-- Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean
theorem gradedCommAux_lof_tmul (i j : ι) (a : 𝒜 i) (b : ℬ j) :
    gradedCommAux R 𝒜 ℬ (lof R _ 𝒜ℬ (i, j) (a ⊗ₜ b)) =
      (-1 : ℤˣ)^(j * i) • lof R _ ℬ𝒜 (j, i) (b ⊗ₜ a)

Proof state:
R : Type u_1
ι : Type u_2
inst✝⁷ : CommSemiring ι
inst✝⁶ : Module ι (Additive ℤˣ)
inst✝⁵ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁴ : CommRing R
inst✝³ : (i : ι) → AddCommGroup (𝒜 i)
inst✝² : (i : ι) → AddCommGroup (ℬ i)
inst✝¹ : (i : ι) → Module R (𝒜 i)
inst✝ : (i : ι) → Module R (ℬ i)
i j : ι
a : 𝒜 i
b : ℬ j
⊢ (DirectSum.toModule R (ι × ι) (⨁ (i : ι × ι), ℬ i.1 ⊗[R] 𝒜 i.2) fun i =>
        let_fun o := lof R (ι × ι) (fun i => ℬ i.1 ⊗[R] 𝒜 i.2) i.swap;
        let_fun s := (-1) ^ (i.1 * i.2);
        (s • o) ∘ₗ ↑(TensorProduct.comm R (𝒜 i.1) (ℬ i.2)))
      ((lof R (ι × ι) (fun i => 𝒜 i.1 ⊗[R] ℬ i.2) (i, j)) (a ⊗ₜ[R] b)) =
    (-1) ^ (j * i) • (lof R (ι × ι) (fun i => ℬ i.1 ⊗[R] 𝒜 i.2) (j, i)) (b ⊗ₜ[R] a)

Tactic:
dsimp

Result:
R : Type u_1
ι : Type u_2
inst✝⁷ : CommSemiring ι
inst✝⁶ : Module ι (Additive ℤˣ)
inst✝⁵ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁴ : CommRing R
inst✝³ : (i : ι) → AddCommGroup (𝒜 i)
inst✝² : (i : ι) → AddCommGroup (ℬ i)
inst✝¹ : (i : ι) → Module R (𝒜 i)
inst✝ : (i : ι) → Module R (ℬ i)
i j : ι
a : 𝒜 i
b : ℬ j
⊢ (DirectSum.toModule R (ι × ι) (⨁ (i : ι × ι), ℬ i.1 ⊗[R] 𝒜 i.2) fun i =>
        ((-1) ^ (i.1 * i.2) • lof R (ι × ι) (fun i => ℬ i.1 ⊗[R] 𝒜 i.2) i.swap) ∘ₗ
          ↑(TensorProduct.comm R (𝒜 i.1) (ℬ i.2)))
      ((lof R (ι × ι) (fun i => 𝒜 i.1 ⊗[R] ℬ i.2) (i, j)) (a ⊗ₜ[R] b)) =
    (-1) ^ (j * i) • (lof R (ι × ι) (fun i => ℬ i.1 ⊗[R] 𝒜 i.2) (j, i)) (b ⊗ₜ[R] a)

Proof state:
R : Type u_1
ι : Type u_2
inst✝⁷ : CommSemiring ι
inst✝⁶ : Module ι (Additive ℤˣ)
inst✝⁵ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁴ : CommRing R
inst✝³ : (i : ι) → AddCommGroup (𝒜 i)
inst✝² : (i : ι) → AddCommGroup (ℬ i)
inst✝¹ : (i : ι) → Module R (𝒜 i)
inst✝ : (i : ι) → Module R (ℬ i)
i j : ι
a : 𝒜 i
b : ℬ j
⊢ (gradedCommAux R 𝒜 ℬ) ((lof R (ι × ι) (fun i => 𝒜 i.1 ⊗[R] ℬ i.2) (i, j)) (a ⊗ₜ[R] b)) =
    (-1) ^ (j * i) • (lof R (ι × ι) (fun i => ℬ i.1 ⊗[R] 𝒜 i.2) (j, i)) (b ⊗ₜ[R] a)

Tactic:
rw [gradedCommAux]

Result:
R : Type u_1
ι : Type u_2
inst✝⁷ : CommSemiring ι
inst✝⁶ : Module ι (Additive ℤˣ)
inst✝⁵ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁴ : CommRing R
inst✝³ : (i : ι) → AddCommGroup (𝒜 i)
inst✝² : (i : ι) → AddCommGroup (ℬ i)
inst✝¹ : (i : ι) → Module R (𝒜 i)
inst✝ : (i : ι) → Module R (ℬ i)
i j : ι
a : 𝒜 i
b : ℬ j
⊢ (DirectSum.toModule R (ι × ι) (⨁ (i : ι × ι), ℬ i.1 ⊗[R] 𝒜 i.2) fun i =>
        let_fun o := lof R (ι × ι) (fun i => ℬ i.1 ⊗[R] 𝒜 i.2) i.swap;
        let_fun s := (-1) ^ (i.1 * i.2);
        (s • o) ∘ₗ ↑(TensorProduct.comm R (𝒜 i.1) (ℬ i.2)))
      ((lof R (ι × ι) (fun i => 𝒜 i.1 ⊗[R] ℬ i.2) (i, j)) (a ⊗ₜ[R] b)) =
    (-1) ^ (j * i) • (lof R (ι × ι) (fun i => ℬ i.1 ⊗[R] 𝒜 i.2) (j, i)) (b ⊗ₜ[R] a)

Proof state:
R : Type u_1
ι : Type u_2
inst✝⁷ : CommSemiring ι
inst✝⁶ : Module ι (Additive ℤˣ)
inst✝⁵ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁴ : CommRing R
inst✝³ : (i : ι) → AddCommGroup (𝒜 i)
inst✝² : (i : ι) → AddCommGroup (ℬ i)
inst✝¹ : (i : ι) → Module R (𝒜 i)
inst✝ : (i : ι) → Module R (ℬ i)
i j : ι
a : 𝒜 i
b : ℬ j
⊢ (DirectSum.toModule R (ι × ι) (⨁ (i : ι × ι), ℬ i.1 ⊗[R] 𝒜 i.2) fun i =>
        ((-1) ^ (i.1 * i.2) • lof R (ι × ι) (fun i => ℬ i.1 ⊗[R] 𝒜 i.2) i.swap) ∘ₗ
          ↑(TensorProduct.comm R (𝒜 i.1) (ℬ i.2)))
      ((lof R (ι × ι) (fun i => 𝒜 i.1 ⊗[R] ℬ i.2) (i, j)) (a ⊗ₜ[R] b)) =
    (-1) ^ (j * i) • (lof R (ι × ι) (fun i => ℬ i.1 ⊗[R] 𝒜 i.2) (j, i)) (b ⊗ₜ[R] a)

Tactic:
simp [mul_comm i j]

Result:
no goals