Theorem: AlgebraicTopology.DoldKan.karoubi_PInfty_f
File: Mathlib/AlgebraicTopology/DoldKan/PInfty.lean

tactic: let Y₁ := (karoubiFunctorCategoryEmbedding _ _).obj Y
state_before: C : Type u_1
inst✝¹ : Category C
inst✝ : Preadditive C
X : SimplicialObject C
Y : Karoubi (SimplicialObject C)
n : ℕ
⊢ (HomologicalComplex.Hom.f PInfty n).f = Y.p.app [n].op ≫ HomologicalComplex.Hom.f PInfty n

tactic: let Y₂ := Y.X
state_before: C : Type u_1
inst✝¹ : Category C
inst✝ : Preadditive C
X : SimplicialObject C
Y : Karoubi (SimplicialObject C)
n : ℕ
Y₁ : SimplexCategoryᵒᵖ ⥤ Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj Y
⊢ (HomologicalComplex.Hom.f PInfty n).f = Y.p.app [n].op ≫ HomologicalComplex.Hom.f PInfty n

tactic: let Y₃ := ((whiskering _ _).obj (toKaroubi C)).obj Y.X
state_before: C : Type u_1
inst✝¹ : Category C
inst✝ : Preadditive C
X : SimplicialObject C
Y : Karoubi (SimplicialObject C)
n : ℕ
Y₁ : SimplexCategoryᵒᵖ ⥤ Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj Y
Y₂ : SimplicialObject C := Y.X
⊢ (HomologicalComplex.Hom.f PInfty n).f = Y.p.app [n].op ≫ HomologicalComplex.Hom.f PInfty n

tactic: let Y₄ := (karoubiFunctorCategoryEmbedding _ _).obj ((toKaroubi _).obj Y.X)
state_before: C : Type u_1
inst✝¹ : Category C
inst✝ : Preadditive C
X : SimplicialObject C
Y : Karoubi (SimplicialObject C)
n : ℕ
Y₁ : SimplexCategoryᵒᵖ ⥤ Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj Y
Y₂ : SimplicialObject C := Y.X
Y₃ : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X
⊢ (HomologicalComplex.Hom.f PInfty n).f = Y.p.app [n].op ≫ HomologicalComplex.Hom.f PInfty n

tactic: let P₁ : K[Y₁] ⟶ _ := PInfty
state_before: C : Type u_1
inst✝¹ : Category C
inst✝ : Preadditive C
X : SimplicialObject C
Y : Karoubi (SimplicialObject C)
n : ℕ
Y₁ : SimplexCategoryᵒᵖ ⥤ Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj Y
Y₂ : SimplicialObject C := Y.X
Y₃ : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X
Y₄ : SimplexCategoryᵒᵖ ⥤ Karoubi C :=
  (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj ((toKaroubi (SimplexCategoryᵒᵖ ⥤ C)).obj Y.X)
⊢ (HomologicalComplex.Hom.f PInfty n).f = Y.p.app [n].op ≫ HomologicalComplex.Hom.f PInfty n

tactic: let P₂ : K[Y₂] ⟶ _ := PInfty
state_before: C : Type u_1
inst✝¹ : Category C
inst✝ : Preadditive C
X : SimplicialObject C
Y : Karoubi (SimplicialObject C)
n : ℕ
Y₁ : SimplexCategoryᵒᵖ ⥤ Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj Y
Y₂ : SimplicialObject C := Y.X
Y₃ : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X
Y₄ : SimplexCategoryᵒᵖ ⥤ Karoubi C :=
  (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj ((toKaroubi (SimplexCategoryᵒᵖ ⥤ C)).obj Y.X)
P₁ : K[Y₁] ⟶ K[Y₁] := PInfty
⊢ (HomologicalComplex.Hom.f PInfty n).f = Y.p.app [n].op ≫ HomologicalComplex.Hom.f PInfty n

