-- Module: Mathlib.LinearAlgebra.TensorProduct.Graded.External
Goal:
R : Type u_1
ι : Type u_2
inst : CommSemiring ι
inst_1 : Module ι (Additive ℤˣ)
inst_2 : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst_3 : CommRing R
inst_4 : (i : ι) → AddCommGroup (𝒜 i)
inst_5 : (i : ι) → AddCommGroup (ℬ i)
inst_6 : (i : ι) → Module.{u_1, u_3} R (𝒜 i)
inst_7 : (i : ι) → Module.{u_1, u_4} R (ℬ i)
inst_8 : DirectSum.GRing 𝒜
inst_9 : DirectSum.GRing ℬ
inst_10 : DirectSum.GAlgebra R 𝒜
inst_11 : DirectSum.GAlgebra R ℬ
x y : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ
i₁ : ι
a₁ : 𝒜 i₁
j₁ : ι
b₁ : ℬ j₁
i₂ : ι
a₂ : 𝒜 i₂
j₂ : ι
b₂ : ℬ j₂
a :
  HSMul.hSMul (β := (⨁ (i : ι), ℬ i) ⊗[R] ⨁ (i : ι), 𝒜 i)
      (HPow.hPow (α := ℤˣ) (-1) (j₁ * i₂) * HPow.hPow (α := ℤˣ) (-1) ((j₁ + j₂) * (i₁ + i₂)))
      (HMul.hMul (α := ⨁ (i : ι), ℬ i) (β := ⨁ (i : ι), ℬ i)
          ((lof R ι ℬ j₁ : (a : ℬ j₁) → ⨁ (i : ι), ℬ i) b₁ : ⨁ (i : ι), ℬ i)
          ((lof R ι ℬ j₂ : (a : ℬ j₂) → ⨁ (i : ι), ℬ i) b₂ : ⨁ (i : ι), ℬ i) ⊗ₜ[R]
        HMul.hMul (α := ⨁ (i : ι), 𝒜 i) (β := ⨁ (i : ι), 𝒜 i)
          ((lof R ι 𝒜 i₁ : (a : 𝒜 i₁) → ⨁ (i : ι), 𝒜 i) a₁ : ⨁ (i : ι), 𝒜 i)
          ((lof R ι 𝒜 i₂ : (a : 𝒜 i₂) → ⨁ (i : ι), 𝒜 i) a₂ : ⨁ (i : ι), 𝒜 i)) =
    HSMul.hSMul (α := R) (↑(↑(HPow.hPow (α := ℤˣ) (-1) (j₂ * i₂)) : ℤ) : R)
      (HSMul.hSMul (β := DirectSum ι ℬ ⊗[R] DirectSum ι 𝒜)
        (HPow.hPow (α := ℤˣ) (-1) (j₁ * i₁) * HPow.hPow (α := ℤˣ) (-1) (i₁ * j₂))
        (HMul.hMul (α := ⨁ (i : ι), ℬ i) (β := ⨁ (i : ι), ℬ i)
            ((lof R ι ℬ j₁ : (a : ℬ j₁) → ⨁ (i : ι), ℬ i) b₁ : ⨁ (i : ι), ℬ i)
            ((lof R ι ℬ j₂ : (a : ℬ j₂) → ⨁ (i : ι), ℬ i) b₂ : ⨁ (i : ι), ℬ i) ⊗ₜ[R]
          HMul.hMul (α := ⨁ (i : ι), 𝒜 i) (β := ⨁ (i : ι), 𝒜 i)
            ((lof R ι 𝒜 i₁ : (a : 𝒜 i₁) → ⨁ (i : ι), 𝒜 i) a₁ : ⨁ (i : ι), 𝒜 i)
            ((lof R ι 𝒜 i₂ : (a : 𝒜 i₂) → ⨁ (i : ι), 𝒜 i) a₂ : ⨁ (i : ι), 𝒜 i)))
⊢ HSMul.hSMul (α := R) (β := (⨁ (i : ι), ℬ i) ⊗[R] ⨁ (i : ι), 𝒜 i) (↑(↑(HPow.hPow (α := ℤˣ) (-1) (j₁ * i₂)) : ℤ) : R)
