Theorem: AlgebraicTopology.DoldKan.N₂Γ₂ToKaroubiIso_hom_app
File: Mathlib/AlgebraicTopology/DoldKan/GammaCompN.lean

Proof steps:
state_before: C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : ChainComplex C ℕ
⊢ (N₂Γ₂ToKaroubiIso.hom.app X).f = PInfty
state_after: case h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : ChainComplex C ℕ
n : ℕ
⊢ (N₂Γ₂ToKaroubiIso.hom.app X).f.f n = PInfty.f n
tactic: ext n
state_before: case h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : ChainComplex C ℕ
n : ℕ
⊢ (N₂Γ₂ToKaroubiIso.hom.app X).f.f n = PInfty.f n
state_after: case h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : ChainComplex C ℕ
n : ℕ
⊢ (((PInfty.f n ≫
            (Γ₀.splitting X).desc (op [n]) fun A =>
              𝟙 (X.X (unop A.fst).len) ≫ ((Γ₀.splitting X).cofan (op [n])).inj A) ≫
          PInfty.f n ≫ 𝟙 (Γ₀.Obj.obj₂ X (op [n]))) ≫
        PInfty.f n ≫ 𝟙 (Γ₀.Obj.obj₂ X (op [n]))) ≫
      PInfty.f n =
    PInfty.f n
tactic: dsimp [N₂Γ₂ToKaroubiIso]
state_before: case h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : ChainComplex C ℕ
n : ℕ
⊢ (((PInfty.f n ≫
            (Γ₀.splitting X).desc (op [n]) fun A =>
              𝟙 (X.X (unop A.fst).len) ≫ ((Γ₀.splitting X).cofan (op [n])).inj A) ≫
          PInfty.f n ≫ 𝟙 (Γ₀.Obj.obj₂ X (op [n]))) ≫
        PInfty.f n ≫ 𝟙 (Γ₀.Obj.obj₂ X (op [n]))) ≫
      PInfty.f n =
    PInfty.f n
state_after: case h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : ChainComplex C ℕ
n : ℕ
⊢ PInfty.f n ≫
      ((Γ₀.splitting X).desc (op [n]) fun A => 𝟙 (X.X (unop A.fst).len) ≫ ((Γ₀.splitting X).cofan (op [n])).inj A) ≫
        PInfty.f n =
    PInfty.f n
tactic: simp only [comp_id, assoc, PInfty_f_idem]
state_before: case h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : ChainComplex C ℕ
n : ℕ
⊢ PInfty.f n ≫
      ((Γ₀.splitting X).desc (op [n]) fun A => 𝟙 (X.X (unop A.fst).len) ≫ ((Γ₀.splitting X).cofan (op [n])).inj A) ≫
        PInfty.f n =
    PInfty.f n
state_after: case h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : ChainComplex C ℕ
n : ℕ
⊢ PInfty.f n ≫
      ((Γ₀.splitting X).desc (op [n]) fun A => 𝟙 (X.X (unop A.fst).len) ≫ ((Γ₀.splitting X).cofan (op [n])).inj A) ≫
        PInfty.f n =
    PInfty.f n ≫ PInfty.f n
tactic: conv_rhs =>
    rw [← PInfty_f_idem]
state_before: case h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : ChainComplex C ℕ
n : ℕ
⊢ PInfty.f n ≫
      ((Γ₀.splitting X).desc (op [n]) fun A => 𝟙 (X.X (unop A.fst).len) ≫ ((Γ₀.splitting X).cofan (op [n])).inj A) ≫
        PInfty.f n =
    PInfty.f n ≫ PInfty.f n
state_after: case h.e_a
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : ChainComplex C ℕ
n : ℕ
⊢ ((Γ₀.splitting X).desc (op [n]) fun A => 𝟙 (X.X (unop A.fst).len) ≫ ((Γ₀.splitting X).cofan (op [n])).inj A) ≫
      PInfty.f n =
    PInfty.f n
tactic: congr 1
state_before: case h.e_a
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : ChainComplex C ℕ
n : ℕ
⊢ ((Γ₀.splitting X).desc (op [n]) fun A => 𝟙 (X.X (unop A.fst).len) ≫ ((Γ₀.splitting X).cofan (op [n])).inj A) ≫
      PInfty.f n =
    PInfty.f n
state_after: case h.e_a.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : ChainComplex C ℕ
n : ℕ
⊢ ∀ (A : Splitting.IndexSet (op [n])),
    ((Γ₀.splitting X).cofan (op [n])).inj A ≫
        ((Γ₀.splitting X).desc (op [n]) fun A => 𝟙 (X.X (unop A.fst).len) ≫ ((Γ₀.splitting X).cofan (op [n])).inj A) ≫
          PInfty.f n =
      ((Γ₀.splitting X).cofan (op [n])).inj A ≫ PInfty.f n
tactic: apply (Γ₀.splitting X).hom_ext'
state_before: case h.e_a.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : ChainComplex C ℕ
n : ℕ
⊢ ∀ (A : Splitting.IndexSet (op [n])),
    ((Γ₀.splitting X).cofan (op [n])).inj A ≫
        ((Γ₀.splitting X).desc (op [n]) fun A => 𝟙 (X.X (unop A.fst).len) ≫ ((Γ₀.splitting X).cofan (op [n])).inj A) ≫
          PInfty.f n =
      ((Γ₀.splitting X).cofan (op [n])).inj A ≫ PInfty.f n
state_after: case h.e_a.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : ChainComplex C ℕ
n : ℕ
A : Splitting.IndexSet (op [n])
⊢ ((Γ₀.splitting X).cofan (op [n])).inj A ≫
      ((Γ₀.splitting X).desc (op [n]) fun A => 𝟙 (X.X (unop A.fst).len) ≫ ((Γ₀.splitting X).cofan (op [n])).inj A) ≫
        PInfty.f n =
    ((Γ₀.splitting X).cofan (op [n])).inj A ≫ PInfty.f n
tactic: intro A
state_before: case h.e_a.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : ChainComplex C ℕ
n : ℕ
A : Splitting.IndexSet (op [n])
⊢ ((Γ₀.splitting X).cofan (op [n])).inj A ≫
      ((Γ₀.splitting X).desc (op [n]) fun A => 𝟙 (X.X (unop A.fst).len) ≫ ((Γ₀.splitting X).cofan (op [n])).inj A) ≫
        PInfty.f n =
    ((Γ₀.splitting X).cofan (op [n])).inj A ≫ PInfty.f n
state_after: case h.e_a.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : ChainComplex C ℕ
n : ℕ
A : Splitting.IndexSet (op [n])
⊢ 𝟙 (X.X (unop A.fst).len) ≫ ((Γ₀.splitting X).cofan (op [n])).inj A ≫ PInfty.f n =
    ((Γ₀.splitting X).cofan (op [n])).inj A ≫ PInfty.f n
tactic: rw [Splitting.ι_desc_assoc, assoc]
state_before: case h.e_a.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : ChainComplex C ℕ
n : ℕ
A : Splitting.IndexSet (op [n])
⊢ 𝟙 (X.X (unop A.fst).len) ≫ ((Γ₀.splitting X).cofan (op [n])).inj A ≫ PInfty.f n =
    ((Γ₀.splitting X).cofan (op [n])).inj A ≫ PInfty.f n
state_after: no goals
tactic: apply id_comp