Theorem: CliffordAlgebra.induction
File: Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean

Proof steps:
state_before: R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
n : ℕ
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
C : CliffordAlgebra Q → Prop
algebraMap : ∀ (r : R), C ((_root_.algebraMap R (CliffordAlgebra Q)) r)
ι : ∀ (x : M), C ((CliffordAlgebra.ι Q) x)
mul : ∀ (a b : CliffordAlgebra Q), C a → C b → C (a * b)
add : ∀ (a b : CliffordAlgebra Q), C a → C b → C (a + b)
a : CliffordAlgebra Q
⊢ C a
state_after: R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
n : ℕ
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
C : CliffordAlgebra Q → Prop
algebraMap : ∀ (r : R), C ((_root_.algebraMap R (CliffordAlgebra Q)) r)
ι : ∀ (x : M), C ((CliffordAlgebra.ι Q) x)
mul : ∀ (a b : CliffordAlgebra Q), C a → C b → C (a * b)
add : ∀ (a b : CliffordAlgebra Q), C a → C b → C (a + b)
a : CliffordAlgebra Q
s : Subalgebra R (CliffordAlgebra Q) :=
  { carrier := C, mul_mem' := mul, one_mem' := ⋯, add_mem' := add, zero_mem' := ⋯, algebraMap_mem' := algebraMap }
⊢ C a
tactic: let s : Subalgebra R (CliffordAlgebra Q) :=
    { carrier := C
      mul_mem' := @mul
      add_mem' := @add
      algebraMap_mem' := algebraMap }
state_before: R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
n : ℕ
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
C : CliffordAlgebra Q → Prop
algebraMap : ∀ (r : R), C ((_root_.algebraMap R (CliffordAlgebra Q)) r)
ι : ∀ (x : M), C ((CliffordAlgebra.ι Q) x)
mul : ∀ (a b : CliffordAlgebra Q), C a → C b → C (a * b)
add : ∀ (a b : CliffordAlgebra Q), C a → C b → C (a + b)
a : CliffordAlgebra Q
s : Subalgebra R (CliffordAlgebra Q) :=
  { carrier := C, mul_mem' := mul, one_mem' := ⋯, add_mem' := add, zero_mem' := ⋯, algebraMap_mem' := algebraMap }
⊢ C a
state_after: R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
n : ℕ
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
C : CliffordAlgebra Q → Prop
algebraMap : ∀ (r : R), C ((_root_.algebraMap R (CliffordAlgebra Q)) r)
ι : ∀ (x : M), C ((CliffordAlgebra.ι Q) x)
mul : ∀ (a b : CliffordAlgebra Q), C a → C b → C (a * b)
add : ∀ (a b : CliffordAlgebra Q), C a → C b → C (a + b)
a : CliffordAlgebra Q
s : Subalgebra R (CliffordAlgebra Q) :=
  { carrier := C, mul_mem' := mul, one_mem' := ⋯, add_mem' := add, zero_mem' := ⋯, algebraMap_mem' := algebraMap }
h : AddCommMonoid ↥s := inferInstanceAs (AddCommMonoid ↥(Subalgebra.toSubmodule s))
⊢ C a
tactic: letI h : AddCommMonoid s := inferInstanceAs (AddCommMonoid (Subalgebra.toSubmodule s))
state_before: R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
n : ℕ
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
C : CliffordAlgebra Q → Prop
algebraMap : ∀ (r : R), C ((_root_.algebraMap R (CliffordAlgebra Q)) r)
ι : ∀ (x : M), C ((CliffordAlgebra.ι Q) x)
mul : ∀ (a b : CliffordAlgebra Q), C a → C b → C (a * b)
add : ∀ (a b : CliffordAlgebra Q), C a → C b → C (a + b)
