Theorem: GradedTensorProduct.tmul_coe_mul_coe_tmul
File: Mathlib/LinearAlgebra/TensorProduct/Graded/Internal.lean

Proof steps:
state_before: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : Module ι (Additive ℤˣ)
inst✝⁷ : DecidableEq ι
inst✝⁶ : CommRing R
inst✝⁵ : Ring A
inst✝⁴ : Ring B
inst✝³ : Algebra R A
inst✝² : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝¹ : GradedAlgebra 𝒜
inst✝ : GradedAlgebra ℬ
j₁ i₂ : ι
a₁ : A
b₁ : ↥(ℬ j₁)
a₂ : ↥(𝒜 i₂)
b₂ : B
⊢ a₁ ᵍ⊗ₜ[R] ↑b₁ * ↑a₂ ᵍ⊗ₜ[R] b₂ = (-1) ^ (j₁ * i₂) • (a₁ * ↑a₂) ᵍ⊗ₜ[R] (↑b₁ * b₂)
state_after: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : Module ι (Additive ℤˣ)
inst✝⁷ : DecidableEq ι
inst✝⁶ : CommRing R
inst✝⁵ : Ring A
inst✝⁴ : Ring B
inst✝³ : Algebra R A
inst✝² : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝¹ : GradedAlgebra 𝒜
inst✝ : GradedAlgebra ℬ
j₁ i₂ : ι
a₁ : A
b₁ : ↥(ℬ j₁)
a₂ : ↥(𝒜 i₂)
b₂ : B
⊢ (auxEquiv R 𝒜 ℬ).symm
      (((gradedMul R (fun x => ↥(𝒜 x)) fun x => ↥(ℬ x)) ((auxEquiv R 𝒜 ℬ) (a₁ ᵍ⊗ₜ[R] ↑b₁)))
        ((auxEquiv R 𝒜 ℬ) (↑a₂ ᵍ⊗ₜ[R] b₂))) =
    (-1) ^ (j₁ * i₂) • (a₁ * ↑a₂) ᵍ⊗ₜ[R] (↑b₁ * b₂)
tactic: dsimp only [mul_def, mulHom_apply, of_symm_of]
state_before: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : Module ι (Additive ℤˣ)
inst✝⁷ : DecidableEq ι
inst✝⁶ : CommRing R
inst✝⁵ : Ring A
inst✝⁴ : Ring B
inst✝³ : Algebra R A
inst✝² : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝¹ : GradedAlgebra 𝒜
inst✝ : GradedAlgebra ℬ
j₁ i₂ : ι
a₁ : A
b₁ : ↥(ℬ j₁)
a₂ : ↥(𝒜 i₂)
b₂ : B
⊢ (auxEquiv R 𝒜 ℬ).symm
      (((gradedMul R (fun x => ↥(𝒜 x)) fun x => ↥(ℬ x)) ((auxEquiv R 𝒜 ℬ) (a₁ ᵍ⊗ₜ[R] ↑b₁)))
        ((auxEquiv R 𝒜 ℬ) (↑a₂ ᵍ⊗ₜ[R] b₂))) =
    (-1) ^ (j₁ * i₂) • (a₁ * ↑a₂) ᵍ⊗ₜ[R] (↑b₁ * b₂)
state_after: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : Module ι (Additive ℤˣ)
inst✝⁷ : DecidableEq ι
inst✝⁶ : CommRing R
inst✝⁵ : Ring A
inst✝⁴ : Ring B
inst✝³ : Algebra R A
inst✝² : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝¹ : GradedAlgebra 𝒜
inst✝ : GradedAlgebra ℬ
j₁ i₂ : ι
a₁ : A
b₁ : ↥(ℬ j₁)
a₂ : ↥(𝒜 i₂)
b₂ : B
⊢ (of R 𝒜 ℬ).symm.symm
      ((TensorProduct.congr (decomposeAlgEquiv 𝒜).toLinearEquiv (decomposeAlgEquiv ℬ).toLinearEquiv).symm
        (((gradedMul R (fun x => ↥(𝒜 x)) fun x => ↥(ℬ x)) ((decompose 𝒜) a₁ ⊗ₜ[R] (decompose ℬ) ↑b₁))
          ((decompose 𝒜) ↑a₂ ⊗ₜ[R] (decompose ℬ) b₂))) =
    (-1) ^ (j₁ * i₂) • (of R 𝒜 ℬ) ((a₁ * ↑a₂) ⊗ₜ[R] (↑b₁ * b₂))
tactic: dsimp [auxEquiv, tmul]