tactic: let P₃ : K[Y₃] ⟶ _ := PInfty
state_before: C : Type u_1
inst✝¹ : Category C
inst✝ : Preadditive C
X : SimplicialObject C
Y : Karoubi (SimplicialObject C)
n : ℕ
Y₁ : SimplexCategoryᵒᵖ ⥤ Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj Y
Y₂ : SimplicialObject C := Y.X
Y₃ : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X
Y₄ : SimplexCategoryᵒᵖ ⥤ Karoubi C :=
  (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj ((toKaroubi (SimplexCategoryᵒᵖ ⥤ C)).obj Y.X)
P₁ : K[Y₁] ⟶ K[Y₁] := PInfty
P₂ : K[Y₂] ⟶ K[Y₂] := PInfty
⊢ (HomologicalComplex.Hom.f PInfty n).f = Y.p.app [n].op ≫ HomologicalComplex.Hom.f PInfty n

tactic: let P₄ : K[Y₄] ⟶ _ := PInfty
state_before: C : Type u_1
inst✝¹ : Category C
inst✝ : Preadditive C
X : SimplicialObject C
Y : Karoubi (SimplicialObject C)
n : ℕ
Y₁ : SimplexCategoryᵒᵖ ⥤ Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj Y
Y₂ : SimplicialObject C := Y.X
Y₃ : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X
Y₄ : SimplexCategoryᵒᵖ ⥤ Karoubi C :=
  (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj ((toKaroubi (SimplexCategoryᵒᵖ ⥤ C)).obj Y.X)
P₁ : K[Y₁] ⟶ K[Y₁] := PInfty
P₂ : K[Y₂] ⟶ K[Y₂] := PInfty
P₃ : K[Y₃] ⟶ K[Y₃] := PInfty
⊢ (HomologicalComplex.Hom.f PInfty n).f = Y.p.app [n].op ≫ HomologicalComplex.Hom.f PInfty n

tactic: change (P₁.f n).f = Y.p.app (op [n]) ≫ P₂.f n
state_before: C : Type u_1
inst✝¹ : Category C
inst✝ : Preadditive C
X : SimplicialObject C
Y : Karoubi (SimplicialObject C)
n : ℕ
Y₁ : SimplexCategoryᵒᵖ ⥤ Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj Y
Y₂ : SimplicialObject C := Y.X
Y₃ : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X
Y₄ : SimplexCategoryᵒᵖ ⥤ Karoubi C :=
  (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj ((toKaroubi (SimplexCategoryᵒᵖ ⥤ C)).obj Y.X)
P₁ : K[Y₁] ⟶ K[Y₁] := PInfty
P₂ : K[Y₂] ⟶ K[Y₂] := PInfty
P₃ : K[Y₃] ⟶ K[Y₃] := PInfty
P₄ : K[Y₄] ⟶ K[Y₄] := PInfty
⊢ (HomologicalComplex.Hom.f PInfty n).f = Y.p.app [n].op ≫ HomologicalComplex.Hom.f PInfty n

tactic: have h₃₂ : (P₃.f n).f = P₂.f n := Karoubi.hom_ext_iff.mp (map_PInfty_f (toKaroubi C) Y₂ n)
state_before: C : Type u_1
inst✝¹ : Category C
inst✝ : Preadditive C
X : SimplicialObject C
Y : Karoubi (SimplicialObject C)
n : ℕ
Y₁ : SimplexCategoryᵒᵖ ⥤ Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj Y
Y₂ : SimplicialObject C := Y.X
Y₃ : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X
Y₄ : SimplexCategoryᵒᵖ ⥤ Karoubi C :=
  (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj ((toKaroubi (SimplexCategoryᵒᵖ ⥤ C)).obj Y.X)
P₁ : K[Y₁] ⟶ K[Y₁] := PInfty
P₂ : K[Y₂] ⟶ K[Y₂] := PInfty
P₃ : K[Y₃] ⟶ K[Y₃] := PInfty
P₄ : K[Y₄] ⟶ K[Y₄] := PInfty
⊢ (HomologicalComplex.Hom.f P₁ n).f = Y.p.app [n].op ≫ HomologicalComplex.Hom.f P₂ n

