mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.External

Goal state before:
pp: R : Type u_1
ι : Type u_2
inst✝¹¹ : CommSemiring ι
inst✝¹⁰ : Module ι (Additive ℤˣ)
inst✝⁹ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)
inst✝⁶ : (i : ι) → AddCommGroup (ℬ i)
inst✝⁵ : (i : ι) → Module R (𝒜 i)
inst✝⁴ : (i : ι) → Module R (ℬ i)
inst✝³ : DirectSum.GRing 𝒜
inst✝² : DirectSum.GRing ℬ
inst✝¹ : DirectSum.GAlgebra R 𝒜
inst✝ : DirectSum.GAlgebra R ℬ
r : R
x : (⨁ (i : ι), 𝒜 i) ⊗[R] ⨁ (i : ι), ℬ i
this :
  (gradedMul R 𝒜 ℬ) ((algebraMap R (DirectSum ι 𝒜)) r ⊗ₜ[R] 1) =
    DistribSMul.toLinearMap R (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) r
⊢ ((gradedMul R 𝒜 ℬ) ((algebraMap R (DirectSum ι 𝒜)) r ⊗ₜ[R] 1)) x = r • x

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.External

Goal state before:
pp: R : Type u_1
ι : Type u_2
inst✝¹¹ : CommSemiring ι
inst✝¹⁰ : Module ι (Additive ℤˣ)
inst✝⁹ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)
inst✝⁶ : (i : ι) → AddCommGroup (ℬ i)
inst✝⁵ : (i : ι) → Module R (𝒜 i)
inst✝⁴ : (i : ι) → Module R (ℬ i)
inst✝³ : DirectSum.GRing 𝒜
inst✝² : DirectSum.GRing ℬ
inst✝¹ : DirectSum.GAlgebra R 𝒜
inst✝ : DirectSum.GAlgebra R ℬ
r : R
x : (⨁ (i : ι), 𝒜 i) ⊗[R] ⨁ (i : ι), ℬ i
⊢ ((gradedMul R 𝒜 ℬ) ((algebraMap R (DirectSum ι 𝒜)) r ⊗ₜ[R] 1)) x = r • x

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.External

Goal state before:
pp: R : Type u_1
ι : Type u_2
inst✝¹¹ : CommSemiring ι
inst✝¹⁰ : Module ι (Additive ℤˣ)
inst✝⁹ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)
inst✝⁶ : (i : ι) → AddCommGroup (ℬ i)
inst✝⁵ : (i : ι) → Module R (𝒜 i)
inst✝⁴ : (i : ι) → Module R (ℬ i)
inst✝³ : DirectSum.GRing 𝒜
inst✝² : DirectSum.GRing ℬ
inst✝¹ : DirectSum.GAlgebra R 𝒜
inst✝ : DirectSum.GAlgebra R ℬ
x : (⨁ (i : ι), 𝒜 i) ⊗[R] ⨁ (i : ι), ℬ i
⊢ ((gradedMul R 𝒜 ℬ) 1) x = x

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.External

Goal state before:
pp: R : Type u_1
ι : Type u_2
inst✝¹¹ : CommSemiring ι
inst✝¹⁰ : Module ι (Additive ℤˣ)
inst✝⁹ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)
inst✝⁶ : (i : ι) → AddCommGroup (ℬ i)
inst✝⁵ : (i : ι) → Module R (𝒜 i)
inst✝⁴ : (i : ι) → Module R (ℬ i)
inst✝³ : DirectSum.GRing 𝒜
inst✝² : DirectSum.GRing ℬ
inst✝¹ : DirectSum.GAlgebra R 𝒜
inst✝ : DirectSum.GAlgebra R ℬ
x : (⨁ (i : ι), 𝒜 i) ⊗[R] ⨁ (i : ι), ℬ i
r : R
this :
  (gradedMul R 𝒜 ℬ).flip ((algebraMap R (DirectSum ι 𝒜)) r ⊗ₜ[R] 1) =
    DistribSMul.toLinearMap R (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) r
⊢ ((gradedMul R 𝒜 ℬ) x) ((algebraMap R (DirectSum ι 𝒜)) r ⊗ₜ[R] 1) = r • x