state_before: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : Module ι (Additive ℤˣ)
inst✝⁷ : DecidableEq ι
inst✝⁶ : CommRing R
inst✝⁵ : Ring A
inst✝⁴ : Ring B
inst✝³ : Algebra R A
inst✝² : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝¹ : GradedAlgebra 𝒜
inst✝ : GradedAlgebra ℬ
j₁ i₂ : ι
a₁ : A
b₁ : ↥(ℬ j₁)
a₂ : ↥(𝒜 i₂)
b₂ : B
⊢ (of R 𝒜 ℬ).symm.symm
      ((TensorProduct.congr (decomposeAlgEquiv 𝒜).toLinearEquiv (decomposeAlgEquiv ℬ).toLinearEquiv).symm
        (((gradedMul R (fun x => ↥(𝒜 x)) fun x => ↥(ℬ x)) ((decompose 𝒜) a₁ ⊗ₜ[R] (decompose ℬ) ↑b₁))
          ((decompose 𝒜) ↑a₂ ⊗ₜ[R] (decompose ℬ) b₂))) =
    (-1) ^ (j₁ * i₂) • (of R 𝒜 ℬ) ((a₁ * ↑a₂) ⊗ₜ[R] (↑b₁ * b₂))
state_after: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : Module ι (Additive ℤˣ)
inst✝⁷ : DecidableEq ι
inst✝⁶ : CommRing R
inst✝⁵ : Ring A
inst✝⁴ : Ring B
inst✝³ : Algebra R A
inst✝² : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝¹ : GradedAlgebra 𝒜
inst✝ : GradedAlgebra ℬ
j₁ i₂ : ι
a₁ : A
b₁ : ↥(ℬ j₁)
a₂ : ↥(𝒜 i₂)
b₂ : B
⊢ (of R 𝒜 ℬ).symm.symm
      ((TensorProduct.congr (decomposeAlgEquiv 𝒜).toLinearEquiv (decomposeAlgEquiv ℬ).toLinearEquiv).symm
        (((gradedMul R (fun x => ↥(𝒜 x)) fun x => ↥(ℬ x))
            ((decompose 𝒜) a₁ ⊗ₜ[R] (DirectSum.of (fun i => ↥(ℬ i)) j₁) b₁))
          ((DirectSum.of (fun i => ↥(𝒜 i)) i₂) a₂ ⊗ₜ[R] (decompose ℬ) b₂))) =
    (-1) ^ (j₁ * i₂) • (of R 𝒜 ℬ) ((a₁ * ↑a₂) ⊗ₜ[R] (↑b₁ * b₂))
tactic: erw [decompose_coe, decompose_coe]
state_before: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : Module ι (Additive ℤˣ)
inst✝⁷ : DecidableEq ι
inst✝⁶ : CommRing R
inst✝⁵ : Ring A
inst✝⁴ : Ring B
inst✝³ : Algebra R A
inst✝² : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝¹ : GradedAlgebra 𝒜
inst✝ : GradedAlgebra ℬ
j₁ i₂ : ι
a₁ : A
b₁ : ↥(ℬ j₁)
a₂ : ↥(𝒜 i₂)
b₂ : B
⊢ (of R 𝒜 ℬ).symm.symm
      ((TensorProduct.congr (decomposeAlgEquiv 𝒜).toLinearEquiv (decomposeAlgEquiv ℬ).toLinearEquiv).symm
        (((gradedMul R (fun x => ↥(𝒜 x)) fun x => ↥(ℬ x))
            ((decompose 𝒜) a₁ ⊗ₜ[R] (DirectSum.of (fun i => ↥(ℬ i)) j₁) b₁))
          ((DirectSum.of (fun i => ↥(𝒜 i)) i₂) a₂ ⊗ₜ[R] (decompose ℬ) b₂))) =
    (-1) ^ (j₁ * i₂) • (of R 𝒜 ℬ) ((a₁ * ↑a₂) ⊗ₜ[R] (↑b₁ * b₂))
state_after: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : Module ι (Additive ℤˣ)
inst✝⁷ : DecidableEq ι
inst✝⁶ : CommRing R
inst✝⁵ : Ring A
inst✝⁴ : Ring B
inst✝³ : Algebra R A
inst✝² : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝¹ : GradedAlgebra 𝒜
