awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.LinearAlgebra.CliffordAlgebra.Prod
Goal:
R : Type u_1
M₁ : Type u_2
M₂ : Type u_3
inst : CommRing R
inst_1 : AddCommGroup M₁
inst_2 : AddCommGroup M₂
inst_3 : Module R M₁
inst_4 : Module R M₂
Q₁ : QuadraticForm R M₁
Q₂ : QuadraticForm R M₂
x : M₂
⊢ Eq (α := evenOdd Q₁ ᵍ⊗[R] evenOdd Q₂)
    (((((ofProd Q₁ Q₂).comp (toProd Q₁ Q₂)).comp
                (GradedTensorProduct.includeRight (evenOdd Q₁) (evenOdd Q₂))).toLinearMap ∘ₗ
            ι Q₂ :
          (a : M₂) → evenOdd Q₁ ᵍ⊗[R] evenOdd Q₂)
        x :
      evenOdd Q₁ ᵍ⊗[R] evenOdd Q₂)
    ((AlgHom.toLinearMap
              (AlgHom.comp (AlgHom.id R (evenOdd Q₁ ᵍ⊗[R] evenOdd Q₂))
                (GradedTensorProduct.includeRight (evenOdd Q₁) (evenOdd Q₂))) ∘ₗ
            ι Q₂ :
          (a : M₂) → evenOdd Q₁ ᵍ⊗[R] evenOdd Q₂)
        x :
      evenOdd Q₁ ᵍ⊗[R] evenOdd Q₂)

Tactic:
simp
-- Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
x : (CliffordAlgebra Q)ˣ
hx :
  Membership.mem (γ := Subgroup (CliffordAlgebra Q)ˣ)
    (Subgroup.closure (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q))) x
inst_3 : Invertible (α := R) 2
m : M
rw :
  HMul.hMul (β := CliffordAlgebra Q) (↑x : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
      (↑x⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range
⊢ HSMul.hSMul (α := ConjAct (CliffordAlgebra Q)ˣ) (β := CliffordAlgebra Q)
      ((ConjAct.toConjAct : (a : (CliffordAlgebra Q)ˣ) → ConjAct (CliffordAlgebra Q)ˣ) x : ConjAct (CliffordAlgebra Q)ˣ)
      ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) ∈
    (ι Q).range

Tactic:
rw [ConjAct.units_smul_def, ConjAct.ofConjAct_toConjAct]

Premises used: ConjAct.units_smul_def, ConjAct.ofConjAct_toConjAct
-- Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
x_1 : (CliffordAlgebra Q)ˣ
inst_3 : Invertible (α := R) 2
x : (CliffordAlgebra Q)ˣ
hx :
  Membership.mem (γ := Set (CliffordAlgebra Q)ˣ)
    (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q)) x
m : M
mem :
  ∀ (a : M) (ha : (ι Q : (a : M) → CliffordAlgebra Q) a = (↑x : CliffordAlgebra Q)),
    HMul.hMul (β := CliffordAlgebra Q) (↑x : CliffordAlgebra Q)
          ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
        (↑x⁻¹ : CliffordAlgebra Q) ∈
      (ι Q).range
⊢ HMul.hMul (β := CliffordAlgebra Q) (↑x : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
      (↑x⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range

Tactic:
obtain ⟨a, ha⟩ := hx
-- Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
x_1 : (CliffordAlgebra Q)ˣ
inst_3 : Invertible (α := R) 2
x : (CliffordAlgebra Q)ˣ
m a : M
ha : (ι Q : (a : M) → CliffordAlgebra Q) a = (↑x : CliffordAlgebra Q)
this : Invertible (↑x : CliffordAlgebra Q) := x.invertible
rwa rwa₁ rwa₂ rwa₃ : Invertible (↑x : CliffordAlgebra Q)
⊢ Invertible (α := CliffordAlgebra Q) ((ι Q : (a : M) → CliffordAlgebra Q) a : CliffordAlgebra Q)

