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

Proof steps:
state_before: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasProjectiveResolutions C
inst✝ : EnoughProjectives C
X Y : C
f : X ⟶ Y
⊢ d f ≫ f = 0
state_after: no goals
tactic: simp
state_before: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasProjectiveResolutions C
inst✝ : EnoughProjectives C
X Y : C
f : X ⟶ Y
⊢ (ShortComplex.mk (d f) f ⋯).Exact
state_after: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasProjectiveResolutions C
inst✝ : EnoughProjectives C
X Y : C
f : X ⟶ Y
α : ShortComplex.mk (d f) f ⋯ ⟶ ShortComplex.mk (kernel.ι f) f ⋯ :=
  { τ₁ := π (kernel f), τ₂ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₂, τ₃ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₃, comm₁₂ := ⋯,
    comm₂₃ := ⋯ }
⊢ (ShortComplex.mk (d f) f ⋯).Exact
tactic: let α : ShortComplex.mk (d f) f (by simp) ⟶ ShortComplex.mk (kernel.ι f) f (by simp) :=
    { τ₁ := Projective.π _
      τ₂ := 𝟙 _
      τ₃ := 𝟙 _ }
state_before: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasProjectiveResolutions C
inst✝ : EnoughProjectives C
X Y : C
f : X ⟶ Y
α : ShortComplex.mk (d f) f ⋯ ⟶ ShortComplex.mk (kernel.ι f) f ⋯ :=
  { τ₁ := π (kernel f), τ₂ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₂, τ₃ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₃, comm₁₂ := ⋯,
    comm₂₃ := ⋯ }
⊢ (ShortComplex.mk (d f) f ⋯).Exact
state_after: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasProjectiveResolutions C
inst✝ : EnoughProjectives C
X Y : C
f : X ⟶ Y
α : ShortComplex.mk (d f) f ⋯ ⟶ ShortComplex.mk (kernel.ι f) f ⋯ :=
  { τ₁ := π (kernel f), τ₂ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₂, τ₃ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₃, comm₁₂ := ⋯,
    comm₂₃ := ⋯ }
this : Epi α.τ₁
⊢ (ShortComplex.mk (d f) f ⋯).Exact
tactic: have : Epi α.τ₁ := by dsimp; infer_instance
state_before: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasProjectiveResolutions C
inst✝ : EnoughProjectives C
X Y : C
f : X ⟶ Y
α : ShortComplex.mk (d f) f ⋯ ⟶ ShortComplex.mk (kernel.ι f) f ⋯ :=
  { τ₁ := π (kernel f), τ₂ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₂, τ₃ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₃, comm₁₂ := ⋯,
    comm₂₃ := ⋯ }
this : Epi α.τ₁
⊢ (ShortComplex.mk (d f) f ⋯).Exact
state_after: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasProjectiveResolutions C
inst✝ : EnoughProjectives C
X Y : C
f : X ⟶ Y
α : ShortComplex.mk (d f) f ⋯ ⟶ ShortComplex.mk (kernel.ι f) f ⋯ :=
  { τ₁ := π (kernel f), τ₂ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₂, τ₃ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₃, comm₁₂ := ⋯,
    comm₂₃ := ⋯ }
this✝ : Epi α.τ₁
this : IsIso α.τ₂
⊢ (ShortComplex.mk (d f) f ⋯).Exact
tactic: have : IsIso α.τ₂ := by dsimp; infer_instance
state_before: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasProjectiveResolutions C
inst✝ : EnoughProjectives C
X Y : C
f : X ⟶ Y
α : ShortComplex.mk (d f) f ⋯ ⟶ ShortComplex.mk (kernel.ι f) f ⋯ :=
  { τ₁ := π (kernel f), τ₂ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₂, τ₃ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₃, comm₁₂ := ⋯,