inst✝ : GradedAlgebra ℬ
j₁ i₂ : ι
a₁ : A
b₁ : ↥(ℬ j₁)
a₂ : ↥(𝒜 i₂)
b₂ : B
⊢ (of R 𝒜 ℬ).symm.symm
      ((TensorProduct.congr (decomposeAlgEquiv 𝒜).toLinearEquiv (decomposeAlgEquiv ℬ).toLinearEquiv).symm
        (((gradedMul R (fun x => ↥(𝒜 x)) fun x => ↥(ℬ x)) ((decompose 𝒜) a₁ ⊗ₜ[R] (lof R ι (fun i => ↥(ℬ i)) j₁) b₁))
          ((lof R ι (fun i => ↥(𝒜 i)) i₂) a₂ ⊗ₜ[R] (decompose ℬ) b₂))) =
    (-1) ^ (j₁ * i₂) • (of R 𝒜 ℬ) ((a₁ * ↑a₂) ⊗ₜ[R] (↑b₁ * b₂))
tactic: simp_rw [← lof_eq_of R]
state_before: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : Module ι (Additive ℤˣ)
inst✝⁷ : DecidableEq ι
inst✝⁶ : CommRing R
inst✝⁵ : Ring A
inst✝⁴ : Ring B
inst✝³ : Algebra R A
inst✝² : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝¹ : GradedAlgebra 𝒜
inst✝ : GradedAlgebra ℬ
j₁ i₂ : ι
a₁ : A
b₁ : ↥(ℬ j₁)
a₂ : ↥(𝒜 i₂)
b₂ : B
⊢ (of R 𝒜 ℬ).symm.symm
      ((TensorProduct.congr (decomposeAlgEquiv 𝒜).toLinearEquiv (decomposeAlgEquiv ℬ).toLinearEquiv).symm
        (((gradedMul R (fun x => ↥(𝒜 x)) fun x => ↥(ℬ x)) ((decompose 𝒜) a₁ ⊗ₜ[R] (lof R ι (fun i => ↥(ℬ i)) j₁) b₁))
          ((lof R ι (fun i => ↥(𝒜 i)) i₂) a₂ ⊗ₜ[R] (decompose ℬ) b₂))) =
    (-1) ^ (j₁ * i₂) • (of R 𝒜 ℬ) ((a₁ * ↑a₂) ⊗ₜ[R] (↑b₁ * b₂))
state_after: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : Module ι (Additive ℤˣ)
inst✝⁷ : DecidableEq ι
inst✝⁶ : CommRing R
inst✝⁵ : Ring A
inst✝⁴ : Ring B
inst✝³ : Algebra R A
inst✝² : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝¹ : GradedAlgebra 𝒜
inst✝ : GradedAlgebra ℬ
j₁ i₂ : ι
a₁ : A
b₁ : ↥(ℬ j₁)
a₂ : ↥(𝒜 i₂)
b₂ : B
⊢ (of R 𝒜 ℬ).symm.symm
      ((TensorProduct.congr (decomposeAlgEquiv 𝒜).toLinearEquiv (decomposeAlgEquiv ℬ).toLinearEquiv).symm
        ((-1) ^ (j₁ * i₂) •
          ((decompose 𝒜) a₁ * (lof R ι (fun x => ↥(𝒜 x)) i₂) a₂) ⊗ₜ[R]
            ((lof R ι (fun x => ↥(ℬ x)) j₁) b₁ * (decompose ℬ) b₂))) =
    (-1) ^ (j₁ * i₂) • (of R 𝒜 ℬ) ((a₁ * ↑a₂) ⊗ₜ[R] (↑b₁ * b₂))
tactic: rw [tmul_of_gradedMul_of_tmul]
state_before: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : Module ι (Additive ℤˣ)
inst✝⁷ : DecidableEq ι
inst✝⁶ : CommRing R
inst✝⁵ : Ring A
inst✝⁴ : Ring B
inst✝³ : Algebra R A
inst✝² : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝¹ : GradedAlgebra 𝒜
inst✝ : GradedAlgebra ℬ
j₁ i₂ : ι
a₁ : A
b₁ : ↥(ℬ j₁)
a₂ : ↥(𝒜 i₂)
b₂ : B
⊢ (of R 𝒜 ℬ).symm.symm
      ((TensorProduct.congr (decomposeAlgEquiv 𝒜).toLinearEquiv (decomposeAlgEquiv ℬ).toLinearEquiv).symm
        ((-1) ^ (j₁ * i₂) •