tactic: have h₄₃ : P₄.f n = P₃.f n := by
  have h := Functor.congr_obj (toKaroubi_comp_karoubiFunctorCategoryEmbedding _ _) Y₂
  simp only [← natTransPInfty_f_app]
  congr 1
state_before: C : Type u_1
inst✝¹ : Category C
inst✝ : Preadditive C
X : SimplicialObject C
Y : Karoubi (SimplicialObject C)
n : ℕ
Y₁ : SimplexCategoryᵒᵖ ⥤ Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj Y
Y₂ : SimplicialObject C := Y.X
Y₃ : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X
Y₄ : SimplexCategoryᵒᵖ ⥤ Karoubi C :=
  (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj ((toKaroubi (SimplexCategoryᵒᵖ ⥤ C)).obj Y.X)
P₁ : K[Y₁] ⟶ K[Y₁] := PInfty
P₂ : K[Y₂] ⟶ K[Y₂] := PInfty
P₃ : K[Y₃] ⟶ K[Y₃] := PInfty
P₄ : K[Y₄] ⟶ K[Y₄] := PInfty
h₃₂ : (HomologicalComplex.Hom.f P₃ n).f = HomologicalComplex.Hom.f P₂ n
⊢ (HomologicalComplex.Hom.f P₁ n).f = Y.p.app [n].op ≫ HomologicalComplex.Hom.f P₂ n

tactic: have h₁₄ := Idempotents.natTrans_eq
  ((𝟙 (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C)) ◫
    (natTransPInfty_f (Karoubi C) n)) Y
state_before: C : Type u_1
inst✝¹ : Category C
inst✝ : Preadditive C
X : SimplicialObject C
Y : Karoubi (SimplicialObject C)
n : ℕ
Y₁ : SimplexCategoryᵒᵖ ⥤ Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj Y
Y₂ : SimplicialObject C := Y.X
Y₃ : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X
Y₄ : SimplexCategoryᵒᵖ ⥤ Karoubi C :=
  (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj ((toKaroubi (SimplexCategoryᵒᵖ ⥤ C)).obj Y.X)
P₁ : K[Y₁] ⟶ K[Y₁] := PInfty
P₂ : K[Y₂] ⟶ K[Y₂] := PInfty
P₃ : K[Y₃] ⟶ K[Y₃] := PInfty
P₄ : K[Y₄] ⟶ K[Y₄] := PInfty
h₃₂ : (HomologicalComplex.Hom.f P₃ n).f = HomologicalComplex.Hom.f P₂ n
h₄₃ : HomologicalComplex.Hom.f P₄ n = HomologicalComplex.Hom.f P₃ n
⊢ (HomologicalComplex.Hom.f P₁ n).f = Y.p.app [n].op ≫ HomologicalComplex.Hom.f P₂ n

tactic: dsimp [natTransPInfty_f] at h₁₄
state_before: C : Type u_1
inst✝¹ : Category C
inst✝ : Preadditive C
X : SimplicialObject C
Y : Karoubi (SimplicialObject C)
n : ℕ
Y₁ : SimplexCategoryᵒᵖ ⥤ Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj Y
Y₂ : SimplicialObject C := Y.X
Y₃ : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X
Y₄ : SimplexCategoryᵒᵖ ⥤ Karoubi C :=
  (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj ((toKaroubi (SimplexCategoryᵒᵖ ⥤ C)).obj Y.X)
P₁ : K[Y₁] ⟶ K[Y₁] := PInfty
P₂ : K[Y₂] ⟶ K[Y₂] := PInfty
P₃ : K[Y₃] ⟶ K[Y₃] := PInfty
P₄ : K[Y₄] ⟶ K[Y₄] := PInfty
h₃₂ : (HomologicalComplex.Hom.f P₃ n).f = HomologicalComplex.Hom.f P₂ n
h₄₃ : HomologicalComplex.Hom.f P₄ n = HomologicalComplex.Hom.f P₃ n
h₁₄ :
  (𝟙 (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C) ◫ natTransPInfty_f (Karoubi C) n).app Y =
    (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C ⋙
            alternatingFaceMapComplex (Karoubi C) ⋙ HomologicalComplex.eval (Karoubi C) (ComplexShape.down ℕ) n).map
        (Karoubi.decompId_i Y) ≫
      (𝟙 (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C) ◫ natTransPInfty_f (Karoubi C) n).app
          (Karoubi.mk Y.X (𝟙 Y.X)) ≫
        (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C ⋙
              alternatingFaceMapComplex (Karoubi C) ⋙ HomologicalComplex.eval (Karoubi C) (ComplexShape.down ℕ) n).map
          (Karoubi.decompId_p Y)
⊢ (HomologicalComplex.Hom.f P₁ n).f = Y.p.app [n].op ≫ HomologicalComplex.Hom.f P₂ n

