-- Mathlib/AlgebraicTopology/DoldKan/Homotopies.lean
theorem map_hσ' {D : Type*} [Category D] [Preadditive D] (G : C ⥤ D) [G.Additive]
    (X : SimplicialObject C) (q n m : ℕ) (hnm : c.Rel m n) :
    (hσ' q n m hnm : K[((whiskering _ _).obj G).obj X].X n ⟶ _) =
      G.map (hσ' q n m hnm : K[X].X n ⟶ _)

Proof state:
case neg
C : Type u_1
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Preadditive 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 m : ℕ
hnm : c.Rel m n
h✝ : ¬n < q
⊢ ((-1) ^ (n - q) • (((whiskering C D).obj G).obj X).σ ⟨n - q, ⋯⟩) ≫ eqToHom ⋯ =
    ((-1) ^ (n - q) • G.map (X.σ ⟨n - q, ⋯⟩)) ≫ eqToHom ⋯

Tactic:
rfl

Result:
no goals

Proof state:
case pos
C : Type u_1
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Preadditive 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 m : ℕ
hnm : c.Rel m n
h✝ : n < q
⊢ 0 ≫ eqToHom ⋯ = G.map (0 ≫ eqToHom ⋯)

Tactic:
simp only [Functor.map_zero, zero_comp]

Result:
no goals

Proof state:
C : Type u_1
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Preadditive 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 m : ℕ
hnm : c.Rel m n
⊢ hσ' q n m hnm = G.map (hσ' q n m hnm)

Tactic:
unfold hσ' hσ

Result:
C : Type u_1
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Preadditive 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 m : ℕ
hnm : c.Rel m n
⊢ (if n < q then 0 else (-1) ^ (n - q) • (((whiskering C D).obj G).obj X).σ ⟨n - q, ⋯⟩) ≫ eqToHom ⋯ =
    G.map ((if n < q then 0 else (-1) ^ (n - q) • X.σ ⟨n - q, ⋯⟩) ≫ eqToHom ⋯)

Proof state:
C : Type u_1
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Preadditive 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 m : ℕ
hnm : c.Rel m n
⊢ (if n < q then 0 else (-1) ^ (n - q) • (((whiskering C D).obj G).obj X).σ ⟨n - q, ⋯⟩) ≫ eqToHom ⋯ =
    G.map ((if n < q then 0 else (-1) ^ (n - q) • X.σ ⟨n - q, ⋯⟩) ≫ eqToHom ⋯)

Tactic:
split_ifs

Result:
case pos
C : Type u_1
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Preadditive 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 m : ℕ
hnm : c.Rel m n
h✝ : n < q
⊢ 0 ≫ eqToHom ⋯ = G.map (0 ≫ eqToHom ⋯)

case neg
C : Type u_1
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Preadditive 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 m : ℕ
hnm : c.Rel m n
h✝ : ¬n < q
⊢ ((-1) ^ (n - q) • (((whiskering C D).obj G).obj X).σ ⟨n - q, ⋯⟩) ≫ eqToHom ⋯ =
    G.map (((-1) ^ (n - q) • X.σ ⟨n - q, ⋯⟩) ≫ eqToHom ⋯)

Proof state:
case neg
C : Type u_1
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Preadditive 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 m : ℕ
hnm : c.Rel m n
h✝ : ¬n < q
⊢ ((-1) ^ (n - q) • (((whiskering C D).obj G).obj X).σ ⟨n - q, ⋯⟩) ≫ eqToHom ⋯ =
    G.map (((-1) ^ (n - q) • X.σ ⟨n - q, ⋯⟩) ≫ eqToHom ⋯)

Tactic:
simp only [eqToHom_map, Functor.map_comp, Functor.map_zsmul]

Result:
case neg
C : Type u_1
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Preadditive 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 m : ℕ
hnm : c.Rel m n
h✝ : ¬n < q
⊢ ((-1) ^ (n - q) • (((whiskering C D).obj G).obj X).σ ⟨n - q, ⋯⟩) ≫ eqToHom ⋯ =
    ((-1) ^ (n - q) • G.map (X.σ ⟨n - q, ⋯⟩)) ≫ eqToHom ⋯