Tactic:
rwa [ha]
-- Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
x_1 : (CliffordAlgebra Q)ˣ
inst_3 : Invertible (α := R) 2
x : (CliffordAlgebra Q)ˣ
m a : M
ha : (ι Q : (a : M) → CliffordAlgebra Q) a = (↑x : CliffordAlgebra Q)
this_1 : Invertible (↑x : CliffordAlgebra Q) := x.invertible
this_2 : Invertible (α := CliffordAlgebra Q) ((ι Q : (a : M) → CliffordAlgebra Q) a : CliffordAlgebra Q) :=
  Eq.mpr ⋯ this_1
this : Invertible (α := R) ((Q : (a : M) → R) a : R) := invertibleOfInvertibleι Q a
mem :
  HMul.hMul (β := CliffordAlgebra Q) (↑x : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
      @Invertible.invOf _ MulOne.toMul MulOne.toOne (↑x : CliffordAlgebra Q) _ ∈
    (ι Q).range
mem₁ :
  HMul.hMul (β := CliffordAlgebra Q)
      (HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) a : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q))
      (@Invertible.invOf _ MulOne.toMul MulOne.toOne ((ι Q : (a : M) → CliffordAlgebra Q) a) _ : CliffordAlgebra Q) ∈
    (ι Q).range
mem₂ :
  Membership.mem (α := CliffordAlgebra Q) (ι Q).range
    ((ι Q : (a : M) → CliffordAlgebra Q)
        (HMul.hMul (α := R) (β := R)
              (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne ((Q : (a : M) → R) a) _ : R)
              (QuadraticMap.polar (⇑Q : (a : M) → R) a m : R) •
            a -
          m) :
      CliffordAlgebra Q)
⊢ HMul.hMul (β := CliffordAlgebra Q) (↑x : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
      (↑x⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range

Tactic:
simp_rw [← invOf_units x, ← ha, ι_mul_ι_mul_invOf_ι, LinearMap.mem_range_self]

Premises used: CliffordAlgebra.ι_mul_ι_mul_invOf_ι, LinearMap.mem_range_self, invOf_units
-- Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
x_1 : (CliffordAlgebra Q)ˣ
inst_3 : Invertible (α := R) 2
x : (CliffordAlgebra Q)ˣ
hx :
  Membership.mem (γ := Set (CliffordAlgebra Q)ˣ)
    (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q)) x
m : M
inv_mem :
  ∀ (a : M) (ha : (ι Q : (a : M) → CliffordAlgebra Q) a = (↑x : CliffordAlgebra Q)),
    HMul.hMul (β := CliffordAlgebra Q) (↑x⁻¹ : CliffordAlgebra Q)
          ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
        (↑x⁻¹⁻¹ : CliffordAlgebra Q) ∈
      (ι Q).range
⊢ HMul.hMul (β := CliffordAlgebra Q) (↑x⁻¹ : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
      (↑x⁻¹⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range

Tactic:
obtain ⟨a, ha⟩ := hx
-- Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
x_1 : (CliffordAlgebra Q)ˣ
inst_3 : Invertible (α := R) 2
x : (CliffordAlgebra Q)ˣ
m a : M
ha : (ι Q : (a : M) → CliffordAlgebra Q) a = (↑x : CliffordAlgebra Q)
this_1 : Invertible (↑x : CliffordAlgebra Q) := x.invertible
this_2 : Invertible (α := CliffordAlgebra Q) ((ι Q : (a : M) → CliffordAlgebra Q) a : CliffordAlgebra Q) :=
  Eq.mpr ⋯ this_1
this : Invertible (α := R) ((Q : (a : M) → R) a : R) := invertibleOfInvertibleι Q a
inv_mem :
  HMul.hMul (β := CliffordAlgebra Q) (@Invertible.invOf _ MulOne.toMul MulOne.toOne (↑x : CliffordAlgebra Q) _)
        ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
      (↑x⁻¹⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range
inv_mem₁ :
  HMul.hMul (β := CliffordAlgebra Q) (@Invertible.invOf _ MulOne.toMul MulOne.toOne (↑x : CliffordAlgebra Q) _)
        ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
      (↑x : CliffordAlgebra Q) ∈
    (ι Q).range
inv_mem₂ :
  HMul.hMul (β := CliffordAlgebra Q)
      (HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        (@Invertible.invOf _ MulOne.toMul MulOne.toOne ((ι Q : (a : M) → CliffordAlgebra Q) a) _ : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q))
      ((ι Q : (a : M) → CliffordAlgebra Q) a : CliffordAlgebra Q) ∈
    (ι Q).range
inv_mem₃ :
  Membership.mem (α := CliffordAlgebra Q) (ι Q).range
    ((ι Q : (a : M) → CliffordAlgebra Q)
        (HMul.hMul (α := R) (β := R)
              (@Invertible.invOf _ Distrib.toMul AddMonoidWithOne.toOne ((Q : (a : M) → R) a) _ : R)
              (QuadraticMap.polar (⇑Q : (a : M) → R) a m : R) •
            a -
          m) :
      CliffordAlgebra Q)
⊢ HMul.hMul (β := CliffordAlgebra Q) (↑x⁻¹ : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
      (↑x⁻¹⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range