tactic: erw [id_comp, id_comp, comp_id, comp_id] at h₁₄
state_before: C : Type u_1
inst✝¹ : Category C
inst✝ : Preadditive C
X : SimplicialObject C
Y : Karoubi (SimplicialObject C)
n : ℕ
Y₁ : SimplexCategoryᵒᵖ ⥤ Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj Y
Y₂ : SimplicialObject C := Y.X
Y₃ : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X
Y₄ : SimplexCategoryᵒᵖ ⥤ Karoubi C :=
  (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj ((toKaroubi (SimplexCategoryᵒᵖ ⥤ C)).obj Y.X)
P₁ : K[Y₁] ⟶ K[Y₁] := PInfty
P₂ : K[Y₂] ⟶ K[Y₂] := PInfty
P₃ : K[Y₃] ⟶ K[Y₃] := PInfty
P₄ : K[Y₄] ⟶ K[Y₄] := PInfty
h₃₂ : (HomologicalComplex.Hom.f P₃ n).f = HomologicalComplex.Hom.f P₂ n
h₄₃ : HomologicalComplex.Hom.f P₄ n = HomologicalComplex.Hom.f P₃ n
h₁₄ :
  (Karoubi.Hom.mk (Y.p.app [n].op) ≫ HomologicalComplex.Hom.f PInfty n) ≫ Karoubi.Hom.mk (Y.p.app [n].op) =
    (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_i Y)).app [n].op ≫
      ((Karoubi.Hom.mk (𝟙 (Y.X.obj [n].op)) ≫ HomologicalComplex.Hom.f PInfty n) ≫
          Karoubi.Hom.mk (𝟙 (Y.X.obj [n].op))) ≫
        (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_p Y)).app [n].op
⊢ (HomologicalComplex.Hom.f P₁ n).f = Y.p.app [n].op ≫ HomologicalComplex.Hom.f P₂ n

tactic: rw [← h₃₂, ← h₄₃, h₁₄]
state_before: C : Type u_1
inst✝¹ : Category C
inst✝ : Preadditive C
X : SimplicialObject C
Y : Karoubi (SimplicialObject C)
n : ℕ
Y₁ : SimplexCategoryᵒᵖ ⥤ Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj Y
Y₂ : SimplicialObject C := Y.X
Y₃ : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X
Y₄ : SimplexCategoryᵒᵖ ⥤ Karoubi C :=
  (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj ((toKaroubi (SimplexCategoryᵒᵖ ⥤ C)).obj Y.X)
P₁ : K[Y₁] ⟶ K[Y₁] := PInfty
P₂ : K[Y₂] ⟶ K[Y₂] := PInfty
P₃ : K[Y₃] ⟶ K[Y₃] := PInfty
P₄ : K[Y₄] ⟶ K[Y₄] := PInfty
h₃₂ : (HomologicalComplex.Hom.f P₃ n).f = HomologicalComplex.Hom.f P₂ n
h₄₃ : HomologicalComplex.Hom.f P₄ n = HomologicalComplex.Hom.f P₃ n
h₁₄ :
  HomologicalComplex.Hom.f PInfty n =
    (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_i Y)).app [n].op ≫
      HomologicalComplex.Hom.f PInfty n ≫ (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_p Y)).app [n].op
⊢ (HomologicalComplex.Hom.f P₁ n).f = Y.p.app [n].op ≫ HomologicalComplex.Hom.f P₂ n

