-- 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 z : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ
mA : 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 :=
  gradedMul R 𝒜 ℬ
ixa : ι
xa : 𝒜 ixa
ixb : ι
xb : ℬ ixb
iya : ι
ya : 𝒜 iya
iyb : ι
yb : ℬ iyb
iza : ι
za : 𝒜 iza
izb : ι
zb : ℬ izb
a :
  HSMul.hSMul (α := R) (↑(↑(HPow.hPow (α := ℤˣ) (-1) (ixb * iya)) : ℤ) : R)
      (HSMul.hSMul (β := (⨁ (i : ι), 𝒜 i) ⊗[R] ⨁ (i : ι), ℬ i) (HPow.hPow (α := ℤˣ) (-1) ((ixb + iyb) * iza))
        (HMul.hMul (α := ⨁ (i : ι), 𝒜 i) ((lof R ι 𝒜 ixa : (a : 𝒜 ixa) → ⨁ (i : ι), 𝒜 i) xa : ⨁ (i : ι), 𝒜 i)
            (HMul.hMul (α := ⨁ (i : ι), 𝒜 i) (β := ⨁ (i : ι), 𝒜 i)
              ((lof R ι 𝒜 iya : (a : 𝒜 iya) → ⨁ (i : ι), 𝒜 i) ya : ⨁ (i : ι), 𝒜 i)
              ((lof R ι 𝒜 iza : (a : 𝒜 iza) → ⨁ (i : ι), 𝒜 i) za : ⨁ (i : ι), 𝒜 i)) ⊗ₜ[R]
          HMul.hMul (α := ⨁ (i : ι), ℬ i) ((lof R ι ℬ ixb : (a : ℬ ixb) → ⨁ (i : ι), ℬ i) xb : ⨁ (i : ι), ℬ i)
            (HMul.hMul (α := ⨁ (i : ι), ℬ i) (β := ⨁ (i : ι), ℬ i)
              ((lof R ι ℬ iyb : (a : ℬ iyb) → ⨁ (i : ι), ℬ i) yb : ⨁ (i : ι), ℬ i)
              ((lof R ι ℬ izb : (a : ℬ izb) → ⨁ (i : ι), ℬ i) zb : ⨁ (i : ι), ℬ i)))) =
    HSMul.hSMul (α := R) (↑(↑(HPow.hPow (α := ℤˣ) (-1) (iyb * iza)) : ℤ) : R)
      (HSMul.hSMul (β := (⨁ (i : ι), 𝒜 i) ⊗[R] ⨁ (i : ι), ℬ i) (HPow.hPow (α := ℤˣ) (-1) (ixb * (iya + iza)))
        (HMul.hMul (α := ⨁ (i : ι), 𝒜 i) ((lof R ι 𝒜 ixa : (a : 𝒜 ixa) → ⨁ (i : ι), 𝒜 i) xa : ⨁ (i : ι), 𝒜 i)
            (HMul.hMul (α := ⨁ (i : ι), 𝒜 i) (β := ⨁ (i : ι), 𝒜 i)
              ((lof R ι 𝒜 iya : (a : 𝒜 iya) → ⨁ (i : ι), 𝒜 i) ya : ⨁ (i : ι), 𝒜 i)
              ((lof R ι 𝒜 iza : (a : 𝒜 iza) → ⨁ (i : ι), 𝒜 i) za : ⨁ (i : ι), 𝒜 i)) ⊗ₜ[R]
          HMul.hMul (α := ⨁ (i : ι), ℬ i) ((lof R ι ℬ ixb : (a : ℬ ixb) → ⨁ (i : ι), ℬ i) xb : ⨁ (i : ι), ℬ i)
            (HMul.hMul (α := ⨁ (i : ι), ℬ i) (β := ⨁ (i : ι), ℬ i)
              ((lof R ι ℬ iyb : (a : ℬ iyb) → ⨁ (i : ι), ℬ i) yb : ⨁ (i : ι), ℬ i)
              ((lof R ι ℬ izb : (a : ℬ izb) → ⨁ (i : ι), ℬ i) zb : ⨁ (i : ι), ℬ i))))
⊢ Eq (α := DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)
    ((((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)
              ((((gradedMul R 𝒜 ℬ :
                        (a : (⨁ (i : ι), 𝒜 i) ⊗[R] ⨁ (i : ι), ℬ i) →
                          DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)
                      ((lof R ι 𝒜 ixa : (a : 𝒜 ixa) → ⨁ (i : ι), 𝒜 i) xa ⊗ₜ[R]
                        (lof R ι ℬ ixb : (a : ℬ ixb) → ⨁ (i : ι), ℬ i) xb) :
                    DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) :
                  (a : (⨁ (i : ι), 𝒜 i) ⊗[R] ⨁ (i : ι), ℬ i) → DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)
                ((lof R ι 𝒜 iya : (a : 𝒜 iya) → ⨁ (i : ι), 𝒜 i) ya ⊗ₜ[R]
                  (lof R ι ℬ iyb : (a : ℬ iyb) → ⨁ (i : ι), ℬ i) yb)) :
            DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) :
          (a : (⨁ (i : ι), 𝒜 i) ⊗[R] ⨁ (i : ι), ℬ i) → DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)
        ((lof R ι 𝒜 iza : (a : 𝒜 iza) → ⨁ (i : ι), 𝒜 i) za ⊗ₜ[R] (lof R ι ℬ izb : (a : ℬ izb) → ⨁ (i : ι), ℬ i) zb) :
      DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)
    ((((gradedMul R 𝒜 ℬ :
                (a : (⨁ (i : ι), 𝒜 i) ⊗[R] ⨁ (i : ι), ℬ i) →
                  DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)
              ((lof R ι 𝒜 ixa : (a : 𝒜 ixa) → ⨁ (i : ι), 𝒜 i) xa ⊗ₜ[R]
                (lof R ι ℬ ixb : (a : ℬ ixb) → ⨁ (i : ι), ℬ i) xb) :
            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)
        ((((gradedMul R 𝒜 ℬ :
                  (a : (⨁ (i : ι), 𝒜 i) ⊗[R] ⨁ (i : ι), ℬ i) →
                    DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)
                ((lof R ι 𝒜 iya : (a : 𝒜 iya) → ⨁ (i : ι), 𝒜 i) ya ⊗ₜ[R]
                  (lof R ι ℬ iyb : (a : ℬ iyb) → ⨁ (i : ι), ℬ i) yb) :
              DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) :
            (a : (⨁ (i : ι), 𝒜 i) ⊗[R] ⨁ (i : ι), ℬ i) → DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)
          ((lof R ι 𝒜 iza : (a : 𝒜 iza) → ⨁ (i : ι), 𝒜 i) za ⊗ₜ[R] (lof R ι ℬ izb : (a : ℬ izb) → ⨁ (i : ι), ℬ i) zb)) :
      DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)

Tactic:
simp_rw [tmul_of_gradedMul_of_tmul, Units.smul_def, ← Int.cast_smul_eq_zsmul R,
    LinearMap.map_smul₂, map_smul, DirectSum.lof_eq_of, DirectSum.of_mul_of,
    ← DirectSum.lof_eq_of R, tmul_of_gradedMul_of_tmul, DirectSum.lof_eq_of, ← DirectSum.of_mul_of,
    ← DirectSum.lof_eq_of R, mul_assoc]

Premises used: mul_assoc, DirectSum.of_mul_of, DirectSum.lof_eq_of, Units.smul_def, LinearMap.map_smul₂, TensorProduct.tmul_of_gradedMul_of_tmul, Int.cast_smul_eq_zsmul, map_smul