.down ℕ)).obj
            ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex))) ≫
      (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
        ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app P.complex)
state_after: C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : ProjectiveResolution X
⊢ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
        ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app ((projectiveResolutions C).obj X).as) ≫
      (G.mapHomotopyCategory (ComplexShape.down ℕ)).map P.iso.hom ≫
        𝟙
          ((HomotopyCategory.quotient D (ComplexShape.down ℕ)).obj
            ((G.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex)) =
    (F.mapHomotopyCategory (ComplexShape.down ℕ)).map P.iso.hom ≫
      𝟙
          ((HomotopyCategory.quotient D (ComplexShape.down ℕ)).obj
            ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex)) ≫
        (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
          ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app P.complex)
tactic: rw [assoc]
state_before: C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : ProjectiveResolution X
⊢ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
        ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app ((projectiveResolutions C).obj X).as) ≫
      (G.mapHomotopyCategory (ComplexShape.down ℕ)).map P.iso.hom ≫
        𝟙
          ((HomotopyCategory.quotient D (ComplexShape.down ℕ)).obj
            ((G.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex)) =
    (F.mapHomotopyCategory (ComplexShape.down ℕ)).map P.iso.hom ≫
      𝟙
          ((HomotopyCategory.quotient D (ComplexShape.down ℕ)).obj
            ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex)) ≫
        (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
          ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app P.complex)
state_after: C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : ProjectiveResolution X
⊢ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
        ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app ((projectiveResolutions C).obj X).as) ≫
      (G.mapHomotopyCategory (ComplexShape.down ℕ)).map P.iso.hom =
    (F.mapHomotopyCategory (ComplexShape.down ℕ)).map P.iso.hom ≫
      (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
        ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app P.complex)
tactic: erw [id_comp, comp_id]
state_before: C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : ProjectiveResolution X
⊢ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
        ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app ((projectiveResolutions C).obj X).as) ≫
      (G.mapHomotopyCategory (ComplexShape.down ℕ)).map P.iso.hom =
    (F.mapHomotopyCategory (ComplexShape.down ℕ)).map P.iso.hom ≫
      (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
        ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app P.complex)
state_after: case intro
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : ProjectiveResolution X
β : (projectiveResolution X).complex ⟶ P.complex
hβ : (HomotopyCategory.quotient C (ComplexShape.down ℕ)).map β = P.iso.hom
⊢ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
        ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app ((projectiveResolutions C).obj X).as) ≫
      (G.mapHomotopyCategory (ComplexShape.down ℕ)).map P.iso.hom =
    (F.mapHomotopyCategory (ComplexShape.down ℕ)).map P.iso.hom ≫
      (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
        ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app P.complex)
tactic: obtain ⟨β, hβ⟩ := (HomotopyCategory.quotient _ _).map_surjective (iso P).hom
state_before: case intro
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : ProjectiveResolution X
β : (projectiveResolution X).complex ⟶ P.complex
hβ : (HomotopyCategory.quotient C (ComplexShape.down ℕ)).map β = P.iso.hom
⊢ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
        ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app ((projectiveResolutions C).obj X).as) ≫
      (G.mapHomotopyCategory (ComplexShape.down ℕ)).map P.iso.hom =
    (F.mapHomotopyCategory (ComplexShape.down ℕ)).map P.iso.hom ≫
      (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
        ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app P.complex)
state_after: case intro
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : ProjectiveResolution X
β : (projectiveResolution X).complex ⟶ P.complex
hβ : (HomotopyCategory.quotient C (ComplexShape.down ℕ)).map β = P.iso.hom
⊢ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
        ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app ((projectiveResolutions C).obj X).as) ≫
      (G.mapHomotopyCategory (ComplexShape.down ℕ)).map ((HomotopyCategory.quotient C (ComplexShape.down ℕ)).map β) =
    (F.mapHomotopyCategory (ComplexShape.down ℕ)).map ((HomotopyCategory.quotient C (ComplexShape.down ℕ)).map β) ≫
      (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
        ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app P.complex)
tactic: rw [← hβ]
state_before: case intro
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : ProjectiveResolution X
β : (projectiveResolution X).complex ⟶ P.complex
hβ : (HomotopyCategory.quotient C (ComplexShape.down ℕ)).map β = P.iso.hom
⊢ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
        ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app ((projectiveResolutions C).obj X).as) ≫
      (G.mapHomotopyCategory (ComplexShape.down ℕ)).map ((HomotopyCategory.quotient C (ComplexShape.down ℕ)).map β) =
    (F.mapHomotopyCategory (ComplexShape.down ℕ)).map ((HomotopyCategory.quotient C (ComplexShape.down ℕ)).map β) ≫
      (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
        ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app P.complex)
state_after: case intro
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : ProjectiveResolution X
β : (projectiveResolution X).complex ⟶ P.complex
hβ : (HomotopyCategory.quotient C (ComplexShape.down ℕ)).map β = P.iso.hom
⊢ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
        ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app ((projectiveResolutions C).obj X).as) ≫
      (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map ((G.mapHomologicalComplex (ComplexShape.down ℕ)).map β) =
    (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map ((F.mapHomologicalComplex (ComplexShape.down ℕ)).map β) ≫
      (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
        ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app P.complex)
tactic: dsimp
state_before: case intro
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : ProjectiveResolution X
β : (projectiveResolution X).complex ⟶ P.complex
hβ : (HomotopyCategory.quotient C (ComplexShape.down ℕ)).map β = P.iso.hom
⊢ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
        ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app ((projectiveResolutions C).obj X).as) ≫
      (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map ((G.mapHomologicalComplex (ComplexShape.down ℕ)).map β) =
    (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map ((F.mapHomologicalComplex (ComplexShape.down ℕ)).map β) ≫
      (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
        ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app P.complex)
state_after: case intro
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : ProjectiveResolution X
β : (projectiveResolution X).complex ⟶ P.complex
hβ : (HomotopyCategory.quotient C (ComplexShape.down ℕ)).map β = P.iso.hom
⊢ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
      ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app ((projectiveResolutions C).obj X).as ≫
        (G.mapHomologicalComplex (ComplexShape.down ℕ)).map β) =
    (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
      ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app (projectiveResolution X).complex ≫
        (G.mapHomologicalComplex (ComplexShape.down ℕ)).map β)
tactic: simp only [← Functor.map_comp, NatTrans.mapHomologicalComplex_naturality]
state_before: case intro
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : ProjectiveResolution X
β : (projectiveResolution X).complex ⟶ P.complex
hβ : (HomotopyCategory.quotient C (ComplexShape.down ℕ)).map β = P.iso.hom
⊢ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
      ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app ((projectiveResolutions C).obj X).as ≫
        (G.mapHomologicalComplex (ComplexShape.down ℕ)).map β) =
    (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map
      ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app (projectiveResolution X).complex ≫
        (G.mapHomologicalComplex (ComplexShape.down ℕ)).map β)
state_after: no goals
tactic: rfl