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
Q : QuadraticForm R M
x✝ : (CliffordAlgebra Q)ˣ
inst✝ : Invertible 2
x : (CliffordAlgebra Q)ˣ
m a : M
ha : (ι Q) a = ↑x
this : Invertible ↑x := x.invertible
⊢ Invertible ((ι Q) a)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.External

Goal state before:
pp: R : Type u_1
ι : Type u_2
inst✝¹¹ : CommSemiring ι
inst✝¹⁰ : Module ι (Additive ℤˣ)
inst✝⁹ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)
inst✝⁶ : (i : ι) → AddCommGroup (ℬ i)
inst✝⁵ : (i : ι) → Module R (𝒜 i)
inst✝⁴ : (i : ι) → Module R (ℬ i)
inst✝³ : DirectSum.GRing 𝒜
inst✝² : DirectSum.GRing ℬ
inst✝¹ : DirectSum.GAlgebra R 𝒜
inst✝ : DirectSum.GAlgebra R ℬ
x : (⨁ (i : ι), 𝒜 i) ⊗[R] ⨁ (i : ι), ℬ i
r : R
⊢ ((gradedMul R 𝒜 ℬ) x) ((algebraMap R (DirectSum ι 𝒜)) r ⊗ₜ[R] 1) = r • x

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Goal state before:
pp: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁸ : CommSemiring ι
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
⊢ (auxEquiv R 𝒜 ℬ) 1 = 1

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.External

Goal state before:
pp: R : Type u_1
ι : Type u_2
inst✝¹¹ : CommSemiring ι
inst✝¹⁰ : Module ι (Additive ℤˣ)
inst✝⁹ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)
inst✝⁶ : (i : ι) → AddCommGroup (ℬ i)
inst✝⁵ : (i : ι) → Module R (𝒜 i)
inst✝⁴ : (i : ι) → Module R (ℬ i)
inst✝³ : DirectSum.GRing 𝒜
inst✝² : DirectSum.GRing ℬ
inst✝¹ : DirectSum.GAlgebra R 𝒜
inst✝ : DirectSum.GAlgebra R ℬ
x : (⨁ (i : ι), 𝒜 i) ⊗[R] ⨁ (i : ι), ℬ i
⊢ ((gradedMul R 𝒜 ℬ) x) 1 = x

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Goal state before:
pp: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : DecidableEq ι
inst✝⁷ : CommRing R
inst✝⁶ : Ring A
inst✝⁵ : Ring B
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝² : GradedAlgebra 𝒜
inst✝¹ : GradedAlgebra ℬ
inst✝ : Module ι (Additive ℤˣ)
⊢ 𝒜 ᵍ⊗[R] ℬ →ₗ[R] 𝒜 ᵍ⊗[R] ℬ →ₗ[R] 𝒜 ᵍ⊗[R] ℬ

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.External

Goal state before:
pp: R : Type u_1
ι : Type u_2
inst✝¹¹ : CommSemiring ι
inst✝¹⁰ : Module ι (Additive ℤˣ)
inst✝⁹ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)
inst✝⁶ : (i : ι) → AddCommGroup (ℬ i)
inst✝⁵ : (i : ι) → Module R (𝒜 i)
inst✝⁴ : (i : ι) → Module R (ℬ i)
inst✝³ : DirectSum.GRing 𝒜
inst✝² : DirectSum.GRing ℬ
inst✝¹ : DirectSum.GAlgebra R 𝒜
inst✝ : 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
this :
  (LinearMap.llcomp R (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)
          (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ))
        mA ∘ₗ
      mA =
    (((LinearMap.llcomp 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)
            (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))
          ↑LinearMap.lflip)
        ((LinearMap.llcomp R (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ) (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ)
              (DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ →ₗ[R] DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ))
            mA.flip ∘ₗ
          mA)).flip
⊢ ((gradedMul R 𝒜 ℬ) (((gradedMul R 𝒜 ℬ) x) y)) z = ((gradedMul R 𝒜 ℬ) x) (((gradedMul R 𝒜 ℬ) y) z)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Goal state before:
pp: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : DecidableEq ι
inst✝⁷ : CommRing R
inst✝⁶ : Ring A
inst✝⁵ : Ring B
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝² : GradedAlgebra 𝒜
inst✝¹ : GradedAlgebra ℬ
inst✝ : Module ι (Additive ℤˣ)
x y z : 𝒜 ᵍ⊗[R] ℬ
⊢ x * y * z = x * (y * z)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Goal state before:
pp: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : DecidableEq ι
inst✝⁷ : CommRing R
inst✝⁶ : Ring A
inst✝⁵ : Ring B
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝² : GradedAlgebra 𝒜
inst✝¹ : GradedAlgebra ℬ
inst✝ : Module ι (Additive ℤˣ)
x : 𝒜 ᵍ⊗[R] ℬ
⊢ 1 * x = x