          ((decompose 𝒜) a₁ * (lof R ι (fun x => ↥(𝒜 x)) i₂) a₂) ⊗ₜ[R]
            ((lof R ι (fun x => ↥(ℬ x)) j₁) b₁ * (decompose ℬ) b₂))) =
    (-1) ^ (j₁ * i₂) • (of R 𝒜 ℬ) ((a₁ * ↑a₂) ⊗ₜ[R] (↑b₁ * b₂))
state_after: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : Module ι (Additive ℤˣ)
inst✝⁷ : DecidableEq ι
inst✝⁶ : CommRing R
inst✝⁵ : Ring A
inst✝⁴ : Ring B
inst✝³ : Algebra R A
inst✝² : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝¹ : GradedAlgebra 𝒜
inst✝ : GradedAlgebra ℬ
j₁ i₂ : ι
a₁ : A
b₁ : ↥(ℬ j₁)
a₂ : ↥(𝒜 i₂)
b₂ : B
⊢ (of R 𝒜 ℬ).symm.symm
      ((TensorProduct.congr (decomposeAlgEquiv 𝒜).toLinearEquiv (decomposeAlgEquiv ℬ).toLinearEquiv).symm
        ((-1) ^ (j₁ * i₂) •
          ((decompose 𝒜) a₁ * (DirectSum.of (fun x => ↥(𝒜 x)) i₂) a₂) ⊗ₜ[R]
            ((DirectSum.of (fun x => ↥(ℬ x)) j₁) b₁ * (decompose ℬ) b₂))) =
    (-1) ^ (j₁ * i₂) • (of R 𝒜 ℬ) ((a₁ * ↑a₂) ⊗ₜ[R] (↑b₁ * b₂))
tactic: simp_rw [lof_eq_of R]
state_before: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : Module ι (Additive ℤˣ)
inst✝⁷ : DecidableEq ι
inst✝⁶ : CommRing R
inst✝⁵ : Ring A
inst✝⁴ : Ring B
inst✝³ : Algebra R A
inst✝² : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝¹ : GradedAlgebra 𝒜
inst✝ : GradedAlgebra ℬ
j₁ i₂ : ι
a₁ : A
b₁ : ↥(ℬ j₁)
a₂ : ↥(𝒜 i₂)
b₂ : B
⊢ (of R 𝒜 ℬ).symm.symm
      ((TensorProduct.congr (decomposeAlgEquiv 𝒜).toLinearEquiv (decomposeAlgEquiv ℬ).toLinearEquiv).symm
        ((-1) ^ (j₁ * i₂) •
          ((decompose 𝒜) a₁ * (DirectSum.of (fun x => ↥(𝒜 x)) i₂) a₂) ⊗ₜ[R]
            ((DirectSum.of (fun x => ↥(ℬ x)) j₁) b₁ * (decompose ℬ) b₂))) =
    (-1) ^ (j₁ * i₂) • (of R 𝒜 ℬ) ((a₁ * ↑a₂) ⊗ₜ[R] (↑b₁ * b₂))
state_after: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : Module ι (Additive ℤˣ)
inst✝⁷ : DecidableEq ι
inst✝⁶ : CommRing R
inst✝⁵ : Ring A
inst✝⁴ : Ring B
inst✝³ : Algebra R A
inst✝² : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝¹ : GradedAlgebra 𝒜
inst✝ : GradedAlgebra ℬ
j₁ i₂ : ι
a₁ : A
b₁ : ↥(ℬ j₁)
a₂ : ↥(𝒜 i₂)
b₂ : B
⊢ (of R 𝒜 ℬ)
      ((TensorProduct.congr (decomposeAlgEquiv 𝒜).toLinearEquiv (decomposeAlgEquiv ℬ).toLinearEquiv).symm
        ((-1) ^ (j₁ * i₂) •
          ((decompose 𝒜) a₁ * (DirectSum.of (fun x => ↥(𝒜 x)) i₂) a₂) ⊗ₜ[R]
            ((DirectSum.of (fun x => ↥(ℬ x)) j₁) b₁ * (decompose ℬ) b₂))) =
    (-1) ^ (j₁ * i₂) • (of R 𝒜 ℬ) ((a₁ * ↑a₂) ⊗ₜ[R] (↑b₁ * b₂))
tactic: rw [LinearEquiv.symm_symm]
state_before: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : Module ι (Additive ℤˣ)