tactic: simp only [KaroubiFunctorCategoryEmbedding.map_app_f, Karoubi.decompId_p_f,
  Karoubi.decompId_i_f, Karoubi.comp_f]
state_before: C : Type u_1
inst✝¹ : Category C
inst✝ : Preadditive C
X : SimplicialObject C
Y : Karoubi (SimplicialObject C)
n : ℕ
Y₁ : SimplexCategoryᵒᵖ ⥤ Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj Y
Y₂ : SimplicialObject C := Y.X
Y₃ : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X
Y₄ : SimplexCategoryᵒᵖ ⥤ Karoubi C :=
  (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj ((toKaroubi (SimplexCategoryᵒᵖ ⥤ C)).obj Y.X)
P₁ : K[Y₁] ⟶ K[Y₁] := PInfty
P₂ : K[Y₂] ⟶ K[Y₂] := PInfty
P₃ : K[Y₃] ⟶ K[Y₃] := PInfty
P₄ : K[Y₄] ⟶ K[Y₄] := PInfty
h₃₂ : (HomologicalComplex.Hom.f P₃ n).f = HomologicalComplex.Hom.f P₂ n
h₄₃ : HomologicalComplex.Hom.f P₄ n = HomologicalComplex.Hom.f P₃ n
h₁₄ :
  HomologicalComplex.Hom.f PInfty n =
    (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_i Y)).app [n].op ≫
      HomologicalComplex.Hom.f PInfty n ≫ (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_p Y)).app [n].op
⊢ ((KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_i Y)).app [n].op ≫
        HomologicalComplex.Hom.f PInfty n ≫ (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_p Y)).app [n].op).f =
    Y.p.app [n].op ≫ (HomologicalComplex.Hom.f P₄ n).f

tactic: let π : Y₄ ⟶ Y₄ := (toKaroubi _ ⋙ karoubiFunctorCategoryEmbedding _ _).map Y.p
state_before: C : Type u_1
inst✝¹ : Category C
inst✝ : Preadditive C
X : SimplicialObject C
Y : Karoubi (SimplicialObject C)
n : ℕ
Y₁ : SimplexCategoryᵒᵖ ⥤ Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj Y
Y₂ : SimplicialObject C := Y.X
Y₃ : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X
Y₄ : SimplexCategoryᵒᵖ ⥤ Karoubi C :=
  (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj ((toKaroubi (SimplexCategoryᵒᵖ ⥤ C)).obj Y.X)
P₁ : K[Y₁] ⟶ K[Y₁] := PInfty
P₂ : K[Y₂] ⟶ K[Y₂] := PInfty
P₃ : K[Y₃] ⟶ K[Y₃] := PInfty
P₄ : K[Y₄] ⟶ K[Y₄] := PInfty
h₃₂ : (HomologicalComplex.Hom.f P₃ n).f = HomologicalComplex.Hom.f P₂ n
h₄₃ : HomologicalComplex.Hom.f P₄ n = HomologicalComplex.Hom.f P₃ n
h₁₄ :
  HomologicalComplex.Hom.f PInfty n =
    (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_i Y)).app [n].op ≫
      HomologicalComplex.Hom.f PInfty n ≫ (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_p Y)).app [n].op
⊢ Y.p.app [n].op ≫ (HomologicalComplex.Hom.f PInfty n).f ≫ Y.p.app [n].op =
    Y.p.app [n].op ≫ (HomologicalComplex.Hom.f PInfty n).f