      ((gradedComm R 𝒜 ℬ : (a : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) → (⨁ (i : ι), ℬ i) ⊗[R] ⨁ (i : ι), 𝒜 i)
          (HMul.hMul (α := ⨁ (i : ι), 𝒜 i) (β := ⨁ (i : ι), 𝒜 i)
              ((lof R ι 𝒜 i₁ : (a : 𝒜 i₁) → ⨁ (i : ι), 𝒜 i) a₁ : ⨁ (i : ι), 𝒜 i)
              ((lof R ι 𝒜 i₂ : (a : 𝒜 i₂) → ⨁ (i : ι), 𝒜 i) a₂ : ⨁ (i : ι), 𝒜 i) ⊗ₜ[R]
            HMul.hMul (α := ⨁ (i : ι), ℬ i) (β := ⨁ (i : ι), ℬ i)
              ((lof R ι ℬ j₁ : (a : ℬ j₁) → ⨁ (i : ι), ℬ i) b₁ : ⨁ (i : ι), ℬ i)
              ((lof R ι ℬ j₂ : (a : ℬ j₂) → ⨁ (i : ι), ℬ i) b₂ : ⨁ (i : ι), ℬ i)) :
        (⨁ (i : ι), ℬ i) ⊗[R] ⨁ (i : ι), 𝒜 i) =
    HSMul.hSMul (α := R) (↑(↑(HPow.hPow (α := ℤˣ) (-1) (j₂ * i₂)) : ℤ) : R)
      (HSMul.hSMul (α := R) (β := DirectSum ι ℬ ⊗[R] DirectSum ι 𝒜) (↑(↑(HPow.hPow (α := ℤˣ) (-1) (j₁ * i₁)) : ℤ) : R)
        ((((gradedMul R ℬ 𝒜 :
                    (a : DirectSum ι ℬ ⊗[R] DirectSum ι 𝒜) →
                      DirectSum ι ℬ ⊗[R] DirectSum ι 𝒜 →ₗ[R] DirectSum ι ℬ ⊗[R] DirectSum ι 𝒜)
                  ((lof R ι ℬ j₁ : (a : ℬ j₁) → ⨁ (i : ι), ℬ i) b₁ ⊗ₜ[R]
                    (lof R ι 𝒜 i₁ : (a : 𝒜 i₁) → ⨁ (i : ι), 𝒜 i) a₁) :
                DirectSum ι ℬ ⊗[R] DirectSum ι 𝒜 →ₗ[R] DirectSum ι ℬ ⊗[R] DirectSum ι 𝒜) :
              (a : DirectSum ι ℬ ⊗[R] DirectSum ι 𝒜) → DirectSum ι ℬ ⊗[R] DirectSum ι 𝒜)
            ((lof R ι ℬ j₂ : (a : ℬ j₂) → ⨁ (i : ι), ℬ i) b₂ ⊗ₜ[R] (lof R ι 𝒜 i₂ : (a : 𝒜 i₂) → ⨁ (i : ι), 𝒜 i) a₂) :
          DirectSum ι ℬ ⊗[R] DirectSum ι 𝒜))

Tactic:
simp_rw [Int.cast_smul_eq_zsmul R, ← Units.smul_def, DirectSum.lof_eq_of, DirectSum.of_mul_of,
    ← DirectSum.lof_eq_of R, gradedComm_of_tmul_of, tmul_of_gradedMul_of_tmul, smul_smul,
    DirectSum.lof_eq_of, ← DirectSum.of_mul_of, ← DirectSum.lof_eq_of R]

Premises used: DirectSum.of_mul_of, smul_smul, DirectSum.lof_eq_of, Units.smul_def, TensorProduct.tmul_of_gradedMul_of_tmul, TensorProduct.gradedComm_of_tmul_of, Int.cast_smul_eq_zsmul