inst✝⁷ : DecidableEq ι
inst✝⁶ : CommRing R
inst✝⁵ : Ring A
inst✝⁴ : Ring B
inst✝³ : Algebra R A
inst✝² : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝¹ : GradedAlgebra 𝒜
inst✝ : GradedAlgebra ℬ
j₁ i₂ : ι
a₁ : A
b₁ : ↥(ℬ j₁)
a₂ : ↥(𝒜 i₂)
b₂ : B
⊢ (of R 𝒜 ℬ)
      ((TensorProduct.congr (decomposeAlgEquiv 𝒜).toLinearEquiv (decomposeAlgEquiv ℬ).toLinearEquiv).symm
        ((-1) ^ (j₁ * i₂) •
          ((decompose 𝒜) a₁ * (DirectSum.of (fun x => ↥(𝒜 x)) i₂) a₂) ⊗ₜ[R]
            ((DirectSum.of (fun x => ↥(ℬ x)) j₁) b₁ * (decompose ℬ) b₂))) =
    (-1) ^ (j₁ * i₂) • (of R 𝒜 ℬ) ((a₁ * ↑a₂) ⊗ₜ[R] (↑b₁ * b₂))
state_after: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : Module ι (Additive ℤˣ)
inst✝⁷ : DecidableEq ι
inst✝⁶ : CommRing R
inst✝⁵ : Ring A
inst✝⁴ : Ring B
inst✝³ : Algebra R A
inst✝² : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝¹ : GradedAlgebra 𝒜
inst✝ : GradedAlgebra ℬ
j₁ i₂ : ι
a₁ : A
b₁ : ↥(ℬ j₁)
a₂ : ↥(𝒜 i₂)
b₂ : B
⊢ (-1) ^ (j₁ * i₂) •
      (of R 𝒜 ℬ)
        ((TensorProduct.congr (decomposeAlgEquiv 𝒜).toLinearEquiv (decomposeAlgEquiv ℬ).toLinearEquiv).symm
          (((decompose 𝒜) a₁ * (DirectSum.of (fun x => ↥(𝒜 x)) i₂) a₂) ⊗ₜ[R]
            ((DirectSum.of (fun x => ↥(ℬ x)) j₁) b₁ * (decompose ℬ) b₂))) =
    (-1) ^ (j₁ * i₂) • (of R 𝒜 ℬ) ((a₁ * ↑a₂) ⊗ₜ[R] (↑b₁ * b₂))
tactic: rw [@Units.smul_def _ _ (_) (_), ← Int.cast_smul_eq_nsmul R, LinearEquiv.map_smul, map_smul,
    Int.cast_smul_eq_nsmul R, ← @Units.smul_def _ _ (_) (_)]
state_before: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : Module ι (Additive ℤˣ)
inst✝⁷ : DecidableEq ι
inst✝⁶ : CommRing R
inst✝⁵ : Ring A
inst✝⁴ : Ring B
inst✝³ : Algebra R A
inst✝² : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝¹ : GradedAlgebra 𝒜
inst✝ : GradedAlgebra ℬ
j₁ i₂ : ι
a₁ : A
b₁ : ↥(ℬ j₁)
a₂ : ↥(𝒜 i₂)
b₂ : B
⊢ (-1) ^ (j₁ * i₂) •
      (of R 𝒜 ℬ)
        ((TensorProduct.congr (decomposeAlgEquiv 𝒜).toLinearEquiv (decomposeAlgEquiv ℬ).toLinearEquiv).symm
          (((decompose 𝒜) a₁ * (DirectSum.of (fun x => ↥(𝒜 x)) i₂) a₂) ⊗ₜ[R]
            ((DirectSum.of (fun x => ↥(ℬ x)) j₁) b₁ * (decompose ℬ) b₂))) =
    (-1) ^ (j₁ * i₂) • (of R 𝒜 ℬ) ((a₁ * ↑a₂) ⊗ₜ[R] (↑b₁ * b₂))
state_after: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : Module ι (Additive ℤˣ)
inst✝⁷ : DecidableEq ι
inst✝⁶ : CommRing R
inst✝⁵ : Ring A
inst✝⁴ : Ring B
inst✝³ : Algebra R A
inst✝² : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝¹ : GradedAlgebra 𝒜