    comm₂₃ := ⋯ }
this✝ : Epi α.τ₁
this : IsIso α.τ₂
⊢ (ShortComplex.mk (d f) f ⋯).Exact
state_after: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasProjectiveResolutions C
inst✝ : EnoughProjectives C
X Y : C
f : X ⟶ Y
α : ShortComplex.mk (d f) f ⋯ ⟶ ShortComplex.mk (kernel.ι f) f ⋯ :=
  { τ₁ := π (kernel f), τ₂ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₂, τ₃ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₃, comm₁₂ := ⋯,
    comm₂₃ := ⋯ }
this✝¹ : Epi α.τ₁
this✝ : IsIso α.τ₂
this : Mono α.τ₃
⊢ (ShortComplex.mk (d f) f ⋯).Exact
tactic: have : Mono α.τ₃ := by dsimp; infer_instance
state_before: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasProjectiveResolutions C
inst✝ : EnoughProjectives C
X Y : C
f : X ⟶ Y
α : ShortComplex.mk (d f) f ⋯ ⟶ ShortComplex.mk (kernel.ι f) f ⋯ :=
  { τ₁ := π (kernel f), τ₂ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₂, τ₃ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₃, comm₁₂ := ⋯,
    comm₂₃ := ⋯ }
this✝¹ : Epi α.τ₁
this✝ : IsIso α.τ₂
this : Mono α.τ₃
⊢ (ShortComplex.mk (d f) f ⋯).Exact
state_after: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasProjectiveResolutions C
inst✝ : EnoughProjectives C
X Y : C
f : X ⟶ Y
α : ShortComplex.mk (d f) f ⋯ ⟶ ShortComplex.mk (kernel.ι f) f ⋯ :=
  { τ₁ := π (kernel f), τ₂ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₂, τ₃ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₃, comm₁₂ := ⋯,
    comm₂₃ := ⋯ }
this✝¹ : Epi α.τ₁
this✝ : IsIso α.τ₂
this : Mono α.τ₃
⊢ (ShortComplex.mk (kernel.ι f) f ⋯).Exact
tactic: rw [ShortComplex.exact_iff_of_epi_of_isIso_of_mono α]
state_before: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasProjectiveResolutions C
inst✝ : EnoughProjectives C
X Y : C
f : X ⟶ Y
α : ShortComplex.mk (d f) f ⋯ ⟶ ShortComplex.mk (kernel.ι f) f ⋯ :=
  { τ₁ := π (kernel f), τ₂ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₂, τ₃ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₃, comm₁₂ := ⋯,
    comm₂₃ := ⋯ }
this✝¹ : Epi α.τ₁
this✝ : IsIso α.τ₂
this : Mono α.τ₃
⊢ (ShortComplex.mk (kernel.ι f) f ⋯).Exact
state_after: case hS
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasProjectiveResolutions C
inst✝ : EnoughProjectives C
X Y : C
f : X ⟶ Y
α : ShortComplex.mk (d f) f ⋯ ⟶ ShortComplex.mk (kernel.ι f) f ⋯ :=
  { τ₁ := π (kernel f), τ₂ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₂, τ₃ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₃, comm₁₂ := ⋯,
    comm₂₃ := ⋯ }
this✝¹ : Epi α.τ₁
this✝ : IsIso α.τ₂
this : Mono α.τ₃
⊢ IsLimit (KernelFork.ofι (ShortComplex.mk (kernel.ι f) f ⋯).f ⋯)
tactic: apply ShortComplex.exact_of_f_is_kernel
state_before: case hS
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasProjectiveResolutions C
inst✝ : EnoughProjectives C
X Y : C
f : X ⟶ Y
α : ShortComplex.mk (d f) f ⋯ ⟶ ShortComplex.mk (kernel.ι f) f ⋯ :=
  { τ₁ := π (kernel f), τ₂ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₂, τ₃ := 𝟙 (ShortComplex.mk (d f) f ⋯).X₃, comm₁₂ := ⋯,
    comm₂₃ := ⋯ }
this✝¹ : Epi α.τ₁
this✝ : IsIso α.τ₂
this : Mono α.τ₃
⊢ IsLimit (KernelFork.ofι (ShortComplex.mk (kernel.ι f) f ⋯).f ⋯)
state_after: no goals
tactic: apply kernelIsKernel