a : CliffordAlgebra Q
s : Subalgebra R (CliffordAlgebra Q) :=
  { carrier := C, mul_mem' := mul, one_mem' := ⋯, add_mem' := add, zero_mem' := ⋯, algebraMap_mem' := algebraMap }
h : AddCommMonoid ↥s := inferInstanceAs (AddCommMonoid ↥(Subalgebra.toSubmodule s))
⊢ C a
state_after: R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
n : ℕ
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
C : CliffordAlgebra Q → Prop
algebraMap : ∀ (r : R), C ((_root_.algebraMap R (CliffordAlgebra Q)) r)
ι : ∀ (x : M), C ((CliffordAlgebra.ι Q) x)
mul : ∀ (a b : CliffordAlgebra Q), C a → C b → C (a * b)
add : ∀ (a b : CliffordAlgebra Q), C a → C b → C (a + b)
a : CliffordAlgebra Q
s : Subalgebra R (CliffordAlgebra Q) :=
  { carrier := C, mul_mem' := mul, one_mem' := ⋯, add_mem' := add, zero_mem' := ⋯, algebraMap_mem' := algebraMap }
h : AddCommMonoid ↥s := inferInstanceAs (AddCommMonoid ↥(Subalgebra.toSubmodule s))
of : { f // ∀ (m : M), f m * f m = (_root_.algebraMap R ↥s) (Q m) } :=
  ⟨LinearMap.codRestrict (Subalgebra.toSubmodule s) (CliffordAlgebra.ι Q) ι, ⋯⟩
⊢ C a
tactic: let of : { f : M →ₗ[R] s // ∀ m, f m * f m = _root_.algebraMap _ _ (Q m) } :=
    ⟨(CliffordAlgebra.ι Q).codRestrict (Subalgebra.toSubmodule s) ι,
      fun m => Subtype.eq <| ι_sq_scalar Q m⟩
state_before: R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
n : ℕ
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
C : CliffordAlgebra Q → Prop
algebraMap : ∀ (r : R), C ((_root_.algebraMap R (CliffordAlgebra Q)) r)
ι : ∀ (x : M), C ((CliffordAlgebra.ι Q) x)
mul : ∀ (a b : CliffordAlgebra Q), C a → C b → C (a * b)
add : ∀ (a b : CliffordAlgebra Q), C a → C b → C (a + b)
a : CliffordAlgebra Q
s : Subalgebra R (CliffordAlgebra Q) :=
  { carrier := C, mul_mem' := mul, one_mem' := ⋯, add_mem' := add, zero_mem' := ⋯, algebraMap_mem' := algebraMap }
h : AddCommMonoid ↥s := inferInstanceAs (AddCommMonoid ↥(Subalgebra.toSubmodule s))
of : { f // ∀ (m : M), f m * f m = (_root_.algebraMap R ↥s) (Q m) } :=
  ⟨LinearMap.codRestrict (Subalgebra.toSubmodule s) (CliffordAlgebra.ι Q) ι, ⋯⟩
⊢ C a
state_after: R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
n : ℕ
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
C : CliffordAlgebra Q → Prop
algebraMap : ∀ (r : R), C ((_root_.algebraMap R (CliffordAlgebra Q)) r)
ι : ∀ (x : M), C ((CliffordAlgebra.ι Q) x)
mul : ∀ (a b : CliffordAlgebra Q), C a → C b → C (a * b)
add : ∀ (a b : CliffordAlgebra Q), C a → C b → C (a + b)
a : CliffordAlgebra Q
s : Subalgebra R (CliffordAlgebra Q) :=
  { carrier := C, mul_mem' := mul, one_mem' := ⋯, add_mem' := add, zero_mem' := ⋯, algebraMap_mem' := algebraMap }
h : AddCommMonoid ↥s := inferInstanceAs (AddCommMonoid ↥(Subalgebra.toSubmodule s))
of : { f // ∀ (m : M), f m * f m = (_root_.algebraMap R ↥s) (Q m) } :=
  ⟨LinearMap.codRestrict (Subalgebra.toSubmodule s) (CliffordAlgebra.ι Q) ι, ⋯⟩
of_id : AlgHom.id R (CliffordAlgebra Q) = s.val.comp ((lift Q) of)
⊢ C a