inst✝ : GradedAlgebra ℬ
j₁ i₂ : ι
a₁ : A
b₁ : ↥(ℬ j₁)
a₂ : ↥(𝒜 i₂)
b₂ : B
⊢ (-1) ^ (j₁ * i₂) •
      (of R 𝒜 ℬ)
        ((decomposeAlgEquiv 𝒜).toLinearEquiv.symm ((decompose 𝒜) a₁ * (DirectSum.of (fun x => ↥(𝒜 x)) i₂) a₂) ⊗ₜ[R]
          (decomposeAlgEquiv ℬ).toLinearEquiv.symm ((DirectSum.of (fun x => ↥(ℬ x)) j₁) b₁ * (decompose ℬ) b₂)) =
    (-1) ^ (j₁ * i₂) • (of R 𝒜 ℬ) ((a₁ * ↑a₂) ⊗ₜ[R] (↑b₁ * b₂))
tactic: rw [congr_symm_tmul]
state_before: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : Module ι (Additive ℤˣ)
inst✝⁷ : DecidableEq ι
inst✝⁶ : CommRing R
inst✝⁵ : Ring A
inst✝⁴ : Ring B
inst✝³ : Algebra R A
inst✝² : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝¹ : GradedAlgebra 𝒜
inst✝ : GradedAlgebra ℬ
j₁ i₂ : ι
a₁ : A
b₁ : ↥(ℬ j₁)
a₂ : ↥(𝒜 i₂)
b₂ : B
⊢ (-1) ^ (j₁ * i₂) •
      (of R 𝒜 ℬ)
        ((decomposeAlgEquiv 𝒜).toLinearEquiv.symm ((decompose 𝒜) a₁ * (DirectSum.of (fun x => ↥(𝒜 x)) i₂) a₂) ⊗ₜ[R]
          (decomposeAlgEquiv ℬ).toLinearEquiv.symm ((DirectSum.of (fun x => ↥(ℬ x)) j₁) b₁ * (decompose ℬ) b₂)) =
    (-1) ^ (j₁ * i₂) • (of R 𝒜 ℬ) ((a₁ * ↑a₂) ⊗ₜ[R] (↑b₁ * b₂))
state_after: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : Module ι (Additive ℤˣ)
inst✝⁷ : DecidableEq ι
inst✝⁶ : CommRing R
inst✝⁵ : Ring A
inst✝⁴ : Ring B
inst✝³ : Algebra R A
inst✝² : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝¹ : GradedAlgebra 𝒜
inst✝ : GradedAlgebra ℬ
j₁ i₂ : ι
a₁ : A
b₁ : ↥(ℬ j₁)
a₂ : ↥(𝒜 i₂)
b₂ : B
⊢ (-1) ^ (j₁ * i₂) •
      (of R 𝒜 ℬ)
        ((decompose 𝒜).symm ((decompose 𝒜) a₁ * (DirectSum.of (fun x => ↥(𝒜 x)) i₂) a₂) ⊗ₜ[R]
          (decompose ℬ).symm ((DirectSum.of (fun x => ↥(ℬ x)) j₁) b₁ * (decompose ℬ) b₂)) =
    (-1) ^ (j₁ * i₂) • (of R 𝒜 ℬ) ((a₁ * ↑a₂) ⊗ₜ[R] (↑b₁ * b₂))
tactic: dsimp
state_before: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : Module ι (Additive ℤˣ)
inst✝⁷ : DecidableEq ι
inst✝⁶ : CommRing R
inst✝⁵ : Ring A
inst✝⁴ : Ring B
inst✝³ : Algebra R A
inst✝² : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝¹ : GradedAlgebra 𝒜
inst✝ : GradedAlgebra ℬ
j₁ i₂ : ι
a₁ : A
b₁ : ↥(ℬ j₁)
a₂ : ↥(𝒜 i₂)
b₂ : B
⊢ (-1) ^ (j₁ * i₂) •
      (of R 𝒜 ℬ)
        ((decompose 𝒜).symm ((decompose 𝒜) a₁ * (DirectSum.of (fun x => ↥(𝒜 x)) i₂) a₂) ⊗ₜ[R]
          (decompose ℬ).symm ((DirectSum.of (fun x => ↥(ℬ x)) j₁) b₁ * (decompose ℬ) b₂)) =
    (-1) ^ (j₁ * i₂) • (of R 𝒜 ℬ) ((a₁ * ↑a₂) ⊗ₜ[R] (↑b₁ * b₂))
state_after: no goals
tactic: simp_rw [decompose_symm_mul, decompose_symm_of, Equiv.symm_apply_apply]