Tactic:
simp_rw [← invOf_units x, inv_inv, ← ha, invOf_ι_mul_ι_mul_ι, LinearMap.mem_range_self]

Premises used: CliffordAlgebra.invOf_ι_mul_ι_mul_ι, LinearMap.mem_range_self, inv_inv, invOf_units
-- Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
x : (CliffordAlgebra Q)ˣ
inst_3 : Invertible (α := R) 2
m : M
one :
  HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q) ((↑1 : CliffordAlgebra Q) : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
      (↑1 : CliffordAlgebra Q) ∈
    (ι Q).range
one₁ :
  HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q) (1 : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
      1 ∈
    (ι Q).range
one₂ :
  HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
      ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) (1 : CliffordAlgebra Q) ∈
    (ι Q).range
one₃ : Membership.mem (α := CliffordAlgebra Q) (ι Q).range ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q)
⊢ HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q) ((↑1 : CliffordAlgebra Q) : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
      (↑1⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range

Tactic:
simp_rw [inv_one, Units.val_one, one_mul, mul_one, LinearMap.mem_range_self]

Premises used: inv_one, LinearMap.mem_range_self, Units.val_one, mul_one, one_mul
-- Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
x : (CliffordAlgebra Q)ˣ
inst_3 : Invertible (α := R) 2
y z : (CliffordAlgebra Q)ˣ
hx :
  Membership.mem (γ := Subgroup (CliffordAlgebra Q)ˣ)
    (Subgroup.closure (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q))) y
hy_1 :
  Membership.mem (γ := Subgroup (CliffordAlgebra Q)ˣ)
    (Subgroup.closure (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q))) z
hy :
  ∀ (m : M),
    HMul.hMul (β := CliffordAlgebra Q) (↑y : CliffordAlgebra Q)
          ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
        (↑y⁻¹ : CliffordAlgebra Q) ∈
      (ι Q).range
hz :
  ∀ (m : M),
    HMul.hMul (β := CliffordAlgebra Q) (↑z : CliffordAlgebra Q)
          ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
        (↑z⁻¹ : CliffordAlgebra Q) ∈
      (ι Q).range
m : M
mul :
  HMul.hMul (β := CliffordAlgebra Q) ((↑y : CliffordAlgebra Q) * (↑z : CliffordAlgebra Q))
        ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
      ((↑z⁻¹ : CliffordAlgebra Q) * (↑y⁻¹ : CliffordAlgebra Q)) ∈
    (ι Q).range
⊢ HMul.hMul (β := CliffordAlgebra Q) (↑(y * z) : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
      (↑(y * z)⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range

Tactic:
simp_rw [mul_inv_rev, Units.val_mul]