tactic: have eq := Karoubi.hom_ext_iff.mp (PInfty_f_naturality n π)
state_before: C : Type u_1
inst✝¹ : Category C
inst✝ : Preadditive C
X : SimplicialObject C
Y : Karoubi (SimplicialObject C)
n : ℕ
Y₁ : SimplexCategoryᵒᵖ ⥤ Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj Y
Y₂ : SimplicialObject C := Y.X
Y₃ : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X
Y₄ : SimplexCategoryᵒᵖ ⥤ Karoubi C :=
  (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj ((toKaroubi (SimplexCategoryᵒᵖ ⥤ C)).obj Y.X)
P₁ : K[Y₁] ⟶ K[Y₁] := PInfty
P₂ : K[Y₂] ⟶ K[Y₂] := PInfty
P₃ : K[Y₃] ⟶ K[Y₃] := PInfty
P₄ : K[Y₄] ⟶ K[Y₄] := PInfty
h₃₂ : (HomologicalComplex.Hom.f P₃ n).f = HomologicalComplex.Hom.f P₂ n
h₄₃ : HomologicalComplex.Hom.f P₄ n = HomologicalComplex.Hom.f P₃ n
h₁₄ :
  HomologicalComplex.Hom.f PInfty n =
    (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_i Y)).app [n].op ≫
      HomologicalComplex.Hom.f PInfty n ≫ (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_p Y)).app [n].op
π : Y₄ ⟶ Y₄ := (toKaroubi (SimplexCategoryᵒᵖ ⥤ C) ⋙ karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).map Y.p
⊢ Y.p.app [n].op ≫ (HomologicalComplex.Hom.f PInfty n).f ≫ Y.p.app [n].op =
    Y.p.app [n].op ≫ (HomologicalComplex.Hom.f PInfty n).f

tactic: simp only [Karoubi.comp_f] at eq
state_before: C : Type u_1
inst✝¹ : Category C
inst✝ : Preadditive C
X : SimplicialObject C
Y : Karoubi (SimplicialObject C)
n : ℕ
Y₁ : SimplexCategoryᵒᵖ ⥤ Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj Y
Y₂ : SimplicialObject C := Y.X
Y₃ : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X
Y₄ : SimplexCategoryᵒᵖ ⥤ Karoubi C :=
  (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj ((toKaroubi (SimplexCategoryᵒᵖ ⥤ C)).obj Y.X)
P₁ : K[Y₁] ⟶ K[Y₁] := PInfty
P₂ : K[Y₂] ⟶ K[Y₂] := PInfty
P₃ : K[Y₃] ⟶ K[Y₃] := PInfty
P₄ : K[Y₄] ⟶ K[Y₄] := PInfty
h₃₂ : (HomologicalComplex.Hom.f P₃ n).f = HomologicalComplex.Hom.f P₂ n
h₄₃ : HomologicalComplex.Hom.f P₄ n = HomologicalComplex.Hom.f P₃ n
h₁₄ :
  HomologicalComplex.Hom.f PInfty n =
    (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_i Y)).app [n].op ≫
      HomologicalComplex.Hom.f PInfty n ≫ (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_p Y)).app [n].op
π : Y₄ ⟶ Y₄ := (toKaroubi (SimplexCategoryᵒᵖ ⥤ C) ⋙ karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).map Y.p
eq : (π.app [n].op ≫ HomologicalComplex.Hom.f PInfty n).f = (HomologicalComplex.Hom.f PInfty n ≫ π.app [n].op).f
⊢ Y.p.app [n].op ≫ (HomologicalComplex.Hom.f PInfty n).f ≫ Y.p.app [n].op =
    Y.p.app [n].op ≫ (HomologicalComplex.Hom.f PInfty n).f

