Theorem: CategoryTheory.ProjectiveResolution.iso_inv_naturality
File: Mathlib/CategoryTheory/Abelian/ProjectiveResolution.lean

tactic: apply HomotopyCategory.eq_of_homotopy
state_before: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasProjectiveResolutions C
X Y : C
f : X ⟶ Y
P : ProjectiveResolution X
Q : ProjectiveResolution Y
φ : P.complex ⟶ Q.complex
comm : φ.f 0 ≫ Q.π.f 0 = P.π.f 0 ≫ f
⊢ P.iso.inv ≫ (projectiveResolutions C).map f = (HomotopyCategory.quotient C (ComplexShape.down ℕ)).map φ ≫ Q.iso.inv
state_after: case h
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasProjectiveResolutions C
X Y : C
f : X ⟶ Y
P : ProjectiveResolution X
Q : ProjectiveResolution Y
φ : P.complex ⟶ Q.complex
comm : φ.f 0 ≫ Q.π.f 0 = P.π.f 0 ≫ f
⊢ Homotopy (((projectiveResolution X).homotopyEquiv P).inv ≫ lift f (projectiveResolution X) (projectiveResolution Y))
    (φ ≫ ((projectiveResolution Y).homotopyEquiv Q).inv)

tactic: apply liftHomotopy f
state_before: case h
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasProjectiveResolutions C
X Y : C
f : X ⟶ Y
P : ProjectiveResolution X
Q : ProjectiveResolution Y
φ : P.complex ⟶ Q.complex
comm : φ.f 0 ≫ Q.π.f 0 = P.π.f 0 ≫ f
⊢ Homotopy (((projectiveResolution X).homotopyEquiv P).inv ≫ lift f (projectiveResolution X) (projectiveResolution Y))
    (φ ≫ ((projectiveResolution Y).homotopyEquiv Q).inv)
state_after: case h.g_comm
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasProjectiveResolutions C
X Y : C
f : X ⟶ Y
P : ProjectiveResolution X
Q : ProjectiveResolution Y
φ : P.complex ⟶ Q.complex
comm : φ.f 0 ≫ Q.π.f 0 = P.π.f 0 ≫ f
⊢ (((projectiveResolution X).homotopyEquiv P).inv ≫ lift f (projectiveResolution X) (projectiveResolution Y)) ≫
      (projectiveResolution Y).π =
    P.π ≫ (ChainComplex.single₀ C).map f

case h.h_comm
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasProjectiveResolutions C
X Y : C
f : X ⟶ Y
P : ProjectiveResolution X
Q : ProjectiveResolution Y
φ : P.complex ⟶ Q.complex
comm : φ.f 0 ≫ Q.π.f 0 = P.π.f 0 ≫ f
⊢ (φ ≫ ((projectiveResolution Y).homotopyEquiv Q).inv) ≫ (projectiveResolution Y).π =
    P.π ≫ (ChainComplex.single₀ C).map f

tactic: all_goals
  aesop_cat
state_before: case h.g_comm
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasProjectiveResolutions C
X Y : C
f : X ⟶ Y
P : ProjectiveResolution X
Q : ProjectiveResolution Y
φ : P.complex ⟶ Q.complex
comm : φ.f 0 ≫ Q.π.f 0 = P.π.f 0 ≫ f
⊢ (((projectiveResolution X).homotopyEquiv P).inv ≫ lift f (projectiveResolution X) (projectiveResolution Y)) ≫
      (projectiveResolution Y).π =
    P.π ≫ (ChainComplex.single₀ C).map f

case h.h_comm
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasProjectiveResolutions C
X Y : C
f : X ⟶ Y
P : ProjectiveResolution X
Q : ProjectiveResolution Y
φ : P.complex ⟶ Q.complex
comm : φ.f 0 ≫ Q.π.f 0 = P.π.f 0 ≫ f
⊢ (φ ≫ ((projectiveResolution Y).homotopyEquiv Q).inv) ≫ (projectiveResolution Y).π =
    P.π ≫ (ChainComplex.single₀ C).map f
state_after: no goals

tactic: aesop_cat
state_before: case h.h_comm
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasProjectiveResolutions C
X Y : C
f : X ⟶ Y
P : ProjectiveResolution X
Q : ProjectiveResolution Y
φ : P.complex ⟶ Q.complex
comm : φ.f 0 ≫ Q.π.f 0 = P.π.f 0 ≫ f
⊢ (φ ≫ ((projectiveResolution Y).homotopyEquiv Q).inv) ≫ (projectiveResolution Y).π =
    P.π ≫ (ChainComplex.single₀ C).map f
state_after: no goals