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Goal state before:
pp: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : DecidableEq ι
inst✝⁷ : CommRing R
inst✝⁶ : Ring A
inst✝⁵ : Ring B
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝² : GradedAlgebra 𝒜
inst✝¹ : GradedAlgebra ℬ
inst✝ : Module ι (Additive ℤˣ)
x : 𝒜 ᵍ⊗[R] ℬ
⊢ x * 1 = x

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Goal state before:
pp: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : DecidableEq ι
inst✝⁷ : CommRing R
inst✝⁶ : Ring A
inst✝⁵ : Ring B
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝² : GradedAlgebra 𝒜
inst✝¹ : GradedAlgebra ℬ
inst✝ : Module ι (Additive ℤˣ)
x : 𝒜 ᵍ⊗[R] ℬ
⊢ 0 * x = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Goal state before:
pp: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : DecidableEq ι
inst✝⁷ : CommRing R
inst✝⁶ : Ring A
inst✝⁵ : Ring B
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝² : GradedAlgebra 𝒜
inst✝¹ : GradedAlgebra ℬ
inst✝ : Module ι (Additive ℤˣ)
x : 𝒜 ᵍ⊗[R] ℬ
⊢ x * 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
Q : QuadraticForm R M
x : (CliffordAlgebra Q)ˣ
inst✝ : Invertible 2
y z : (CliffordAlgebra Q)ˣ
hx✝ : y ∈ Subgroup.closure (Units.val ⁻¹' Set.range ⇑(ι Q))
hy✝ : z ∈ Subgroup.closure (Units.val ⁻¹' Set.range ⇑(ι Q))
hy : ∀ (m : M), ↑y * (ι Q) m * ↑y⁻¹ ∈ (ι Q).range
hz : ∀ (m : M), ↑z * (ι Q) m * ↑z⁻¹ ∈ (ι Q).range
m : M
this : ↑y * (↑z * (ι Q) m * ↑z⁻¹) * ↑y⁻¹ ∈ ?m.199
⊢ ↑y * ↑z * (ι Q) m * (↑z⁻¹ * ↑y⁻¹) ∈ (ι Q).range

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Goal state before:
pp: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : DecidableEq ι
inst✝⁷ : CommRing R
inst✝⁶ : Ring A
inst✝⁵ : Ring B
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝² : GradedAlgebra 𝒜
inst✝¹ : GradedAlgebra ℬ
inst✝ : Module ι (Additive ℤˣ)
x y z : 𝒜 ᵍ⊗[R] ℬ
⊢ x * (y + z) = x * y + x * z

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Goal state before:
pp: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : DecidableEq ι
inst✝⁷ : CommRing R
inst✝⁶ : Ring A
inst✝⁵ : Ring B
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝² : GradedAlgebra 𝒜
inst✝¹ : GradedAlgebra ℬ
inst✝ : Module ι (Additive ℤˣ)
x y z : 𝒜 ᵍ⊗[R] ℬ
⊢ (x + y) * z = x * z + y * z