tactic: dsimp at eq
state_before: C : Type u_1
inst✝¹ : Category C
inst✝ : Preadditive C
X : SimplicialObject C
Y : Karoubi (SimplicialObject C)
n : ℕ
Y₁ : SimplexCategoryᵒᵖ ⥤ Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj Y
Y₂ : SimplicialObject C := Y.X
Y₃ : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X
Y₄ : SimplexCategoryᵒᵖ ⥤ Karoubi C :=
  (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj ((toKaroubi (SimplexCategoryᵒᵖ ⥤ C)).obj Y.X)
P₁ : K[Y₁] ⟶ K[Y₁] := PInfty
P₂ : K[Y₂] ⟶ K[Y₂] := PInfty
P₃ : K[Y₃] ⟶ K[Y₃] := PInfty
P₄ : K[Y₄] ⟶ K[Y₄] := PInfty
h₃₂ : (HomologicalComplex.Hom.f P₃ n).f = HomologicalComplex.Hom.f P₂ n
h₄₃ : HomologicalComplex.Hom.f P₄ n = HomologicalComplex.Hom.f P₃ n
h₁₄ :
  HomologicalComplex.Hom.f PInfty n =
    (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_i Y)).app [n].op ≫
      HomologicalComplex.Hom.f PInfty n ≫ (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_p Y)).app [n].op
π : Y₄ ⟶ Y₄ := (toKaroubi (SimplexCategoryᵒᵖ ⥤ C) ⋙ karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).map Y.p
eq :
  (((toKaroubi (SimplexCategoryᵒᵖ ⥤ C) ⋙ karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).map Y.p).app [n].op).f ≫
      (HomologicalComplex.Hom.f PInfty n).f =
    (HomologicalComplex.Hom.f PInfty n).f ≫
      (((toKaroubi (SimplexCategoryᵒᵖ ⥤ C) ⋙ karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).map Y.p).app [n].op).f
⊢ Y.p.app [n].op ≫ (HomologicalComplex.Hom.f PInfty n).f ≫ Y.p.app [n].op =
    Y.p.app [n].op ≫ (HomologicalComplex.Hom.f PInfty n).f

tactic: rw [← eq, app_idem_assoc Y (op [n])]
state_before: C : Type u_1
inst✝¹ : Category C
inst✝ : Preadditive C
X : SimplicialObject C
Y : Karoubi (SimplicialObject C)
n : ℕ
Y₁ : SimplexCategoryᵒᵖ ⥤ Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj Y
Y₂ : SimplicialObject C := Y.X
Y₃ : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X
Y₄ : SimplexCategoryᵒᵖ ⥤ Karoubi C :=
  (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj ((toKaroubi (SimplexCategoryᵒᵖ ⥤ C)).obj Y.X)
P₁ : K[Y₁] ⟶ K[Y₁] := PInfty
P₂ : K[Y₂] ⟶ K[Y₂] := PInfty
P₃ : K[Y₃] ⟶ K[Y₃] := PInfty
P₄ : K[Y₄] ⟶ K[Y₄] := PInfty
h₃₂ : (HomologicalComplex.Hom.f P₃ n).f = HomologicalComplex.Hom.f P₂ n
h₄₃ : HomologicalComplex.Hom.f P₄ n = HomologicalComplex.Hom.f P₃ n
h₁₄ :
  HomologicalComplex.Hom.f PInfty n =
    (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_i Y)).app [n].op ≫
      HomologicalComplex.Hom.f PInfty n ≫ (KaroubiFunctorCategoryEmbedding.map (Karoubi.decompId_p Y)).app [n].op
π : Y₄ ⟶ Y₄ := (toKaroubi (SimplexCategoryᵒᵖ ⥤ C) ⋙ karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).map Y.p
eq : Y.p.app [n].op ≫ (HomologicalComplex.Hom.f PInfty n).f = (HomologicalComplex.Hom.f PInfty n).f ≫ Y.p.app [n].op
⊢ Y.p.app [n].op ≫ (HomologicalComplex.Hom.f PInfty n).f ≫ Y.p.app [n].op =
    Y.p.app [n].op ≫ (HomologicalComplex.Hom.f PInfty n).f