Premises used: mul_inv_rev, Units.val_mul
-- Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
x : (CliffordAlgebra Q)ˣ
inst_3 : Invertible (α := R) 2
y z : (CliffordAlgebra Q)ˣ
hx :
  Membership.mem (γ := Subgroup (CliffordAlgebra Q)ˣ)
    (Subgroup.closure (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q))) y
hy_1 :
  Membership.mem (γ := Subgroup (CliffordAlgebra Q)ˣ)
    (Subgroup.closure (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q))) z
hy :
  ∀ (m : M),
    HMul.hMul (β := CliffordAlgebra Q) (↑y : CliffordAlgebra Q)
          ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
        (↑y⁻¹ : CliffordAlgebra Q) ∈
      (ι Q).range
hz :
  ∀ (m : M),
    HMul.hMul (β := CliffordAlgebra Q) (↑z : CliffordAlgebra Q)
          ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
        (↑z⁻¹ : CliffordAlgebra Q) ∈
      (ι Q).range
m : M
this :
  (↑y : CliffordAlgebra Q) *
        (HMul.hMul (β := CliffordAlgebra Q) (↑z : CliffordAlgebra Q)
            ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
          (↑z⁻¹ : CliffordAlgebra Q)) *
      (↑y⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range
⊢ HMul.hMul (β := CliffordAlgebra Q) ((↑y : CliffordAlgebra Q) * (↑z : CliffordAlgebra Q))
        ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
      ((↑z⁻¹ : CliffordAlgebra Q) * (↑y⁻¹ : CliffordAlgebra Q)) ∈
    (ι Q).range

Tactic:
simpa only [mul_assoc] using this

Premises used: mul_assoc
-- Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
x : (CliffordAlgebra Q)ˣ
inst_3 : Invertible (α := R) 2
y z : (CliffordAlgebra Q)ˣ
hx :
  Membership.mem (γ := Subgroup (CliffordAlgebra Q)ˣ)
    (Subgroup.closure (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q))) y
hy_1 :
  Membership.mem (γ := Subgroup (CliffordAlgebra Q)ˣ)
    (Subgroup.closure (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q))) z
hy :
  ∀ (m : M),
    HMul.hMul (β := CliffordAlgebra Q) (↑y : CliffordAlgebra Q)
          ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
        (↑y⁻¹ : CliffordAlgebra Q) ∈
      (ι Q).range
hz :
  ∀ (m : M),
    HMul.hMul (β := CliffordAlgebra Q) (↑z : CliffordAlgebra Q)
          ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
        (↑z⁻¹ : CliffordAlgebra Q) ∈
      (ι Q).range
m : M
mul :
  (↑y : CliffordAlgebra Q) *
        (HMul.hMul (β := CliffordAlgebra Q) (↑z : CliffordAlgebra Q)
            ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
          (↑z⁻¹ : CliffordAlgebra Q)) *
      (↑y⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range
⊢ HMul.hMul (β := CliffordAlgebra Q) ((↑y : CliffordAlgebra Q) * (↑z : CliffordAlgebra Q))
        ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
      ((↑z⁻¹ : CliffordAlgebra Q) * (↑y⁻¹ : CliffordAlgebra Q)) ∈
    (ι Q).range

Tactic:
suffices ↑y * (↑z * ι Q m * ↑z⁻¹) * ↑y⁻¹ ∈ _ by
      simpa only [mul_assoc] using this
-- Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
x : (CliffordAlgebra Q)ˣ
inst_3 : Invertible (α := R) 2
y z : (CliffordAlgebra Q)ˣ
hx :
  Membership.mem (γ := Subgroup (CliffordAlgebra Q)ˣ)
    (Subgroup.closure (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q))) y
hy_1 :
  Membership.mem (γ := Subgroup (CliffordAlgebra Q)ˣ)
    (Subgroup.closure (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q))) z
hy :
  ∀ (m : M),
    HMul.hMul (β := CliffordAlgebra Q) (↑y : CliffordAlgebra Q)
          ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
        (↑y⁻¹ : CliffordAlgebra Q) ∈
      (ι Q).range
hz :
  ∀ (m : M),
    HMul.hMul (β := CliffordAlgebra Q) (↑z : CliffordAlgebra Q)
          ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
        (↑z⁻¹ : CliffordAlgebra Q) ∈
      (ι Q).range
m : M
mul :
  ∀ (z' : M)
    (hz' :
      (ι Q : (a : M) → CliffordAlgebra Q) z' =
        HMul.hMul (β := CliffordAlgebra Q) (↑z : CliffordAlgebra Q)
            ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
          (↑z⁻¹ : CliffordAlgebra Q)),
    (↑y : CliffordAlgebra Q) *
          (HMul.hMul (β := CliffordAlgebra Q) (↑z : CliffordAlgebra Q)
              ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
            (↑z⁻¹ : CliffordAlgebra Q)) *
        (↑y⁻¹ : CliffordAlgebra Q) ∈
      (ι Q).range
⊢ (↑y : CliffordAlgebra Q) *
        (HMul.hMul (β := CliffordAlgebra Q) (↑z : CliffordAlgebra Q)
            ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
          (↑z⁻¹ : CliffordAlgebra Q)) *
      (↑y⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range

