-- Lean 4 proof step from Mathlib.LinearAlgebra.TensorProduct.Graded.External (goal)
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
j₁ i₂ : ι
a₁ : ⨁ (i : ι), 𝒜 i
b₁ : ℬ j₁
a₂ : 𝒜 i₂
b₂ : ⨁ (i : ι), ℬ i
⊢ (((gradedMul R 𝒜 ℬ :
              (a : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) →
                DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)
            (a₁ ⊗ₜ[R] (lof R ι ℬ j₁ : (a : ℬ j₁) → ⨁ (i : ι), ℬ i) b₁) :
          DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) :
        (a : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) → DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)
      ((lof R ι 𝒜 i₂ : (a : 𝒜 i₂) → ⨁ (i : ι), 𝒜 i) a₂ ⊗ₜ[R] b₂) =
    HSMul.hSMul (β := (⨁ (i : ι), 𝒜 i) ⊗[R] ⨁ (i : ι), ℬ i) (HPow.hPow (α := ℤˣ) (-1) (j₁ * i₂))
      (HMul.hMul (β := ⨁ (i : ι), 𝒜 i) a₁ ((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) ((lof R ι ℬ j₁ : (a : ℬ j₁) → ⨁ (i : ι), ℬ i) b₁ : ⨁ (i : ι), ℬ i) b₂)
have rw : ((((curry
                (map (LinearMap.mul' R (⨁ (i : ι), 𝒜 i)) (LinearMap.mul' R (⨁ (i : ι), ℬ i)) ∘ₗ
                  (↑(LinearEquiv.symm
                          (TensorProduct.assoc R (⨁ (i : ι), 𝒜 i) (⨁ (i : ι), 𝒜 i)
                            ((⨁ (i : ι), ℬ i) ⊗[R] ⨁ (i : ι), ℬ i))) :
                      (⨁ (i : ι), 𝒜 i) ⊗[R] ((⨁ (i : ι), 𝒜 i) ⊗[R] ((⨁ (i : ι), ℬ i) ⊗[R] ⨁ (i : ι), ℬ i)) →ₗ[R]
                        ((⨁ (i : ι), 𝒜 i) ⊗[R] ⨁ (i : ι), 𝒜 i) ⊗[R] ((⨁ (i : ι), ℬ i) ⊗[R] ⨁ (i : ι), ℬ i)) ∘ₗ
                    LinearMap.lTensor (⨁ (i : ι), 𝒜 i)
                        ((↑(TensorProduct.assoc R (⨁ (i : ι), 𝒜 i) (⨁ (i : ι), ℬ i) (⨁ (i : ι), ℬ i)) :
                            (((⨁ (i : ι), 𝒜 i) ⊗[R] ⨁ (i : ι), ℬ i) ⊗[R] ⨁ (i : ι), ℬ i) →ₗ[R]
                              (⨁ (i : ι), 𝒜 i) ⊗[R] ((⨁ (i : ι), ℬ i) ⊗[R] ⨁ (i : ι), ℬ i)) ∘ₗ
                          LinearMap.rTensor (⨁ (i : ι), ℬ i)
                              (↑(gradedComm R ℬ 𝒜) :
                                ((⨁ (i : ι), ℬ i) ⊗[R] ⨁ (i : ι), 𝒜 i) →ₗ[R] (⨁ (i : ι), 𝒜 i) ⊗[R] ⨁ (i : ι), ℬ i) ∘ₗ
                            (↑(LinearEquiv.symm
                                  (TensorProduct.assoc R (DirectSum ι ℬ) (DirectSum ι 𝒜) (DirectSum ι ℬ))) :
                              DirectSum ι ℬ ⊗[R] (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) →ₗ[R]
                                DirectSum ι ℬ ⊗[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)) ∘ₗ
                      (↑(TensorProduct.assoc R (DirectSum ι 𝒜) (DirectSum ι ℬ) (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)) :
                        DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ ⊗[R] (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) →ₗ[R]
                          DirectSum ι 𝒜 ⊗[R] (DirectSum ι ℬ ⊗[R] (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)))) :
              DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R]
                DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) :
            (a : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) →
              DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)
          (a₁ ⊗ₜ[R] (lof R ι ℬ j₁ : (a : ℬ j₁) → ⨁ (i : ι), ℬ i) b₁) :
        DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) :
      (a : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) → DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)
    ((lof R ι 𝒜 i₂ : (a : 𝒜 i₂) → ⨁ (i : ι), 𝒜 i) a₂ ⊗ₜ[R] b₂) =
  HSMul.hSMul (β := (⨁ (i : ι), 𝒜 i) ⊗[R] ⨁ (i : ι), ℬ i) (HPow.hPow (α := ℤˣ) (-1) (j₁ * i₂))
    (HMul.hMul (β := ⨁ (i : ι), 𝒜 i) a₁ ((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) ((lof R ι ℬ j₁ : (a : ℬ j₁) → ⨁ (i : ι), ℬ i) b₁ : ⨁ (i : ι), ℬ i) b₂)
Tactic: rw [gradedMul]