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
Q : QuadraticForm R M
x : (CliffordAlgebra Q)ˣ
hx : x ∈ lipschitzGroup Q
inst✝ : Invertible 2
m : M
⊢ ConjAct.toConjAct x • (ι Q) m ∈ (ι Q).range

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
Q : QuadraticForm R M
inst✝ : Invertible 2
x✝ x : (CliffordAlgebra Q)ˣ
b a : M
ha : (ι Q) a = ↑x
this : Invertible ↑x := x.invertible
⊢ Invertible ((ι Q) a)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.External

Goal state before:
pp: R : Type u_1
ι : Type u_2
inst✝¹¹ : CommSemiring ι
inst✝¹⁰ : Module ι (Additive ℤˣ)
inst✝⁹ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)
inst✝⁶ : (i : ι) → AddCommGroup (ℬ i)
inst✝⁵ : (i : ι) → Module R (𝒜 i)
inst✝⁴ : (i : ι) → Module R (ℬ i)
inst✝³ : DirectSum.GRing 𝒜
inst✝² : DirectSum.GRing ℬ
inst✝¹ : DirectSum.GAlgebra R 𝒜
inst✝ : DirectSum.GAlgebra R ℬ
x y z : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ
⊢ ((gradedMul R 𝒜 ℬ) (((gradedMul R 𝒜 ℬ) x) y)) z = ((gradedMul R 𝒜 ℬ) x) (((gradedMul R 𝒜 ℬ) y) z)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Goal state before:
pp: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : DecidableEq ι
inst✝⁷ : CommRing R
inst✝⁶ : Ring A
inst✝⁵ : Ring B
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝² : GradedAlgebra 𝒜
inst✝¹ : GradedAlgebra ℬ
inst✝ : Module ι (Additive ℤˣ)
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)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Goal state before:
pp: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : DecidableEq ι
inst✝⁷ : CommRing R
inst✝⁶ : Ring A
inst✝⁵ : Ring B
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝² : GradedAlgebra 𝒜
inst✝¹ : GradedAlgebra ℬ
inst✝ : Module ι (Additive ℤˣ)
i₂ : ι
a₁ : A
b₁ : ↥(ℬ 0)
a₂ : ↥(𝒜 i₂)
b₂ : B
⊢ a₁ ᵍ⊗ₜ[R] ↑b₁ * ↑a₂ ᵍ⊗ₜ[R] b₂ = (a₁ * ↑a₂) ᵍ⊗ₜ[R] (↑b₁ * b₂)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Goal state before:
pp: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : DecidableEq ι
inst✝⁷ : CommRing R
inst✝⁶ : Ring A
inst✝⁵ : Ring B
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝² : GradedAlgebra 𝒜
inst✝¹ : GradedAlgebra ℬ
inst✝ : Module ι (Additive ℤˣ)
j₁ : ι
a₁ : A
b₁ : ↥(ℬ j₁)
a₂ : ↥(𝒜 0)
b₂ : B
⊢ a₁ ᵍ⊗ₜ[R] ↑b₁ * ↑a₂ ᵍ⊗ₜ[R] b₂ = (a₁ * ↑a₂) ᵍ⊗ₜ[R] (↑b₁ * b₂)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Goal state before:
pp: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : DecidableEq ι
inst✝⁷ : CommRing R
inst✝⁶ : Ring A
inst✝⁵ : Ring B
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝² : GradedAlgebra 𝒜
inst✝¹ : GradedAlgebra ℬ
inst✝ : Module ι (Additive ℤˣ)
i₂ : ι
a₁ : A
a₂ : ↥(𝒜 i₂)
b₂ : B
⊢ a₁ ᵍ⊗ₜ[R] 1 * ↑a₂ ᵍ⊗ₜ[R] b₂ = (a₁ * ↑a₂) ᵍ⊗ₜ[R] b₂