Tactic:
obtain ⟨z', hz'⟩ := hz m
-- Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
x : (CliffordAlgebra Q)ˣ
inst_3 : Invertible (α := R) 2
y z : (CliffordAlgebra Q)ˣ
hx :
  Membership.mem (γ := Subgroup (CliffordAlgebra Q)ˣ)
    (Subgroup.closure (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q))) y
hy_1 :
  Membership.mem (γ := Subgroup (CliffordAlgebra Q)ˣ)
    (Subgroup.closure (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q))) z
hy :
  ∀ (m : M),
    HMul.hMul (β := CliffordAlgebra Q) (↑y : CliffordAlgebra Q)
          ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
        (↑y⁻¹ : CliffordAlgebra Q) ∈
      (ι Q).range
hz :
  ∀ (m : M),
    HMul.hMul (β := CliffordAlgebra Q) (↑z : CliffordAlgebra Q)
          ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
        (↑z⁻¹ : CliffordAlgebra Q) ∈
      (ι Q).range
m z' : M
hz' :
  (ι Q : (a : M) → CliffordAlgebra Q) z' =
    HMul.hMul (β := CliffordAlgebra Q) (↑z : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
      (↑z⁻¹ : CliffordAlgebra Q)
mul :
  ∀ (y' : M)
    (hy' :
      (ι Q : (a : M) → CliffordAlgebra Q) y' =
        HMul.hMul (β := CliffordAlgebra Q) (↑y : CliffordAlgebra Q)
            ((ι Q : (a : M) → CliffordAlgebra Q) z' : CliffordAlgebra Q) *
          (↑y⁻¹ : CliffordAlgebra Q)),
    (↑y : CliffordAlgebra Q) *
          (HMul.hMul (β := CliffordAlgebra Q) (↑z : CliffordAlgebra Q)
              ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
            (↑z⁻¹ : CliffordAlgebra Q)) *
        (↑y⁻¹ : CliffordAlgebra Q) ∈
      (ι Q).range
⊢ (↑y : CliffordAlgebra Q) *
        (HMul.hMul (β := CliffordAlgebra Q) (↑z : CliffordAlgebra Q)
            ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
          (↑z⁻¹ : CliffordAlgebra Q)) *
      (↑y⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range

