Theorem: AlgebraicTopology.DoldKan.map_Hσ
File: Mathlib/AlgebraicTopology/DoldKan/Homotopies.lean

tactic: unfold Hσ
state_before: C : Type u_1
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Preadditive C
X✝ : SimplicialObject C
D : Type u_2
inst✝² : Category.{u_3, u_2} D
inst✝¹ : Preadditive D
G : C ⥤ D
inst✝ : G.Additive
X : SimplicialObject C
q n : ℕ
⊢ (Hσ q).f n = G.map ((Hσ q).f n)
state_after: C : Type u_1
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Preadditive C
X✝ : SimplicialObject C
D : Type u_2
inst✝² : Category.{u_3, u_2} D
inst✝¹ : Preadditive D
G : C ⥤ D
inst✝ : G.Additive
X : SimplicialObject C
q n : ℕ
⊢ (nullHomotopicMap' (hσ' q)).f n = G.map ((nullHomotopicMap' (hσ' q)).f n)

tactic: have eq := HomologicalComplex.congr_hom (map_nullHomotopicMap' G (@hσ' _ _ _ X q)) n
state_before: C : Type u_1
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Preadditive C
X✝ : SimplicialObject C
D : Type u_2
inst✝² : Category.{u_3, u_2} D
inst✝¹ : Preadditive D
G : C ⥤ D
inst✝ : G.Additive
X : SimplicialObject C
q n : ℕ
⊢ (nullHomotopicMap' (hσ' q)).f n = G.map ((nullHomotopicMap' (hσ' q)).f n)
state_after: C : Type u_1
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Preadditive C
X✝ : SimplicialObject C
D : Type u_2
inst✝² : Category.{u_3, u_2} D
inst✝¹ : Preadditive D
G : C ⥤ D
inst✝ : G.Additive
X : SimplicialObject C
q n : ℕ
eq :
  ((G.mapHomologicalComplex c).map (nullHomotopicMap' (hσ' q))).f n =
    (nullHomotopicMap' fun i j hij => G.map (hσ' q i j hij)).f n
⊢ (nullHomotopicMap' (hσ' q)).f n = G.map ((nullHomotopicMap' (hσ' q)).f n)

tactic: simp only [Functor.mapHomologicalComplex_map_f, ← map_hσ'] at eq
state_before: C : Type u_1
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Preadditive C
X✝ : SimplicialObject C
D : Type u_2
inst✝² : Category.{u_3, u_2} D
inst✝¹ : Preadditive D
G : C ⥤ D
inst✝ : G.Additive
X : SimplicialObject C
q n : ℕ
eq :
  ((G.mapHomologicalComplex c).map (nullHomotopicMap' (hσ' q))).f n =
    (nullHomotopicMap' fun i j hij => G.map (hσ' q i j hij)).f n
⊢ (nullHomotopicMap' (hσ' q)).f n = G.map ((nullHomotopicMap' (hσ' q)).f n)
state_after: C : Type u_1
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Preadditive C
X✝ : SimplicialObject C
D : Type u_2
inst✝² : Category.{u_3, u_2} D
inst✝¹ : Preadditive D
G : C ⥤ D
inst✝ : G.Additive
X : SimplicialObject C
q n : ℕ
eq : G.map ((nullHomotopicMap' (hσ' q)).f n) = (nullHomotopicMap' fun i j hij => hσ' q i j hij).f n
⊢ (nullHomotopicMap' (hσ' q)).f n = G.map ((nullHomotopicMap' (hσ' q)).f n)

tactic: rw [eq]
state_before: C : Type u_1
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Preadditive C
X✝ : SimplicialObject C
D : Type u_2
inst✝² : Category.{u_3, u_2} D
inst✝¹ : Preadditive D
G : C ⥤ D
inst✝ : G.Additive
X : SimplicialObject C
q n : ℕ
eq : G.map ((nullHomotopicMap' (hσ' q)).f n) = (nullHomotopicMap' fun i j hij => hσ' q i j hij).f n
⊢ (nullHomotopicMap' (hσ' q)).f n = G.map ((nullHomotopicMap' (hσ' q)).f n)
state_after: C : Type u_1
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Preadditive C
X✝ : SimplicialObject C
D : Type u_2
inst✝² : Category.{u_3, u_2} D
inst✝¹ : Preadditive D
G : C ⥤ D
inst✝ : G.Additive
X : SimplicialObject C
q n : ℕ
eq : G.map ((nullHomotopicMap' (hσ' q)).f n) = (nullHomotopicMap' fun i j hij => hσ' q i j hij).f n
⊢ (nullHomotopicMap' (hσ' q)).f n = (nullHomotopicMap' fun i j hij => hσ' q i j hij).f n

tactic: let h := (Functor.congr_obj (map_alternatingFaceMapComplex G) X).symm
state_before: C : Type u_1
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Preadditive C
X✝ : SimplicialObject C
D : Type u_2
inst✝² : Category.{u_3, u_2} D
inst✝¹ : Preadditive D
G : C ⥤ D
inst✝ : G.Additive
X : SimplicialObject C
q n : ℕ
eq : G.map ((nullHomotopicMap' (hσ' q)).f n) = (nullHomotopicMap' fun i j hij => hσ' q i j hij).f n
⊢ (nullHomotopicMap' (hσ' q)).f n = (nullHomotopicMap' fun i j hij => hσ' q i j hij).f n
state_after: C : Type u_1
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Preadditive C
X✝ : SimplicialObject C
D : Type u_2
inst✝² : Category.{u_3, u_2} D
inst✝¹ : Preadditive D
G : C ⥤ D
inst✝ : G.Additive
X : SimplicialObject C
q n : ℕ
eq : G.map ((nullHomotopicMap' (hσ' q)).f n) = (nullHomotopicMap' fun i j hij => hσ' q i j hij).f n
h : ((whiskering C D).obj G ⋙ alternatingFaceMapComplex D).obj X =
  (alternatingFaceMapComplex C ⋙ G.mapHomologicalComplex (ComplexShape.down ℕ)).obj X :=
  Eq.symm (Functor.congr_obj (map_alternatingFaceMapComplex G) X)
⊢ (nullHomotopicMap' (hσ' q)).f n = (nullHomotopicMap' fun i j hij => hσ' q i j hij).f n

tactic: congr
state_before: C : Type u_1
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Preadditive C
X✝ : SimplicialObject C
D : Type u_2
inst✝² : Category.{u_3, u_2} D
inst✝¹ : Preadditive D
G : C ⥤ D
inst✝ : G.Additive
X : SimplicialObject C
q n : ℕ
eq : G.map ((nullHomotopicMap' (hσ' q)).f n) = (nullHomotopicMap' fun i j hij => hσ' q i j hij).f n
h : ((whiskering C D).obj G ⋙ alternatingFaceMapComplex D).obj X =
  (alternatingFaceMapComplex C ⋙ G.mapHomologicalComplex (ComplexShape.down ℕ)).obj X :=
  Eq.symm (Functor.congr_obj (map_alternatingFaceMapComplex G) X)
⊢ (nullHomotopicMap' (hσ' q)).f n = (nullHomotopicMap' fun i j hij => hσ' q i j hij).f n
state_after: no goals