Theorem: CategoryTheory.ShortComplex.Homotopy.smul_h₂
File: Mathlib/Algebra/Homology/ShortComplex/Linear.lean

Proof steps:
state_before: R : Type u_1
C : Type u_2
inst✝³ : Semiring R
inst✝² : Category.{?u.88315, u_2} C
inst✝¹ : Preadditive C
inst✝ : Linear R C
S₁ S₂ : ShortComplex C
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
a : R
⊢ (a • φ₁).τ₁ = S₁.f ≫ (a • h.h₁) + a • h.h₀ + (a • φ₂).τ₁
state_after: R : Type u_1
C : Type u_2
inst✝³ : Semiring R
inst✝² : Category.{?u.88315, u_2} C
inst✝¹ : Preadditive C
inst✝ : Linear R C
S₁ S₂ : ShortComplex C
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
a : R
⊢ a • φ₁.τ₁ = S₁.f ≫ (a • h.h₁) + a • h.h₀ + a • φ₂.τ₁
tactic: dsimp
state_before: R : Type u_1
C : Type u_2
inst✝³ : Semiring R
inst✝² : Category.{?u.88315, u_2} C
inst✝¹ : Preadditive C
inst✝ : Linear R C
S₁ S₂ : ShortComplex C
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
a : R
⊢ a • φ₁.τ₁ = S₁.f ≫ (a • h.h₁) + a • h.h₀ + a • φ₂.τ₁
state_after: R : Type u_1
C : Type u_2
inst✝³ : Semiring R
inst✝² : Category.{?u.88315, u_2} C
inst✝¹ : Preadditive C
inst✝ : Linear R C
S₁ S₂ : ShortComplex C
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
a : R
⊢ a • (S₁.f ≫ h.h₁ + h.h₀ + φ₂.τ₁) = S₁.f ≫ (a • h.h₁) + a • h.h₀ + a • φ₂.τ₁
tactic: rw [h.comm₁]
state_before: R : Type u_1
C : Type u_2
inst✝³ : Semiring R
inst✝² : Category.{?u.88315, u_2} C
inst✝¹ : Preadditive C
inst✝ : Linear R C
S₁ S₂ : ShortComplex C
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
a : R
⊢ a • (S₁.f ≫ h.h₁ + h.h₀ + φ₂.τ₁) = S₁.f ≫ (a • h.h₁) + a • h.h₀ + a • φ₂.τ₁
state_after: no goals
tactic: simp only [smul_add, Linear.comp_smul]
state_before: R : Type u_1
C : Type u_2
inst✝³ : Semiring R
inst✝² : Category.{?u.88315, u_2} C
inst✝¹ : Preadditive C
inst✝ : Linear R C
S₁ S₂ : ShortComplex C
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
a : R
⊢ (a • φ₁).τ₂ = S₁.g ≫ (a • h.h₂) + (a • h.h₁) ≫ S₂.f + (a • φ₂).τ₂
state_after: R : Type u_1
C : Type u_2
inst✝³ : Semiring R
inst✝² : Category.{?u.88315, u_2} C
inst✝¹ : Preadditive C
inst✝ : Linear R C
S₁ S₂ : ShortComplex C
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
a : R
⊢ a • φ₁.τ₂ = S₁.g ≫ (a • h.h₂) + (a • h.h₁) ≫ S₂.f + a • φ₂.τ₂
tactic: dsimp
state_before: R : Type u_1
C : Type u_2
inst✝³ : Semiring R
inst✝² : Category.{?u.88315, u_2} C
inst✝¹ : Preadditive C
inst✝ : Linear R C
S₁ S₂ : ShortComplex C
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
a : R
⊢ a • φ₁.τ₂ = S₁.g ≫ (a • h.h₂) + (a • h.h₁) ≫ S₂.f + a • φ₂.τ₂
state_after: R : Type u_1
C : Type u_2
inst✝³ : Semiring R
inst✝² : Category.{?u.88315, u_2} C
inst✝¹ : Preadditive C
inst✝ : Linear R C
S₁ S₂ : ShortComplex C
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
a : R
⊢ a • (S₁.g ≫ h.h₂ + h.h₁ ≫ S₂.f + φ₂.τ₂) = S₁.g ≫ (a • h.h₂) + (a • h.h₁) ≫ S₂.f + a • φ₂.τ₂
tactic: rw [h.comm₂]
state_before: R : Type u_1
C : Type u_2
inst✝³ : Semiring R
inst✝² : Category.{?u.88315, u_2} C
inst✝¹ : Preadditive C
inst✝ : Linear R C
S₁ S₂ : ShortComplex C
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
a : R
⊢ a • (S₁.g ≫ h.h₂ + h.h₁ ≫ S₂.f + φ₂.τ₂) = S₁.g ≫ (a • h.h₂) + (a • h.h₁) ≫ S₂.f + a • φ₂.τ₂
state_after: no goals
tactic: simp only [smul_add, Linear.comp_smul, Linear.smul_comp]
state_before: R : Type u_1
C : Type u_2
inst✝³ : Semiring R
inst✝² : Category.{?u.88315, u_2} C
inst✝¹ : Preadditive C
inst✝ : Linear R C
S₁ S₂ : ShortComplex C
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
a : R
⊢ (a • φ₁).τ₃ = a • h.h₃ + (a • h.h₂) ≫ S₂.g + (a • φ₂).τ₃
state_after: R : Type u_1
C : Type u_2
inst✝³ : Semiring R
inst✝² : Category.{?u.88315, u_2} C
inst✝¹ : Preadditive C
inst✝ : Linear R C
S₁ S₂ : ShortComplex C
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
a : R
⊢ a • φ₁.τ₃ = a • h.h₃ + (a • h.h₂) ≫ S₂.g + a • φ₂.τ₃
tactic: dsimp
state_before: R : Type u_1
C : Type u_2
inst✝³ : Semiring R
inst✝² : Category.{?u.88315, u_2} C
inst✝¹ : Preadditive C
inst✝ : Linear R C
S₁ S₂ : ShortComplex C
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
a : R
⊢ a • φ₁.τ₃ = a • h.h₃ + (a • h.h₂) ≫ S₂.g + a • φ₂.τ₃
state_after: R : Type u_1
C : Type u_2
inst✝³ : Semiring R
inst✝² : Category.{?u.88315, u_2} C
inst✝¹ : Preadditive C
inst✝ : Linear R C
S₁ S₂ : ShortComplex C
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
a : R
⊢ a • (h.h₃ + h.h₂ ≫ S₂.g + φ₂.τ₃) = a • h.h₃ + (a • h.h₂) ≫ S₂.g + a • φ₂.τ₃
tactic: rw [h.comm₃]
state_before: R : Type u_1
C : Type u_2
inst✝³ : Semiring R
inst✝² : Category.{?u.88315, u_2} C
inst✝¹ : Preadditive C
inst✝ : Linear R C
S₁ S₂ : ShortComplex C
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
a : R
⊢ a • (h.h₃ + h.h₂ ≫ S₂.g + φ₂.τ₃) = a • h.h₃ + (a • h.h₂) ≫ S₂.g + a • φ₂.τ₃
state_after: no goals
tactic: simp only [smul_add, Linear.smul_comp]