Tactic:
obtain ⟨y', hy'⟩ := hy z'
-- Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
x : (CliffordAlgebra Q)ˣ
inst_3 : Invertible (α := R) 2
y z : (CliffordAlgebra Q)ˣ
hx :
  Membership.mem (γ := Subgroup (CliffordAlgebra Q)ˣ)
    (Subgroup.closure (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q))) y
hy_1 :
  Membership.mem (γ := Subgroup (CliffordAlgebra Q)ˣ)
    (Subgroup.closure (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q))) z
hy :
  ∀ (m : M),
    HMul.hMul (β := CliffordAlgebra Q) (↑y : CliffordAlgebra Q)
          ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
        (↑y⁻¹ : CliffordAlgebra Q) ∈
      (ι Q).range
hz :
  ∀ (m : M),
    HMul.hMul (β := CliffordAlgebra Q) (↑z : CliffordAlgebra Q)
          ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
        (↑z⁻¹ : CliffordAlgebra Q) ∈
      (ι Q).range
m z' : M
hz' :
  (ι Q : (a : M) → CliffordAlgebra Q) z' =
    HMul.hMul (β := CliffordAlgebra Q) (↑z : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
      (↑z⁻¹ : CliffordAlgebra Q)
y' : M
hy' :
  (ι Q : (a : M) → CliffordAlgebra Q) y' =
    HMul.hMul (β := CliffordAlgebra Q) (↑y : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) z' : CliffordAlgebra Q) *
      (↑y⁻¹ : CliffordAlgebra Q)
mul :
  HMul.hMul (β := CliffordAlgebra Q) (↑y : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) z' : CliffordAlgebra Q) *
      (↑y⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range
mul₁ : Membership.mem (α := CliffordAlgebra Q) (ι Q).range ((ι Q : (a : M) → CliffordAlgebra Q) y' : CliffordAlgebra Q)
⊢ (↑y : CliffordAlgebra Q) *
        (HMul.hMul (β := CliffordAlgebra Q) (↑z : CliffordAlgebra Q)
            ((ι Q : (a : M) → CliffordAlgebra Q) m : CliffordAlgebra Q) *
          (↑z⁻¹ : CliffordAlgebra Q)) *
      (↑y⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range

Tactic:
simp_rw [← hz', ← hy', LinearMap.mem_range_self]

Premises used: LinearMap.mem_range_self
-- Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
inst_3 : Invertible (α := R) 2
x_1 x : (CliffordAlgebra Q)ˣ
hx :
  Membership.mem (γ := Set (CliffordAlgebra Q)ˣ)
    (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q)) x
b : M
mem :
  ∀ (a : M) (ha : (ι Q : (a : M) → CliffordAlgebra Q) a = (↑x : CliffordAlgebra Q)),
    HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
          ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) (↑x : CliffordAlgebra Q) : CliffordAlgebra Q)
          ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
        (↑x⁻¹ : CliffordAlgebra Q) ∈
      (ι Q).range
⊢ HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) (↑x : CliffordAlgebra Q) : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
      (↑x⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range

Tactic:
obtain ⟨a, ha⟩ := hx
-- Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
inst_3 : Invertible (α := R) 2
x_1 x : (CliffordAlgebra Q)ˣ
b a : M
ha : (ι Q : (a : M) → CliffordAlgebra Q) a = (↑x : CliffordAlgebra Q)
this : Invertible (↑x : CliffordAlgebra Q) := x.invertible
rwa rwa₁ rwa₂ rwa₃ : Invertible (↑x : CliffordAlgebra Q)
⊢ Invertible (α := CliffordAlgebra Q) ((ι Q : (a : M) → CliffordAlgebra Q) a : CliffordAlgebra Q)

Tactic:
rwa [ha]
-- Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup
Goal:
R : Type u_1
inst : CommRing R
M : Type u_2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
inst_3 : Invertible (α := R) 2
x_1 x : (CliffordAlgebra Q)ˣ
hx :
  Membership.mem (γ := Set (CliffordAlgebra Q)ˣ)
    (Units.val ⁻¹' Set.range (ι := M) (⇑(ι Q) : (a : M) → CliffordAlgebra Q)) x
b : M
inv_mem :
  ∀ (a : M) (ha : (ι Q : (a : M) → CliffordAlgebra Q) a = (↑x : CliffordAlgebra Q)),
    HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
          ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) (↑x⁻¹ : CliffordAlgebra Q) : CliffordAlgebra Q)
          ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
        (↑x⁻¹⁻¹ : CliffordAlgebra Q) ∈
      (ι Q).range
⊢ HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        ((involute : (a : CliffordAlgebra Q) → CliffordAlgebra Q) (↑x⁻¹ : CliffordAlgebra Q) : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) b : CliffordAlgebra Q) *
      (↑x⁻¹⁻¹ : CliffordAlgebra Q) ∈
    (ι Q).range

Tactic:
obtain ⟨a, ha⟩ := hx