Theorem: AlgebraicTopology.DoldKan.identity_N₂_objectwise
File: Mathlib/AlgebraicTopology/DoldKan/NCompGamma.lean

Proof steps:
state_before: C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
P : Karoubi (SimplicialObject C)
⊢ N₂Γ₂.inv.app (N₂.obj P) ≫ N₂.map (Γ₂N₂.natTrans.app P) = 𝟙 (N₂.obj P)
state_after: case h.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
P : Karoubi (SimplicialObject C)
n : ℕ
⊢ (N₂Γ₂.inv.app (N₂.obj P) ≫ N₂.map (Γ₂N₂.natTrans.app P)).f.f n = (𝟙 (N₂.obj P)).f.f n
tactic: ext n
state_before: case h.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
P : Karoubi (SimplicialObject C)
n : ℕ
⊢ (N₂Γ₂.inv.app (N₂.obj P) ≫ N₂.map (Γ₂N₂.natTrans.app P)).f.f n = (𝟙 (N₂.obj P)).f.f n
state_after: case h.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
P : Karoubi (SimplicialObject C)
n : ℕ
eq₁ :
  (N₂Γ₂.inv.app (N₂.obj P)).f.f n =
    PInfty.f n ≫ P.p.app (op [n]) ≫ ((Γ₀.splitting (N₂.obj P).X).cofan (op [n])).inj (Splitting.IndexSet.id (op [n]))
⊢ (N₂Γ₂.inv.app (N₂.obj P) ≫ N₂.map (Γ₂N₂.natTrans.app P)).f.f n = (𝟙 (N₂.obj P)).f.f n
tactic: have eq₁ : (N₂Γ₂.inv.app (N₂.obj P)).f.f n = PInfty.f n ≫ P.p.app (op [n]) ≫
      ((Γ₀.splitting (N₂.obj P).X).cofan _).inj (Splitting.IndexSet.id (op [n])) := by
    simp only [N₂Γ₂_inv_app_f_f, N₂_obj_p_f, assoc]
state_before: case h.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
P : Karoubi (SimplicialObject C)
n : ℕ
eq₁ :
  (N₂Γ₂.inv.app (N₂.obj P)).f.f n =
    PInfty.f n ≫ P.p.app (op [n]) ≫ ((Γ₀.splitting (N₂.obj P).X).cofan (op [n])).inj (Splitting.IndexSet.id (op [n]))
⊢ (N₂Γ₂.inv.app (N₂.obj P) ≫ N₂.map (Γ₂N₂.natTrans.app P)).f.f n = (𝟙 (N₂.obj P)).f.f n
state_after: case h.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
P : Karoubi (SimplicialObject C)
n : ℕ
eq₁ :
  (N₂Γ₂.inv.app (N₂.obj P)).f.f n =
    PInfty.f n ≫ P.p.app (op [n]) ≫ ((Γ₀.splitting (N₂.obj P).X).cofan (op [n])).inj (Splitting.IndexSet.id (op [n]))
eq₂ :
  ((Γ₀.splitting (N₂.obj P).X).cofan (op [n])).inj (Splitting.IndexSet.id (op [n])) ≫
      (N₂.map (Γ₂N₂.natTrans.app P)).f.f n =
    PInfty.f n ≫ P.p.app (op [n])
⊢ (N₂Γ₂.inv.app (N₂.obj P) ≫ N₂.map (Γ₂N₂.natTrans.app P)).f.f n = (𝟙 (N₂.obj P)).f.f n
tactic: have eq₂ : ((Γ₀.splitting (N₂.obj P).X).cofan _).inj (Splitting.IndexSet.id (op [n])) ≫
      (N₂.map (Γ₂N₂.natTrans.app P)).f.f n = PInfty.f n ≫ P.p.app (op [n]) := by
    dsimp
    rw [PInfty_on_Γ₀_splitting_summand_eq_self_assoc, Γ₂N₂.natTrans_app_f_app]
    dsimp
    rw [Γ₂N₂ToKaroubiIso_hom_app, assoc, Splitting.ι_desc_assoc, assoc, assoc]
    dsimp [toKaroubi]
    rw [Splitting.ι_desc_assoc]
    dsimp
    simp only [assoc, Splitting.ι_desc_assoc, unop_op, Splitting.IndexSet.id_fst,
      len_mk, NatTrans.naturality, PInfty_f_idem_assoc,
      PInfty_f_naturality_assoc, app_idem_assoc]
    erw [P.X.map_id, comp_id]
state_before: case h.h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
P : Karoubi (SimplicialObject C)
n : ℕ
eq₁ :
  (N₂Γ₂.inv.app (N₂.obj P)).f.f n =
    PInfty.f n ≫ P.p.app (op [n]) ≫ ((Γ₀.splitting (N₂.obj P).X).cofan (op [n])).inj (Splitting.IndexSet.id (op [n]))
eq₂ :
  ((Γ₀.splitting (N₂.obj P).X).cofan (op [n])).inj (Splitting.IndexSet.id (op [n])) ≫
      (N₂.map (Γ₂N₂.natTrans.app P)).f.f n =
    PInfty.f n ≫ P.p.app (op [n])
⊢ (N₂Γ₂.inv.app (N₂.obj P) ≫ N₂.map (Γ₂N₂.natTrans.app P)).f.f n = (𝟙 (N₂.obj P)).f.f n
state_after: no goals
tactic: simp only [Karoubi.comp_f, HomologicalComplex.comp_f, Karoubi.id_f, N₂_obj_p_f, assoc,
    eq₁, eq₂, PInfty_f_naturality_assoc, app_idem, PInfty_f_idem_assoc]