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Goal state before:
pp: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : DecidableEq ι
inst✝⁷ : CommRing R
inst✝⁶ : Ring A
inst✝⁵ : Ring B
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝² : GradedAlgebra 𝒜
inst✝¹ : GradedAlgebra ℬ
inst✝ : Module ι (Additive ℤˣ)
j₁ : ι
a₁ : A
b₁ : ↥(ℬ j₁)
b₂ : B
⊢ a₁ ᵍ⊗ₜ[R] ↑b₁ * 1 ᵍ⊗ₜ[R] b₂ = a₁ ᵍ⊗ₜ[R] (↑b₁ * b₂)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
Q : QuadraticForm R M
inst✝ : Invertible 2
x y z : (CliffordAlgebra Q)ˣ
hx✝ : y ∈ Subgroup.closure (Units.val ⁻¹' Set.range ⇑(ι Q))
hy✝ : z ∈ Subgroup.closure (Units.val ⁻¹' Set.range ⇑(ι Q))
hy : ∀ (b : M), involute ↑y * (ι Q) b * ↑y⁻¹ ∈ (ι Q).range
hz : ∀ (b : M), involute ↑z * (ι Q) b * ↑z⁻¹ ∈ (ι Q).range
b : M
this : involute ↑y * (involute ↑z * (ι Q) b * ↑z⁻¹) * ↑y⁻¹ ∈ ?m.248
⊢ involute ↑y * involute ↑z * (ι Q) b * (↑z⁻¹ * ↑y⁻¹) ∈ (ι Q).range

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Goal state before:
pp: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : DecidableEq ι
inst✝⁷ : CommRing R
inst✝⁶ : Ring A
inst✝⁵ : Ring B
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝² : GradedAlgebra 𝒜
inst✝¹ : GradedAlgebra ℬ
inst✝ : Module ι (Additive ℤˣ)
a₁ : A
b₂ : B
⊢ a₁ ᵍ⊗ₜ[R] 1 * 1 ᵍ⊗ₜ[R] b₂ = a₁ ᵍ⊗ₜ[R] b₂

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Goal state before:
pp: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : DecidableEq ι
inst✝⁷ : CommRing R
inst✝⁶ : Ring A
inst✝⁵ : Ring B
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝² : GradedAlgebra 𝒜
inst✝¹ : GradedAlgebra ℬ
inst✝ : Module ι (Additive ℤˣ)
a₁ a₂ : A
⊢ (∑ i ∈ DFinsupp.support ((decompose 𝒜) a₂), a₁ * ↑(((decompose 𝒜) a₂) i)) ᵍ⊗ₜ[R] 1 =
    a₁ ᵍ⊗ₜ[R] 1 * (∑ i ∈ DFinsupp.support ((decompose 𝒜) a₂), ↑(((decompose 𝒜) a₂) i)) ᵍ⊗ₜ[R] 1

Tactic:
simp_rw [tmul, sum_tmul, map_sum, Finset.mul_sum]

Goal state after:
R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : DecidableEq ι
inst✝⁷ : CommRing R
inst✝⁶ : Ring A
inst✝⁵ : Ring B
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝² : GradedAlgebra 𝒜
inst✝¹ : GradedAlgebra ℬ
inst✝ : Module ι (Additive ℤˣ)
a₁ a₂ : A
⊢ ∑ x ∈ DFinsupp.support ((decompose 𝒜) a₂), (of R 𝒜 ℬ) ((a₁ * ↑(((decompose 𝒜) a₂) x)) ⊗ₜ[R] 1) =
    ∑ i ∈ DFinsupp.support ((decompose 𝒜) a₂), (of R 𝒜 ℬ) (a₁ ⊗ₜ[R] 1) * (of R 𝒜 ℬ) (↑(((decompose 𝒜) a₂) i) ⊗ₜ[R] 1)
Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
Q : QuadraticForm R M
inst✝ : Invertible 2
x : (CliffordAlgebra Q)ˣ
hx : x ∈ lipschitzGroup Q
b : M
⊢ involute ↑x * (ι Q) b * ↑x⁻¹ ∈ (ι Q).range

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
Q : QuadraticForm R M
x : (CliffordAlgebra Q)ˣ
hx : x ∈ lipschitzGroup Q
inst✝ : Invertible 2
this : ∀ x ∈ lipschitzGroup Q, ConjAct.toConjAct x • (ι Q).range ≤ (ι Q).range
⊢ ConjAct.toConjAct x • (ι Q).range = (ι Q).range