tactic: have h := Functor.congr_obj (toKaroubi_comp_karoubiFunctorCategoryEmbedding _ _) Y₂
state_before: C : Type u_1
inst✝¹ : Category C
inst✝ : Preadditive C
X : SimplicialObject C
Y : Karoubi (SimplicialObject C)
n : ℕ
Y₁ : SimplexCategoryᵒᵖ ⥤ Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj Y
Y₂ : SimplicialObject C := Y.X
Y₃ : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X
Y₄ : SimplexCategoryᵒᵖ ⥤ Karoubi C :=
  (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj ((toKaroubi (SimplexCategoryᵒᵖ ⥤ C)).obj Y.X)
P₁ : K[Y₁] ⟶ K[Y₁] := PInfty
P₂ : K[Y₂] ⟶ K[Y₂] := PInfty
P₃ : K[Y₃] ⟶ K[Y₃] := PInfty
P₄ : K[Y₄] ⟶ K[Y₄] := PInfty
h₃₂ : (HomologicalComplex.Hom.f P₃ n).f = HomologicalComplex.Hom.f P₂ n
⊢ HomologicalComplex.Hom.f P₄ n = HomologicalComplex.Hom.f P₃ n

tactic: simp only [← natTransPInfty_f_app]
state_before: C : Type u_1
inst✝¹ : Category C
inst✝ : Preadditive C
X : SimplicialObject C
Y : Karoubi (SimplicialObject C)
n : ℕ
Y₁ : SimplexCategoryᵒᵖ ⥤ Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj Y
Y₂ : SimplicialObject C := Y.X
Y₃ : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X
Y₄ : SimplexCategoryᵒᵖ ⥤ Karoubi C :=
  (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj ((toKaroubi (SimplexCategoryᵒᵖ ⥤ C)).obj Y.X)
P₁ : K[Y₁] ⟶ K[Y₁] := PInfty
P₂ : K[Y₂] ⟶ K[Y₂] := PInfty
P₃ : K[Y₃] ⟶ K[Y₃] := PInfty
P₄ : K[Y₄] ⟶ K[Y₄] := PInfty
h₃₂ : (HomologicalComplex.Hom.f P₃ n).f = HomologicalComplex.Hom.f P₂ n
h :
  (toKaroubi (SimplexCategoryᵒᵖ ⥤ C) ⋙ karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj Y₂ =
    ((whiskeringRight SimplexCategoryᵒᵖ C (Karoubi C)).obj (toKaroubi C)).obj Y₂
⊢ HomologicalComplex.Hom.f P₄ n = HomologicalComplex.Hom.f P₃ n

tactic: congr 1
state_before: C : Type u_1
inst✝¹ : Category C
inst✝ : Preadditive C
X : SimplicialObject C
Y : Karoubi (SimplicialObject C)
n : ℕ
Y₁ : SimplexCategoryᵒᵖ ⥤ Karoubi C := (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj Y
Y₂ : SimplicialObject C := Y.X
Y₃ : SimplicialObject (Karoubi C) := ((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X
Y₄ : SimplexCategoryᵒᵖ ⥤ Karoubi C :=
  (karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj ((toKaroubi (SimplexCategoryᵒᵖ ⥤ C)).obj Y.X)
P₁ : K[Y₁] ⟶ K[Y₁] := PInfty
P₂ : K[Y₂] ⟶ K[Y₂] := PInfty
P₃ : K[Y₃] ⟶ K[Y₃] := PInfty
P₄ : K[Y₄] ⟶ K[Y₄] := PInfty
h₃₂ : (HomologicalComplex.Hom.f P₃ n).f = HomologicalComplex.Hom.f P₂ n
h :
  (toKaroubi (SimplexCategoryᵒᵖ ⥤ C) ⋙ karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj Y₂ =
    ((whiskeringRight SimplexCategoryᵒᵖ C (Karoubi C)).obj (toKaroubi C)).obj Y₂
⊢ (natTransPInfty_f (Karoubi C) n).app
      ((karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C).obj ((toKaroubi (SimplexCategoryᵒᵖ ⥤ C)).obj Y.X)) =
    (natTransPInfty_f (Karoubi C) n).app (((whiskering C (Karoubi C)).obj (toKaroubi C)).obj Y.X)