tactic: have of_id : AlgHom.id R (CliffordAlgebra Q) = s.val.comp (lift Q of) := by
    ext
    simp [of]
    -- Porting note: `simp` can't apply this
    erw [LinearMap.codRestrict_apply]
state_before: R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
n : ℕ
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
C : CliffordAlgebra Q → Prop
algebraMap : ∀ (r : R), C ((_root_.algebraMap R (CliffordAlgebra Q)) r)
ι : ∀ (x : M), C ((CliffordAlgebra.ι Q) x)
mul : ∀ (a b : CliffordAlgebra Q), C a → C b → C (a * b)
add : ∀ (a b : CliffordAlgebra Q), C a → C b → C (a + b)
a : CliffordAlgebra Q
s : Subalgebra R (CliffordAlgebra Q) :=
  { carrier := C, mul_mem' := mul, one_mem' := ⋯, add_mem' := add, zero_mem' := ⋯, algebraMap_mem' := algebraMap }
h : AddCommMonoid ↥s := inferInstanceAs (AddCommMonoid ↥(Subalgebra.toSubmodule s))
of : { f // ∀ (m : M), f m * f m = (_root_.algebraMap R ↥s) (Q m) } :=
  ⟨LinearMap.codRestrict (Subalgebra.toSubmodule s) (CliffordAlgebra.ι Q) ι, ⋯⟩
of_id : AlgHom.id R (CliffordAlgebra Q) = s.val.comp ((lift Q) of)
⊢ C a
state_after: R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
n : ℕ
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
C : CliffordAlgebra Q → Prop
algebraMap : ∀ (r : R), C ((_root_.algebraMap R (CliffordAlgebra Q)) r)
ι : ∀ (x : M), C ((CliffordAlgebra.ι Q) x)
mul : ∀ (a b : CliffordAlgebra Q), C a → C b → C (a * b)
add : ∀ (a b : CliffordAlgebra Q), C a → C b → C (a + b)
a : CliffordAlgebra Q
s : Subalgebra R (CliffordAlgebra Q) :=
  { carrier := C, mul_mem' := mul, one_mem' := ⋯, add_mem' := add, zero_mem' := ⋯, algebraMap_mem' := algebraMap }
h : AddCommMonoid ↥s := inferInstanceAs (AddCommMonoid ↥(Subalgebra.toSubmodule s))
of : { f // ∀ (m : M), f m * f m = (_root_.algebraMap R ↥s) (Q m) } :=
  ⟨LinearMap.codRestrict (Subalgebra.toSubmodule s) (CliffordAlgebra.ι Q) ι, ⋯⟩
of_id : AlgHom.id R (CliffordAlgebra Q) = s.val.comp ((lift Q) of)
⊢ C ((s.val.comp ((lift Q) of)) a)
tactic: rw [← AlgHom.id_apply (R := R) a, of_id]
state_before: R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
n : ℕ
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
C : CliffordAlgebra Q → Prop
algebraMap : ∀ (r : R), C ((_root_.algebraMap R (CliffordAlgebra Q)) r)
ι : ∀ (x : M), C ((CliffordAlgebra.ι Q) x)
mul : ∀ (a b : CliffordAlgebra Q), C a → C b → C (a * b)
add : ∀ (a b : CliffordAlgebra Q), C a → C b → C (a + b)
a : CliffordAlgebra Q
s : Subalgebra R (CliffordAlgebra Q) :=
  { carrier := C, mul_mem' := mul, one_mem' := ⋯, add_mem' := add, zero_mem' := ⋯, algebraMap_mem' := algebraMap }
h : AddCommMonoid ↥s := inferInstanceAs (AddCommMonoid ↥(Subalgebra.toSubmodule s))
of : { f // ∀ (m : M), f m * f m = (_root_.algebraMap R ↥s) (Q m) } :=
  ⟨LinearMap.codRestrict (Subalgebra.toSubmodule s) (CliffordAlgebra.ι Q) ι, ⋯⟩
of_id : AlgHom.id R (CliffordAlgebra Q) = s.val.comp ((lift Q) of)
⊢ C ((s.val.comp ((lift Q) of)) a)
state_after: no goals
tactic: exact Subtype.prop (lift Q of a)