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Goal state before:
pp: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : DecidableEq ι
inst✝⁷ : CommRing R
inst✝⁶ : Ring A
inst✝⁵ : Ring B
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝² : GradedAlgebra 𝒜
inst✝¹ : GradedAlgebra ℬ
inst✝ : Module ι (Additive ℤˣ)
a₁ a₂ : A
⊢ (a₁ * a₂) ᵍ⊗ₜ[R] 1 = a₁ ᵍ⊗ₜ[R] 1 * a₂ ᵍ⊗ₜ[R] 1

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Prod

Goal state before:
pp: R : Type u_1
M₁ : Type u_2
M₂ : Type u_3
N : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M₁
inst✝⁴ : AddCommGroup M₂
inst✝³ : AddCommGroup N
inst✝² : Module R M₁
inst✝¹ : Module R M₂
inst✝ : Module R N
Q₁ : QuadraticForm R M₁
Q₂ : QuadraticForm R M₂
Qₙ : QuadraticForm R N
f₁ : Q₁ →qᵢ Qₙ
f₂ : Q₂ →qᵢ Qₙ
hf : ∀ (x : M₁) (y : M₂), QuadraticMap.IsOrtho Qₙ (f₁ x) (f₂ y)
m₁ : CliffordAlgebra Q₁
m₂ : CliffordAlgebra Q₂
i₁ i₂ : ZMod 2
hm₁ : m₁ ∈ evenOdd Q₁ i₁
hm₂ : m₂ ∈ evenOdd Q₂ i₂
⊢ (map f₁) m₁ * (map f₂) m₂ = (-1) ^ (i₂ * i₁) • ((map f₂) m₂ * (map f₁) m₁)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Goal state before:
pp: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : DecidableEq ι
inst✝⁷ : CommRing R
inst✝⁶ : Ring A
inst✝⁵ : Ring B
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝² : GradedAlgebra 𝒜
inst✝¹ : GradedAlgebra ℬ
inst✝ : Module ι (Additive ℤˣ)
⊢ 0 ᵍ⊗ₜ[R] 1 = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Goal state before:
pp: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : DecidableEq ι
inst✝⁷ : CommRing R
inst✝⁶ : Ring A
inst✝⁵ : Ring B
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝² : GradedAlgebra 𝒜
inst✝¹ : GradedAlgebra ℬ
inst✝ : Module ι (Additive ℤˣ)
⊢ ∀ (x y : A), (x + y) ᵍ⊗ₜ[R] 1 = x ᵍ⊗ₜ[R] 1 + y ᵍ⊗ₜ[R] 1

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.External

Goal state before:
pp: R : Type u_1
ι : Type u_2
inst✝¹¹ : CommSemiring ι
inst✝¹⁰ : Module ι (Additive ℤˣ)
inst✝⁹ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)
inst✝⁶ : (i : ι) → AddCommGroup (ℬ i)
inst✝⁵ : (i : ι) → Module R (𝒜 i)
inst✝⁴ : (i : ι) → Module R (ℬ i)
inst✝³ : DirectSum.GRing 𝒜
inst✝² : DirectSum.GRing ℬ
inst✝¹ : DirectSum.GAlgebra R 𝒜
inst✝ : DirectSum.GAlgebra R ℬ
x y : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ
⊢ (gradedComm R 𝒜 ℬ) (((gradedMul R 𝒜 ℬ) x) y) = ((gradedMul R ℬ 𝒜) ((gradedComm R 𝒜 ℬ) x)) ((gradedComm R 𝒜 ℬ) y)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Prod

Goal state before:
pp: R : Type u_1
M₁ : Type u_2
M₂ : Type u_3
N : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M₁
inst✝⁴ : AddCommGroup M₂
inst✝³ : AddCommGroup N
inst✝² : Module R M₁
inst✝¹ : Module R M₂
inst✝ : Module R N
Q₁ : QuadraticForm R M₁
Q₂ : QuadraticForm R M₂
Qₙ : QuadraticForm R N
f₁ : Q₁ →qᵢ Qₙ
f₂ : Q₂ →qᵢ Qₙ
hf : ∀ (x : M₁) (y : M₂), QuadraticMap.IsOrtho Qₙ (f₁ x) (f₂ y)
m₁ : CliffordAlgebra Q₁
m₂ : CliffordAlgebra Q₂
i₂ : ZMod 2
hm₁ : m₁ ∈ evenOdd Q₁ 0
hm₂ : m₂ ∈ evenOdd Q₂ i₂
⊢ (-1) ^ (i₂ * 0) • ((map f₂) m₂ * (map f₁) m₁) = (map f₂) m₂ * (map f₁) m₁

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Prod

Goal state before:
pp: R : Type u_1
M₁ : Type u_2
M₂ : Type u_3
N : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M₁
inst✝⁴ : AddCommGroup M₂
inst✝³ : AddCommGroup N
inst✝² : Module R M₁
inst✝¹ : Module R M₂
inst✝ : Module R N
Q₁ : QuadraticForm R M₁
Q₂ : QuadraticForm R M₂
Qₙ : QuadraticForm R N
f₁ : Q₁ →qᵢ Qₙ
f₂ : Q₂ →qᵢ Qₙ
hf : ∀ (x : M₁) (y : M₂), QuadraticMap.IsOrtho Qₙ (f₁ x) (f₂ y)
m₁ : CliffordAlgebra Q₁
m₂ : CliffordAlgebra Q₂
i₁ : ZMod 2
hm₁ : m₁ ∈ evenOdd Q₁ i₁
hm₂ : m₂ ∈ evenOdd Q₂ 0
⊢ (-1) ^ (0 * i₁) • ((map f₂) m₂ * (map f₁) m₁) = (map f₂) m₂ * (map f₁) m₁

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
Q : QuadraticForm R M
x : (CliffordAlgebra Q)ˣ
hx : x ∈ lipschitzGroup Q
inst✝ : Invertible 2
⊢ ConjAct.toConjAct x • (ι Q).range = (ι Q).range

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Prod

Goal state before:
pp: R : Type u_1
M₁ : Type u_2
M₂ : Type u_3
N : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M₁
inst✝⁴ : AddCommGroup M₂
inst✝³ : AddCommGroup N
inst✝² : Module R M₁
inst✝¹ : Module R M₂
inst✝ : Module R N
Q₁ : QuadraticForm R M₁
Q₂ : QuadraticForm R M₂
Qₙ : QuadraticForm R N
f₁ : Q₁ →qᵢ Qₙ
f₂ : Q₂ →qᵢ Qₙ
hf : ∀ (x : M₁) (y : M₂), QuadraticMap.IsOrtho Qₙ (f₁ x) (f₂ y)
m₁ : CliffordAlgebra Q₁
m₂ : CliffordAlgebra Q₂
hm₁ : m₁ ∈ evenOdd Q₁ 1
hm₂ : m₂ ∈ evenOdd Q₂ 1
⊢ (map f₁) m₁ * (map f₂) m₂ = -(map f₂) m₂ * (map f₁) m₁

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x : (CliffordAlgebra Q)ˣ
⊢ ↑x ∈ Submonoid.map (Units.coeHom (CliffordAlgebra Q)) (lipschitzGroup Q).toSubmonoid ↔ x ∈ lipschitzGroup Q

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x : (CliffordAlgebra Q)ˣ
⊢ ↑x ∈ pinGroup Q ↔ x ∈ lipschitzGroup Q ∧ ↑x ∈ unitary (CliffordAlgebra Q)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x : CliffordAlgebra Q
y : (CliffordAlgebra Q)ˣ
hy₂ : ↑y = x
_hx₂ : star x * x = 1
hx₃ : x * star x = 1
hy₁ : y ∈ lipschitzGroup Q
⊢ ↑(star y) = star x

Tactic:
by

Goal state after: