state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : CommGroupWithZero G₀
a✝ b✝ c d a b : G₀
hc : c ≠ 0
⊢ c / a / (c / b) = b / a | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Hom.Basic
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Units.... | rw [div_div_div_eq, mul_comm, mul_div_mul_right _ _ hc] | theorem div_div_div_cancel_left' (a b : G₀) (hc : c ≠ 0) : c / a / (c / b) = b / a := by
| Mathlib.Algebra.GroupWithZero.Units.Lemmas.203_0.ICkvbDoLYHVogsB | theorem div_div_div_cancel_left' (a b : G₀) (hc : c ≠ 0) : c / a / (c / b) = b / a | Mathlib_Algebra_GroupWithZero_Units_Lemmas |
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝⁵ : MonoidWithZero M₀
inst✝⁴ : GroupWithZero G₀
inst✝³ : Nontrivial M₀
inst✝² : MonoidWithZero M₀'
inst✝¹ : MonoidWithZeroHomClass F G₀ M₀
inst✝ : MonoidWithZeroHomClass F' G₀ M₀'
f✝ : F
a : G₀
f g : F'
h : f a = g a
... | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Hom.Basic
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Units.... | rcases eq_or_ne a 0 with (rfl | ha) | theorem eq_on_inv₀ (f g : F') (h : f a = g a) : f a⁻¹ = g a⁻¹ := by
| Mathlib.Algebra.GroupWithZero.Units.Lemmas.224_0.ICkvbDoLYHVogsB | theorem eq_on_inv₀ (f g : F') (h : f a = g a) : f a⁻¹ = g a⁻¹ | Mathlib_Algebra_GroupWithZero_Units_Lemmas |
case inl
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝⁵ : MonoidWithZero M₀
inst✝⁴ : GroupWithZero G₀
inst✝³ : Nontrivial M₀
inst✝² : MonoidWithZero M₀'
inst✝¹ : MonoidWithZeroHomClass F G₀ M₀
inst✝ : MonoidWithZeroHomClass F' G₀ M₀'
f✝ : F
f g : F'
h : f 0 = g ... | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Hom.Basic
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Units.... | rw [inv_zero, map_zero, map_zero] | theorem eq_on_inv₀ (f g : F') (h : f a = g a) : f a⁻¹ = g a⁻¹ := by
rcases eq_or_ne a 0 with (rfl | ha)
· | Mathlib.Algebra.GroupWithZero.Units.Lemmas.224_0.ICkvbDoLYHVogsB | theorem eq_on_inv₀ (f g : F') (h : f a = g a) : f a⁻¹ = g a⁻¹ | Mathlib_Algebra_GroupWithZero_Units_Lemmas |
case inr
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝⁵ : MonoidWithZero M₀
inst✝⁴ : GroupWithZero G₀
inst✝³ : Nontrivial M₀
inst✝² : MonoidWithZero M₀'
inst✝¹ : MonoidWithZeroHomClass F G₀ M₀
inst✝ : MonoidWithZeroHomClass F' G₀ M₀'
f✝ : F
a : G₀
f g : F'
h : f... | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Hom.Basic
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Units.... | exact (IsUnit.mk0 a ha).eq_on_inv f g h | theorem eq_on_inv₀ (f g : F') (h : f a = g a) : f a⁻¹ = g a⁻¹ := by
rcases eq_or_ne a 0 with (rfl | ha)
· rw [inv_zero, map_zero, map_zero]
· | Mathlib.Algebra.GroupWithZero.Units.Lemmas.224_0.ICkvbDoLYHVogsB | theorem eq_on_inv₀ (f g : F') (h : f a = g a) : f a⁻¹ = g a⁻¹ | Mathlib_Algebra_GroupWithZero_Units_Lemmas |
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝³ : MonoidWithZero M₀
inst✝² : GroupWithZero G₀
inst✝¹ : GroupWithZero G₀'
inst✝ : MonoidWithZeroHomClass F G₀ G₀'
f : F
a b : G₀
⊢ f a⁻¹ = (f a)⁻¹ | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Hom.Basic
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Units.... | by_cases h : a = 0 | /-- A monoid homomorphism between groups with zeros sending `0` to `0` sends `a⁻¹` to `(f a)⁻¹`. -/
@[simp]
theorem map_inv₀ : f a⁻¹ = (f a)⁻¹ := by
| Mathlib.Algebra.GroupWithZero.Units.Lemmas.236_0.ICkvbDoLYHVogsB | /-- A monoid homomorphism between groups with zeros sending `0` to `0` sends `a⁻¹` to `(f a)⁻¹`. -/
@[simp]
theorem map_inv₀ : f a⁻¹ = (f a)⁻¹ | Mathlib_Algebra_GroupWithZero_Units_Lemmas |
case pos
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝³ : MonoidWithZero M₀
inst✝² : GroupWithZero G₀
inst✝¹ : GroupWithZero G₀'
inst✝ : MonoidWithZeroHomClass F G₀ G₀'
f : F
a b : G₀
h : a = 0
⊢ f a⁻¹ = (f a)⁻¹ | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Hom.Basic
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Units.... | simp [h, map_zero f] | /-- A monoid homomorphism between groups with zeros sending `0` to `0` sends `a⁻¹` to `(f a)⁻¹`. -/
@[simp]
theorem map_inv₀ : f a⁻¹ = (f a)⁻¹ := by
by_cases h : a = 0
· | Mathlib.Algebra.GroupWithZero.Units.Lemmas.236_0.ICkvbDoLYHVogsB | /-- A monoid homomorphism between groups with zeros sending `0` to `0` sends `a⁻¹` to `(f a)⁻¹`. -/
@[simp]
theorem map_inv₀ : f a⁻¹ = (f a)⁻¹ | Mathlib_Algebra_GroupWithZero_Units_Lemmas |
case neg
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝³ : MonoidWithZero M₀
inst✝² : GroupWithZero G₀
inst✝¹ : GroupWithZero G₀'
inst✝ : MonoidWithZeroHomClass F G₀ G₀'
f : F
a b : G₀
h : ¬a = 0
⊢ f a⁻¹ = (f a)⁻¹ | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Hom.Basic
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Units.... | apply eq_inv_of_mul_eq_one_left | /-- A monoid homomorphism between groups with zeros sending `0` to `0` sends `a⁻¹` to `(f a)⁻¹`. -/
@[simp]
theorem map_inv₀ : f a⁻¹ = (f a)⁻¹ := by
by_cases h : a = 0
· simp [h, map_zero f]
· | Mathlib.Algebra.GroupWithZero.Units.Lemmas.236_0.ICkvbDoLYHVogsB | /-- A monoid homomorphism between groups with zeros sending `0` to `0` sends `a⁻¹` to `(f a)⁻¹`. -/
@[simp]
theorem map_inv₀ : f a⁻¹ = (f a)⁻¹ | Mathlib_Algebra_GroupWithZero_Units_Lemmas |
case neg.h
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝³ : MonoidWithZero M₀
inst✝² : GroupWithZero G₀
inst✝¹ : GroupWithZero G₀'
inst✝ : MonoidWithZeroHomClass F G₀ G₀'
f : F
a b : G₀
h : ¬a = 0
⊢ f a⁻¹ * f a = 1 | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Hom.Basic
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Units.... | rw [← map_mul, inv_mul_cancel h, map_one] | /-- A monoid homomorphism between groups with zeros sending `0` to `0` sends `a⁻¹` to `(f a)⁻¹`. -/
@[simp]
theorem map_inv₀ : f a⁻¹ = (f a)⁻¹ := by
by_cases h : a = 0
· simp [h, map_zero f]
· apply eq_inv_of_mul_eq_one_left
| Mathlib.Algebra.GroupWithZero.Units.Lemmas.236_0.ICkvbDoLYHVogsB | /-- A monoid homomorphism between groups with zeros sending `0` to `0` sends `a⁻¹` to `(f a)⁻¹`. -/
@[simp]
theorem map_inv₀ : f a⁻¹ = (f a)⁻¹ | Mathlib_Algebra_GroupWithZero_Units_Lemmas |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
r : R
⊢ (ι (Q' Q)) (m, r) = (v Q) m + r • e0 Q | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | rw [e0, v, LinearMap.comp_apply, LinearMap.inl_apply, ← LinearMap.map_smul, Prod.smul_mk,
smul_zero, smul_eq_mul, mul_one, ← LinearMap.map_add, Prod.mk_add_mk, zero_add, add_zero] | theorem ι_eq_v_add_smul_e0 (m : M) (r : R) : ι (Q' Q) (m, r) = v Q m + r • e0 Q := by
| Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.71_0.ip5LZLfGcNzFdsI | theorem ι_eq_v_add_smul_e0 (m : M) (r : R) : ι (Q' Q) (m, r) = v Q m + r • e0 Q | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
⊢ (algebraMap R (CliffordAlgebra (Q' Q))) ((Q' Q) (0, 1)) = -1 | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | simp | theorem e0_mul_e0 : e0 Q * e0 Q = -1 :=
(ι_sq_scalar _ _).trans <| by | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.76_0.ip5LZLfGcNzFdsI | theorem e0_mul_e0 : e0 Q * e0 Q = -1 | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ (algebraMap R (CliffordAlgebra (Q' Q))) ((Q' Q) ((LinearMap.inl R M R) m)) =
(algebraMap R (CliffordAlgebra (Q' Q))) (Q m) | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | simp | theorem v_sq_scalar (m : M) : v Q m * v Q m = algebraMap _ _ (Q m) :=
(ι_sq_scalar _ _).trans <| by | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.80_0.ip5LZLfGcNzFdsI | theorem v_sq_scalar (m : M) : v Q m * v Q m = algebraMap _ _ (Q m) | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ -(e0 Q * (v Q) m) = (v Q) m * e0 Q | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | refine' neg_eq_of_add_eq_zero_right ((ι_mul_ι_add_swap _ _).trans _) | theorem neg_e0_mul_v (m : M) : -(e0 Q * v Q m) = v Q m * e0 Q := by
| Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.84_0.ip5LZLfGcNzFdsI | theorem neg_e0_mul_v (m : M) : -(e0 Q * v Q m) = v Q m * e0 Q | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ (algebraMap R (CliffordAlgebra (Q' Q))) (QuadraticForm.polar ⇑(Q' Q) (0, 1) ((LinearMap.inl R M R) m)) = 0 | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | dsimp [QuadraticForm.polar] | theorem neg_e0_mul_v (m : M) : -(e0 Q * v Q m) = v Q m * e0 Q := by
refine' neg_eq_of_add_eq_zero_right ((ι_mul_ι_add_swap _ _).trans _)
| Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.84_0.ip5LZLfGcNzFdsI | theorem neg_e0_mul_v (m : M) : -(e0 Q * v Q m) = v Q m * e0 Q | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ (algebraMap R (CliffordAlgebra (Q' Q))) (Q (0 + m) + -((1 + 0) * (1 + 0)) - (Q 0 + -(1 * 1)) - (Q m + -(0 * 0))) = 0 | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | simp only [add_zero, mul_zero, mul_one, zero_add, neg_zero, QuadraticForm.map_zero,
add_sub_cancel, sub_self, map_zero, zero_sub] | theorem neg_e0_mul_v (m : M) : -(e0 Q * v Q m) = v Q m * e0 Q := by
refine' neg_eq_of_add_eq_zero_right ((ι_mul_ι_add_swap _ _).trans _)
dsimp [QuadraticForm.polar]
| Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.84_0.ip5LZLfGcNzFdsI | theorem neg_e0_mul_v (m : M) : -(e0 Q * v Q m) = v Q m * e0 Q | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ -((v Q) m * e0 Q) = e0 Q * (v Q) m | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | rw [neg_eq_iff_eq_neg] | theorem neg_v_mul_e0 (m : M) : -(v Q m * e0 Q) = e0 Q * v Q m := by
| Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.91_0.ip5LZLfGcNzFdsI | theorem neg_v_mul_e0 (m : M) : -(v Q m * e0 Q) = e0 Q * v Q m | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ (v Q) m * e0 Q = -(e0 Q * (v Q) m) | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | exact (neg_e0_mul_v _ m).symm | theorem neg_v_mul_e0 (m : M) : -(v Q m * e0 Q) = e0 Q * v Q m := by
rw [neg_eq_iff_eq_neg]
| Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.91_0.ip5LZLfGcNzFdsI | theorem neg_v_mul_e0 (m : M) : -(v Q m * e0 Q) = e0 Q * v Q m | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ e0 Q * (v Q) m * e0 Q = (v Q) m | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | rw [← neg_v_mul_e0, ← neg_mul, mul_assoc, e0_mul_e0, mul_neg_one, neg_neg] | @[simp]
theorem e0_mul_v_mul_e0 (m : M) : e0 Q * v Q m * e0 Q = v Q m := by
| Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.96_0.ip5LZLfGcNzFdsI | @[simp]
theorem e0_mul_v_mul_e0 (m : M) : e0 Q * v Q m * e0 Q = v Q m | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
⊢ CliffordAlgebra Q →ₐ[R] ↥(even (Q' Q)) | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | refine' CliffordAlgebra.lift Q ⟨_, fun m => _⟩ | /-- The embedding from the smaller algebra into the new larger one. -/
def toEven : CliffordAlgebra Q →ₐ[R] CliffordAlgebra.even (Q' Q) := by
| Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.125_0.ip5LZLfGcNzFdsI | /-- The embedding from the smaller algebra into the new larger one. -/
def toEven : CliffordAlgebra Q →ₐ[R] CliffordAlgebra.even (Q' Q) | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
case refine'_1
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
⊢ M →ₗ[R] ↥(even (Q' Q)) | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | refine' LinearMap.codRestrict _ _ fun m => Submodule.mem_iSup_of_mem ⟨2, rfl⟩ _ | /-- The embedding from the smaller algebra into the new larger one. -/
def toEven : CliffordAlgebra Q →ₐ[R] CliffordAlgebra.even (Q' Q) := by
refine' CliffordAlgebra.lift Q ⟨_, fun m => _⟩
· | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.125_0.ip5LZLfGcNzFdsI | /-- The embedding from the smaller algebra into the new larger one. -/
def toEven : CliffordAlgebra Q →ₐ[R] CliffordAlgebra.even (Q' Q) | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
case refine'_1.refine'_1
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
⊢ M →ₗ[R] CliffordAlgebra (Q' Q) | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | exact (LinearMap.mulLeft R <| e0 Q).comp (v Q) | /-- The embedding from the smaller algebra into the new larger one. -/
def toEven : CliffordAlgebra Q →ₐ[R] CliffordAlgebra.even (Q' Q) := by
refine' CliffordAlgebra.lift Q ⟨_, fun m => _⟩
· refine' LinearMap.codRestrict _ _ fun m => Submodule.mem_iSup_of_mem ⟨2, rfl⟩ _
· | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.125_0.ip5LZLfGcNzFdsI | /-- The embedding from the smaller algebra into the new larger one. -/
def toEven : CliffordAlgebra Q →ₐ[R] CliffordAlgebra.even (Q' Q) | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
case refine'_1.refine'_2
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ (LinearMap.mulLeft R (e0 Q) ∘ₗ v Q) m ∈ LinearMap.range (ι (Q' Q)) ^ ↑{ val := 2, property := (_ : ↑2 = ↑2) } | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | rw [Subtype.coe_mk, pow_two] | /-- The embedding from the smaller algebra into the new larger one. -/
def toEven : CliffordAlgebra Q →ₐ[R] CliffordAlgebra.even (Q' Q) := by
refine' CliffordAlgebra.lift Q ⟨_, fun m => _⟩
· refine' LinearMap.codRestrict _ _ fun m => Submodule.mem_iSup_of_mem ⟨2, rfl⟩ _
· exact (LinearMap.mulLeft R <| e0 Q).com... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.125_0.ip5LZLfGcNzFdsI | /-- The embedding from the smaller algebra into the new larger one. -/
def toEven : CliffordAlgebra Q →ₐ[R] CliffordAlgebra.even (Q' Q) | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
case refine'_1.refine'_2
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ (LinearMap.mulLeft R (e0 Q) ∘ₗ v Q) m ∈ LinearMap.range (ι (Q' Q)) * LinearMap.range (ι (Q' Q)) | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | exact Submodule.mul_mem_mul (LinearMap.mem_range_self _ _) (LinearMap.mem_range_self _ _) | /-- The embedding from the smaller algebra into the new larger one. -/
def toEven : CliffordAlgebra Q →ₐ[R] CliffordAlgebra.even (Q' Q) := by
refine' CliffordAlgebra.lift Q ⟨_, fun m => _⟩
· refine' LinearMap.codRestrict _ _ fun m => Submodule.mem_iSup_of_mem ⟨2, rfl⟩ _
· exact (LinearMap.mulLeft R <| e0 Q).com... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.125_0.ip5LZLfGcNzFdsI | /-- The embedding from the smaller algebra into the new larger one. -/
def toEven : CliffordAlgebra Q →ₐ[R] CliffordAlgebra.even (Q' Q) | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
case refine'_2
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ (LinearMap.codRestrict (⨆ i, LinearMap.range (ι (Q' Q)) ^ ↑i) (LinearMap.mulLeft R (e0 Q) ∘ₗ v Q)
(_ : ∀ (m : M), (LinearMap.mulLeft R (e0 Q) ∘ₗ v Q) m ∈ ⨆ i, LinearMap.range (... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | ext1 | /-- The embedding from the smaller algebra into the new larger one. -/
def toEven : CliffordAlgebra Q →ₐ[R] CliffordAlgebra.even (Q' Q) := by
refine' CliffordAlgebra.lift Q ⟨_, fun m => _⟩
· refine' LinearMap.codRestrict _ _ fun m => Submodule.mem_iSup_of_mem ⟨2, rfl⟩ _
· exact (LinearMap.mulLeft R <| e0 Q).com... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.125_0.ip5LZLfGcNzFdsI | /-- The embedding from the smaller algebra into the new larger one. -/
def toEven : CliffordAlgebra Q →ₐ[R] CliffordAlgebra.even (Q' Q) | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
case refine'_2.a
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ ↑((LinearMap.codRestrict (⨆ i, LinearMap.range (ι (Q' Q)) ^ ↑i) (LinearMap.mulLeft R (e0 Q) ∘ₗ v Q)
(_ : ∀ (m : M), (LinearMap.mulLeft R (e0 Q) ∘ₗ v Q) m ∈ ⨆ i, LinearMap.r... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | rw [Subalgebra.coe_mul] | /-- The embedding from the smaller algebra into the new larger one. -/
def toEven : CliffordAlgebra Q →ₐ[R] CliffordAlgebra.even (Q' Q) := by
refine' CliffordAlgebra.lift Q ⟨_, fun m => _⟩
· refine' LinearMap.codRestrict _ _ fun m => Submodule.mem_iSup_of_mem ⟨2, rfl⟩ _
· exact (LinearMap.mulLeft R <| e0 Q).com... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.125_0.ip5LZLfGcNzFdsI | /-- The embedding from the smaller algebra into the new larger one. -/
def toEven : CliffordAlgebra Q →ₐ[R] CliffordAlgebra.even (Q' Q) | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
case refine'_2.a
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ ↑((LinearMap.codRestrict (⨆ i, LinearMap.range (ι (Q' Q)) ^ ↑i) (LinearMap.mulLeft R (e0 Q) ∘ₗ v Q)
(_ : ∀ (m : M), (LinearMap.mulLeft R (e0 Q) ∘ₗ v Q) m ∈ ⨆ i, LinearMap.r... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | erw [LinearMap.codRestrict_apply] | /-- The embedding from the smaller algebra into the new larger one. -/
def toEven : CliffordAlgebra Q →ₐ[R] CliffordAlgebra.even (Q' Q) := by
refine' CliffordAlgebra.lift Q ⟨_, fun m => _⟩
· refine' LinearMap.codRestrict _ _ fun m => Submodule.mem_iSup_of_mem ⟨2, rfl⟩ _
· exact (LinearMap.mulLeft R <| e0 Q).com... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.125_0.ip5LZLfGcNzFdsI | /-- The embedding from the smaller algebra into the new larger one. -/
def toEven : CliffordAlgebra Q →ₐ[R] CliffordAlgebra.even (Q' Q) | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
case refine'_2.a
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ (LinearMap.mulLeft R (e0 Q) ∘ₗ v Q) m * (LinearMap.mulLeft R (e0 Q) ∘ₗ v Q) m = ↑((algebraMap R ↥(even (Q' Q))) (Q m)) | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | dsimp only [LinearMap.comp_apply, LinearMap.mulLeft_apply, Subalgebra.coe_algebraMap] | /-- The embedding from the smaller algebra into the new larger one. -/
def toEven : CliffordAlgebra Q →ₐ[R] CliffordAlgebra.even (Q' Q) := by
refine' CliffordAlgebra.lift Q ⟨_, fun m => _⟩
· refine' LinearMap.codRestrict _ _ fun m => Submodule.mem_iSup_of_mem ⟨2, rfl⟩ _
· exact (LinearMap.mulLeft R <| e0 Q).com... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.125_0.ip5LZLfGcNzFdsI | /-- The embedding from the smaller algebra into the new larger one. -/
def toEven : CliffordAlgebra Q →ₐ[R] CliffordAlgebra.even (Q' Q) | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
case refine'_2.a
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ e0 Q * (v Q) m * (e0 Q * (v Q) m) = (algebraMap R (CliffordAlgebra (Q' Q))) (Q m) | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | rw [← mul_assoc, e0_mul_v_mul_e0, v_sq_scalar] | /-- The embedding from the smaller algebra into the new larger one. -/
def toEven : CliffordAlgebra Q →ₐ[R] CliffordAlgebra.even (Q' Q) := by
refine' CliffordAlgebra.lift Q ⟨_, fun m => _⟩
· refine' LinearMap.codRestrict _ _ fun m => Submodule.mem_iSup_of_mem ⟨2, rfl⟩ _
· exact (LinearMap.mulLeft R <| e0 Q).com... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.125_0.ip5LZLfGcNzFdsI | /-- The embedding from the smaller algebra into the new larger one. -/
def toEven : CliffordAlgebra Q →ₐ[R] CliffordAlgebra.even (Q' Q) | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ ↑((toEven Q) ((ι Q) m)) = e0 Q * (v Q) m | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | rw [toEven, CliffordAlgebra.lift_ι_apply] | @[simp]
theorem toEven_ι (m : M) : (toEven Q (ι Q m) : CliffordAlgebra (Q' Q)) = e0 Q * v Q m := by
| Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.139_0.ip5LZLfGcNzFdsI | @[simp]
theorem toEven_ι (m : M) : (toEven Q (ι Q m) : CliffordAlgebra (Q' Q)) = e0 Q * v Q m | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ ↑((LinearMap.codRestrict (⨆ i, LinearMap.range (ι (Q' Q)) ^ ↑i) (LinearMap.comp (LinearMap.mulLeft R (e0 Q)) (v Q))
(_ : ∀ (m : M), (LinearMap.mulLeft R (e0 Q) ∘ₗ v Q) m ∈ ⨆ i, LinearMap.rang... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | erw [LinearMap.codRestrict_apply] | @[simp]
theorem toEven_ι (m : M) : (toEven Q (ι Q m) : CliffordAlgebra (Q' Q)) = e0 Q * v Q m := by
rw [toEven, CliffordAlgebra.lift_ι_apply]
-- porting note: was `rw`
| Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.139_0.ip5LZLfGcNzFdsI | @[simp]
theorem toEven_ι (m : M) : (toEven Q (ι Q m) : CliffordAlgebra (Q' Q)) = e0 Q * v Q m | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ (LinearMap.comp (LinearMap.mulLeft R (e0 Q)) (v Q)) m = e0 Q * (v Q) m | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | rw [LinearMap.coe_comp, Function.comp_apply, LinearMap.mulLeft_apply] | @[simp]
theorem toEven_ι (m : M) : (toEven Q (ι Q m) : CliffordAlgebra (Q' Q)) = e0 Q * v Q m := by
rw [toEven, CliffordAlgebra.lift_ι_apply]
-- porting note: was `rw`
erw [LinearMap.codRestrict_apply]
| Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.139_0.ip5LZLfGcNzFdsI | @[simp]
theorem toEven_ι (m : M) : (toEven Q (ι Q m) : CliffordAlgebra (Q' Q)) = e0 Q * v Q m | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
⊢ ↥(even (Q' Q)) →ₐ[R] CliffordAlgebra Q | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | let f : M × R →ₗ[R] M × R →ₗ[R] CliffordAlgebra Q :=
((Algebra.lmul R (CliffordAlgebra Q)).toLinearMap.comp <|
(ι Q).comp (LinearMap.fst _ _ _) +
(Algebra.linearMap R _).comp (LinearMap.snd _ _ _)).compl₂
((ι Q).comp (LinearMap.fst _ _ _) - (Algebra.linearMap R _).comp (LinearMap.snd _ _... | /-- The embedding from the even subalgebra with an extra dimension into the original algebra. -/
def ofEven : CliffordAlgebra.even (Q' Q) →ₐ[R] CliffordAlgebra Q := by
/-
Recall that we need:
* `f ⟨0,1⟩ ⟨x,0⟩ = ι x`
* `f ⟨x,0⟩ ⟨0,1⟩ = -ι x`
* `f ⟨x,0⟩ ⟨y,0⟩ = ι x * ι y`
* `f ⟨0,1⟩ ⟨0,1⟩ = -1`
... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.147_0.ip5LZLfGcNzFdsI | /-- The embedding from the even subalgebra with an extra dimension into the original algebra. -/
def ofEven : CliffordAlgebra.even (Q' Q) →ₐ[R] CliffordAlgebra Q | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
f : M × R →ₗ[R] M × R →ₗ[R] CliffordAlgebra Q :=
LinearMap.compl₂
(AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)) ∘ₗ
(ι Q ∘ₗ LinearMap.fst R M R + Algebra.linearMap R (CliffordAlgebra Q) ∘... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | haveI f_apply : ∀ x y, f x y = (ι Q x.1 + algebraMap R _ x.2) * (ι Q y.1 - algebraMap R _ y.2) :=
fun x y => by rfl | /-- The embedding from the even subalgebra with an extra dimension into the original algebra. -/
def ofEven : CliffordAlgebra.even (Q' Q) →ₐ[R] CliffordAlgebra Q := by
/-
Recall that we need:
* `f ⟨0,1⟩ ⟨x,0⟩ = ι x`
* `f ⟨x,0⟩ ⟨0,1⟩ = -ι x`
* `f ⟨x,0⟩ ⟨y,0⟩ = ι x * ι y`
* `f ⟨0,1⟩ ⟨0,1⟩ = -1`
... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.147_0.ip5LZLfGcNzFdsI | /-- The embedding from the even subalgebra with an extra dimension into the original algebra. -/
def ofEven : CliffordAlgebra.even (Q' Q) →ₐ[R] CliffordAlgebra Q | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
f : M × R →ₗ[R] M × R →ₗ[R] CliffordAlgebra Q :=
LinearMap.compl₂
(AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)) ∘ₗ
(ι Q ∘ₗ LinearMap.fst R M R + Algebra.linearMap R (CliffordAlgebra Q) ∘... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | rfl | /-- The embedding from the even subalgebra with an extra dimension into the original algebra. -/
def ofEven : CliffordAlgebra.even (Q' Q) →ₐ[R] CliffordAlgebra Q := by
/-
Recall that we need:
* `f ⟨0,1⟩ ⟨x,0⟩ = ι x`
* `f ⟨x,0⟩ ⟨0,1⟩ = -ι x`
* `f ⟨x,0⟩ ⟨y,0⟩ = ι x * ι y`
* `f ⟨0,1⟩ ⟨0,1⟩ = -1`
... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.147_0.ip5LZLfGcNzFdsI | /-- The embedding from the even subalgebra with an extra dimension into the original algebra. -/
def ofEven : CliffordAlgebra.even (Q' Q) →ₐ[R] CliffordAlgebra Q | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
f : M × R →ₗ[R] M × R →ₗ[R] CliffordAlgebra Q :=
LinearMap.compl₂
(AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)) ∘ₗ
(ι Q ∘ₗ LinearMap.fst R M R + Algebra.linearMap R (CliffordAlgebra Q) ∘... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | haveI hc : ∀ (r : R) (x : CliffordAlgebra Q), Commute (algebraMap _ _ r) x := Algebra.commutes | /-- The embedding from the even subalgebra with an extra dimension into the original algebra. -/
def ofEven : CliffordAlgebra.even (Q' Q) →ₐ[R] CliffordAlgebra Q := by
/-
Recall that we need:
* `f ⟨0,1⟩ ⟨x,0⟩ = ι x`
* `f ⟨x,0⟩ ⟨0,1⟩ = -ι x`
* `f ⟨x,0⟩ ⟨y,0⟩ = ι x * ι y`
* `f ⟨0,1⟩ ⟨0,1⟩ = -1`
... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.147_0.ip5LZLfGcNzFdsI | /-- The embedding from the even subalgebra with an extra dimension into the original algebra. -/
def ofEven : CliffordAlgebra.even (Q' Q) →ₐ[R] CliffordAlgebra Q | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
f : M × R →ₗ[R] M × R →ₗ[R] CliffordAlgebra Q :=
LinearMap.compl₂
(AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)) ∘ₗ
(ι Q ∘ₗ LinearMap.fst R M R + Algebra.linearMap R (CliffordAlgebra Q) ∘... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | haveI hm :
∀ m : M × R,
ι Q m.1 * ι Q m.1 - algebraMap R _ m.2 * algebraMap R _ m.2 = algebraMap R _ (Q' Q m) := by
intro m
rw [ι_sq_scalar, ← RingHom.map_mul, ← RingHom.map_sub, sub_eq_add_neg, Q'_apply, sub_eq_add_neg] | /-- The embedding from the even subalgebra with an extra dimension into the original algebra. -/
def ofEven : CliffordAlgebra.even (Q' Q) →ₐ[R] CliffordAlgebra Q := by
/-
Recall that we need:
* `f ⟨0,1⟩ ⟨x,0⟩ = ι x`
* `f ⟨x,0⟩ ⟨0,1⟩ = -ι x`
* `f ⟨x,0⟩ ⟨y,0⟩ = ι x * ι y`
* `f ⟨0,1⟩ ⟨0,1⟩ = -1`
... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.147_0.ip5LZLfGcNzFdsI | /-- The embedding from the even subalgebra with an extra dimension into the original algebra. -/
def ofEven : CliffordAlgebra.even (Q' Q) →ₐ[R] CliffordAlgebra Q | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
f : M × R →ₗ[R] M × R →ₗ[R] CliffordAlgebra Q :=
LinearMap.compl₂
(AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)) ∘ₗ
(ι Q ∘ₗ LinearMap.fst R M R + Algebra.linearMap R (CliffordAlgebra Q) ∘... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | intro m | /-- The embedding from the even subalgebra with an extra dimension into the original algebra. -/
def ofEven : CliffordAlgebra.even (Q' Q) →ₐ[R] CliffordAlgebra Q := by
/-
Recall that we need:
* `f ⟨0,1⟩ ⟨x,0⟩ = ι x`
* `f ⟨x,0⟩ ⟨0,1⟩ = -ι x`
* `f ⟨x,0⟩ ⟨y,0⟩ = ι x * ι y`
* `f ⟨0,1⟩ ⟨0,1⟩ = -1`
... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.147_0.ip5LZLfGcNzFdsI | /-- The embedding from the even subalgebra with an extra dimension into the original algebra. -/
def ofEven : CliffordAlgebra.even (Q' Q) →ₐ[R] CliffordAlgebra Q | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
f : M × R →ₗ[R] M × R →ₗ[R] CliffordAlgebra Q :=
LinearMap.compl₂
(AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)) ∘ₗ
(ι Q ∘ₗ LinearMap.fst R M R + Algebra.linearMap R (CliffordAlgebra Q) ∘... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | rw [ι_sq_scalar, ← RingHom.map_mul, ← RingHom.map_sub, sub_eq_add_neg, Q'_apply, sub_eq_add_neg] | /-- The embedding from the even subalgebra with an extra dimension into the original algebra. -/
def ofEven : CliffordAlgebra.even (Q' Q) →ₐ[R] CliffordAlgebra Q := by
/-
Recall that we need:
* `f ⟨0,1⟩ ⟨x,0⟩ = ι x`
* `f ⟨x,0⟩ ⟨0,1⟩ = -ι x`
* `f ⟨x,0⟩ ⟨y,0⟩ = ι x * ι y`
* `f ⟨0,1⟩ ⟨0,1⟩ = -1`
... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.147_0.ip5LZLfGcNzFdsI | /-- The embedding from the even subalgebra with an extra dimension into the original algebra. -/
def ofEven : CliffordAlgebra.even (Q' Q) →ₐ[R] CliffordAlgebra Q | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
f : M × R →ₗ[R] M × R →ₗ[R] CliffordAlgebra Q :=
LinearMap.compl₂
(AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)) ∘ₗ
(ι Q ∘ₗ LinearMap.fst R M R + Algebra.linearMap R (CliffordAlgebra Q) ∘... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | refine' even.lift (Q' Q) ⟨f, _, _⟩ | /-- The embedding from the even subalgebra with an extra dimension into the original algebra. -/
def ofEven : CliffordAlgebra.even (Q' Q) →ₐ[R] CliffordAlgebra Q := by
/-
Recall that we need:
* `f ⟨0,1⟩ ⟨x,0⟩ = ι x`
* `f ⟨x,0⟩ ⟨0,1⟩ = -ι x`
* `f ⟨x,0⟩ ⟨y,0⟩ = ι x * ι y`
* `f ⟨0,1⟩ ⟨0,1⟩ = -1`
... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.147_0.ip5LZLfGcNzFdsI | /-- The embedding from the even subalgebra with an extra dimension into the original algebra. -/
def ofEven : CliffordAlgebra.even (Q' Q) →ₐ[R] CliffordAlgebra Q | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
case refine'_1
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
f : M × R →ₗ[R] M × R →ₗ[R] CliffordAlgebra Q :=
LinearMap.compl₂
(AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)) ∘ₗ
(ι Q ∘ₗ LinearMap.fst R M R + Algebra.linearMap R (Cliff... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | simp_rw [f_apply] | /-- The embedding from the even subalgebra with an extra dimension into the original algebra. -/
def ofEven : CliffordAlgebra.even (Q' Q) →ₐ[R] CliffordAlgebra Q := by
/-
Recall that we need:
* `f ⟨0,1⟩ ⟨x,0⟩ = ι x`
* `f ⟨x,0⟩ ⟨0,1⟩ = -ι x`
* `f ⟨x,0⟩ ⟨y,0⟩ = ι x * ι y`
* `f ⟨0,1⟩ ⟨0,1⟩ = -1`
... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.147_0.ip5LZLfGcNzFdsI | /-- The embedding from the even subalgebra with an extra dimension into the original algebra. -/
def ofEven : CliffordAlgebra.even (Q' Q) →ₐ[R] CliffordAlgebra Q | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
case refine'_2
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
f : M × R →ₗ[R] M × R →ₗ[R] CliffordAlgebra Q :=
LinearMap.compl₂
(AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)) ∘ₗ
(ι Q ∘ₗ LinearMap.fst R M R + Algebra.linearMap R (Cliff... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | simp_rw [f_apply] | /-- The embedding from the even subalgebra with an extra dimension into the original algebra. -/
def ofEven : CliffordAlgebra.even (Q' Q) →ₐ[R] CliffordAlgebra Q := by
/-
Recall that we need:
* `f ⟨0,1⟩ ⟨x,0⟩ = ι x`
* `f ⟨x,0⟩ ⟨0,1⟩ = -ι x`
* `f ⟨x,0⟩ ⟨y,0⟩ = ι x * ι y`
* `f ⟨0,1⟩ ⟨0,1⟩ = -1`
... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.147_0.ip5LZLfGcNzFdsI | /-- The embedding from the even subalgebra with an extra dimension into the original algebra. -/
def ofEven : CliffordAlgebra.even (Q' Q) →ₐ[R] CliffordAlgebra Q | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
case refine'_1
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
f : M × R →ₗ[R] M × R →ₗ[R] CliffordAlgebra Q :=
LinearMap.compl₂
(AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)) ∘ₗ
(ι Q ∘ₗ LinearMap.fst R M R + Algebra.linearMap R (Cliff... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | intro m | /-- The embedding from the even subalgebra with an extra dimension into the original algebra. -/
def ofEven : CliffordAlgebra.even (Q' Q) →ₐ[R] CliffordAlgebra Q := by
/-
Recall that we need:
* `f ⟨0,1⟩ ⟨x,0⟩ = ι x`
* `f ⟨x,0⟩ ⟨0,1⟩ = -ι x`
* `f ⟨x,0⟩ ⟨y,0⟩ = ι x * ι y`
* `f ⟨0,1⟩ ⟨0,1⟩ = -1`
... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.147_0.ip5LZLfGcNzFdsI | /-- The embedding from the even subalgebra with an extra dimension into the original algebra. -/
def ofEven : CliffordAlgebra.even (Q' Q) →ₐ[R] CliffordAlgebra Q | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
case refine'_1
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
f : M × R →ₗ[R] M × R →ₗ[R] CliffordAlgebra Q :=
LinearMap.compl₂
(AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)) ∘ₗ
(ι Q ∘ₗ LinearMap.fst R M R + Algebra.linearMap R (Cliff... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | rw [← (hc _ _).symm.mul_self_sub_mul_self_eq, hm] | /-- The embedding from the even subalgebra with an extra dimension into the original algebra. -/
def ofEven : CliffordAlgebra.even (Q' Q) →ₐ[R] CliffordAlgebra Q := by
/-
Recall that we need:
* `f ⟨0,1⟩ ⟨x,0⟩ = ι x`
* `f ⟨x,0⟩ ⟨0,1⟩ = -ι x`
* `f ⟨x,0⟩ ⟨y,0⟩ = ι x * ι y`
* `f ⟨0,1⟩ ⟨0,1⟩ = -1`
... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.147_0.ip5LZLfGcNzFdsI | /-- The embedding from the even subalgebra with an extra dimension into the original algebra. -/
def ofEven : CliffordAlgebra.even (Q' Q) →ₐ[R] CliffordAlgebra Q | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
case refine'_2
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
f : M × R →ₗ[R] M × R →ₗ[R] CliffordAlgebra Q :=
LinearMap.compl₂
(AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)) ∘ₗ
(ι Q ∘ₗ LinearMap.fst R M R + Algebra.linearMap R (Cliff... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | intro m₁ m₂ m₃ | /-- The embedding from the even subalgebra with an extra dimension into the original algebra. -/
def ofEven : CliffordAlgebra.even (Q' Q) →ₐ[R] CliffordAlgebra Q := by
/-
Recall that we need:
* `f ⟨0,1⟩ ⟨x,0⟩ = ι x`
* `f ⟨x,0⟩ ⟨0,1⟩ = -ι x`
* `f ⟨x,0⟩ ⟨y,0⟩ = ι x * ι y`
* `f ⟨0,1⟩ ⟨0,1⟩ = -1`
... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.147_0.ip5LZLfGcNzFdsI | /-- The embedding from the even subalgebra with an extra dimension into the original algebra. -/
def ofEven : CliffordAlgebra.even (Q' Q) →ₐ[R] CliffordAlgebra Q | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
case refine'_2
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
f : M × R →ₗ[R] M × R →ₗ[R] CliffordAlgebra Q :=
LinearMap.compl₂
(AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)) ∘ₗ
(ι Q ∘ₗ LinearMap.fst R M R + Algebra.linearMap R (Cliff... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | rw [← mul_smul_comm, ← mul_assoc, mul_assoc (_ + _), ← (hc _ _).symm.mul_self_sub_mul_self_eq',
Algebra.smul_def, ← mul_assoc, hm] | /-- The embedding from the even subalgebra with an extra dimension into the original algebra. -/
def ofEven : CliffordAlgebra.even (Q' Q) →ₐ[R] CliffordAlgebra Q := by
/-
Recall that we need:
* `f ⟨0,1⟩ ⟨x,0⟩ = ι x`
* `f ⟨x,0⟩ ⟨0,1⟩ = -ι x`
* `f ⟨x,0⟩ ⟨y,0⟩ = ι x * ι y`
* `f ⟨0,1⟩ ⟨0,1⟩ = -1`
... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.147_0.ip5LZLfGcNzFdsI | /-- The embedding from the even subalgebra with an extra dimension into the original algebra. -/
def ofEven : CliffordAlgebra.even (Q' Q) →ₐ[R] CliffordAlgebra Q | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x y : M × R
⊢ (ofEven Q) (((even.ι (Q' Q)).bilin x) y) =
((ι Q) x.1 + (algebraMap R (CliffordAlgebra Q)) x.2) * ((ι Q) y.1 - (algebraMap R (CliffordAlgebra Q)) y.2) | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | unfold ofEven | theorem ofEven_ι (x y : M × R) :
ofEven Q ((even.ι (Q' Q)).bilin x y) =
(ι Q x.1 + algebraMap R _ x.2) * (ι Q y.1 - algebraMap R _ y.2) := by
-- porting note: entire proof was the term-mode `even.lift_ι (Q' Q) _ x y`
| Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.177_0.ip5LZLfGcNzFdsI | theorem ofEven_ι (x y : M × R) :
ofEven Q ((even.ι (Q' Q)).bilin x y) =
(ι Q x.1 + algebraMap R _ x.2) * (ι Q y.1 - algebraMap R _ y.2) | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x y : M × R
⊢ (let f :=
LinearMap.compl₂
(LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))
(LinearMap.comp (ι Q) (LinearMap.fst R M R) +
Line... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | lift_lets | theorem ofEven_ι (x y : M × R) :
ofEven Q ((even.ι (Q' Q)).bilin x y) =
(ι Q x.1 + algebraMap R _ x.2) * (ι Q y.1 - algebraMap R _ y.2) := by
-- porting note: entire proof was the term-mode `even.lift_ι (Q' Q) _ x y`
unfold ofEven
| Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.177_0.ip5LZLfGcNzFdsI | theorem ofEven_ι (x y : M × R) :
ofEven Q ((even.ι (Q' Q)).bilin x y) =
(ι Q x.1 + algebraMap R _ x.2) * (ι Q y.1 - algebraMap R _ y.2) | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x y : M × R
⊢ let f :=
LinearMap.compl₂
(LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))
(LinearMap.comp (ι Q) (LinearMap.fst R M R) +
LinearMap.comp (Algeb... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | intro f | theorem ofEven_ι (x y : M × R) :
ofEven Q ((even.ι (Q' Q)).bilin x y) =
(ι Q x.1 + algebraMap R _ x.2) * (ι Q y.1 - algebraMap R _ y.2) := by
-- porting note: entire proof was the term-mode `even.lift_ι (Q' Q) _ x y`
unfold ofEven
lift_lets
| Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.177_0.ip5LZLfGcNzFdsI | theorem ofEven_ι (x y : M × R) :
ofEven Q ((even.ι (Q' Q)).bilin x y) =
(ι Q x.1 + algebraMap R _ x.2) * (ι Q y.1 - algebraMap R _ y.2) | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x y : M × R
f : M × R →ₗ[R] M × R →ₗ[R] CliffordAlgebra Q :=
LinearMap.compl₂
(LinearMap.comp (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)))
(LinearMap.comp (ι Q) (LinearMap.fst R M R) +... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | refine @even.lift_ι R (M × R) _ _ _ (Q' Q) _ _ _ ⟨f, ?_, ?_⟩ x y | theorem ofEven_ι (x y : M × R) :
ofEven Q ((even.ι (Q' Q)).bilin x y) =
(ι Q x.1 + algebraMap R _ x.2) * (ι Q y.1 - algebraMap R _ y.2) := by
-- porting note: entire proof was the term-mode `even.lift_ι (Q' Q) _ x y`
unfold ofEven
lift_lets
intro f
-- TODO: replacing `?_` with `_` takes way longer?
... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.177_0.ip5LZLfGcNzFdsI | theorem ofEven_ι (x y : M × R) :
ofEven Q ((even.ι (Q' Q)).bilin x y) =
(ι Q x.1 + algebraMap R _ x.2) * (ι Q y.1 - algebraMap R _ y.2) | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m₁✝ m₂✝ : M × R
m₁ : M
r₁ : R
m₂ : M
r₂ : R
⊢ ↑((toEven Q) ((ofEven Q) (((even.ι (Q' Q)).bilin (m₁, r₁)) (m₂, r₂)))) =
(e0 Q * (v Q) m₁ + (algebraMap R (CliffordAlgebra (Q' Q))) r₁) *
(e0 Q * (v Q) m... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | rw [ofEven_ι, AlgHom.map_mul, AlgHom.map_add, AlgHom.map_sub, AlgHom.commutes,
AlgHom.commutes, Subalgebra.coe_mul, Subalgebra.coe_add, Subalgebra.coe_sub,
toEven_ι, toEven_ι, Subalgebra.coe_algebraMap, Subalgebra.coe_algebraMap] | theorem toEven_comp_ofEven : (toEven Q).comp (ofEven Q) = AlgHom.id R _ :=
even.algHom_ext (Q' Q) <|
EvenHom.ext _ _ <|
LinearMap.ext fun m₁ =>
LinearMap.ext fun m₂ =>
Subtype.ext <|
let ⟨m₁, r₁⟩ := m₁
let ⟨m₂, r₂⟩ := m₂
calc
↑(toEven Q (of... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.188_0.ip5LZLfGcNzFdsI | theorem toEven_comp_ofEven : (toEven Q).comp (ofEven Q) = AlgHom.id R _ | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m₁✝ m₂✝ : M × R
m₁ : M
r₁ : R
m₂ : M
r₂ : R
⊢ (e0 Q * (v Q) m₁ + (algebraMap R (CliffordAlgebra (Q' Q))) r₁) *
(e0 Q * (v Q) m₂ - (algebraMap R (CliffordAlgebra (Q' Q))) r₂) =
e0 Q * (v Q) m₁ * (e0 Q... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | rw [mul_sub, add_mul, add_mul, ← Algebra.commutes, ← Algebra.smul_def, ← map_mul, ←
Algebra.smul_def, sub_add_eq_sub_sub, smul_mul_assoc, smul_mul_assoc] | theorem toEven_comp_ofEven : (toEven Q).comp (ofEven Q) = AlgHom.id R _ :=
even.algHom_ext (Q' Q) <|
EvenHom.ext _ _ <|
LinearMap.ext fun m₁ =>
LinearMap.ext fun m₂ =>
Subtype.ext <|
let ⟨m₁, r₁⟩ := m₁
let ⟨m₂, r₂⟩ := m₂
calc
↑(toEven Q (of... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.188_0.ip5LZLfGcNzFdsI | theorem toEven_comp_ofEven : (toEven Q).comp (ofEven Q) = AlgHom.id R _ | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m₁✝ m₂✝ : M × R
m₁ : M
r₁ : R
m₂ : M
r₂ : R
⊢ e0 Q * (v Q) m₁ * (e0 Q * (v Q) m₂) + r₁ • e0 Q * (v Q) m₂ - r₂ • e0 Q * (v Q) m₁ -
(algebraMap R (CliffordAlgebra (Q' Q))) (r₁ * r₂) =
(v Q) m₁ * (v Q) ... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | have h1 : e0 Q * v Q m₁ * (e0 Q * v Q m₂) = v Q m₁ * v Q m₂ := by
rw [← mul_assoc, e0_mul_v_mul_e0] | theorem toEven_comp_ofEven : (toEven Q).comp (ofEven Q) = AlgHom.id R _ :=
even.algHom_ext (Q' Q) <|
EvenHom.ext _ _ <|
LinearMap.ext fun m₁ =>
LinearMap.ext fun m₂ =>
Subtype.ext <|
let ⟨m₁, r₁⟩ := m₁
let ⟨m₂, r₂⟩ := m₂
calc
↑(toEven Q (of... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.188_0.ip5LZLfGcNzFdsI | theorem toEven_comp_ofEven : (toEven Q).comp (ofEven Q) = AlgHom.id R _ | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m₁✝ m₂✝ : M × R
m₁ : M
r₁ : R
m₂ : M
r₂ : R
⊢ e0 Q * (v Q) m₁ * (e0 Q * (v Q) m₂) = (v Q) m₁ * (v Q) m₂ | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | rw [← mul_assoc, e0_mul_v_mul_e0] | theorem toEven_comp_ofEven : (toEven Q).comp (ofEven Q) = AlgHom.id R _ :=
even.algHom_ext (Q' Q) <|
EvenHom.ext _ _ <|
LinearMap.ext fun m₁ =>
LinearMap.ext fun m₂ =>
Subtype.ext <|
let ⟨m₁, r₁⟩ := m₁
let ⟨m₂, r₂⟩ := m₂
calc
↑(toEven Q (of... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.188_0.ip5LZLfGcNzFdsI | theorem toEven_comp_ofEven : (toEven Q).comp (ofEven Q) = AlgHom.id R _ | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m₁✝ m₂✝ : M × R
m₁ : M
r₁ : R
m₂ : M
r₂ : R
h1 : e0 Q * (v Q) m₁ * (e0 Q * (v Q) m₂) = (v Q) m₁ * (v Q) m₂
⊢ e0 Q * (v Q) m₁ * (e0 Q * (v Q) m₂) + r₁ • e0 Q * (v Q) m₂ - r₂ • e0 Q * (v Q) m₁ -
(algebraMa... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | have h2 : -(r₂ • e0 Q * v Q m₁) = v Q m₁ * r₂ • e0 Q := by
rw [mul_smul_comm, smul_mul_assoc, ← smul_neg, neg_e0_mul_v] | theorem toEven_comp_ofEven : (toEven Q).comp (ofEven Q) = AlgHom.id R _ :=
even.algHom_ext (Q' Q) <|
EvenHom.ext _ _ <|
LinearMap.ext fun m₁ =>
LinearMap.ext fun m₂ =>
Subtype.ext <|
let ⟨m₁, r₁⟩ := m₁
let ⟨m₂, r₂⟩ := m₂
calc
↑(toEven Q (of... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.188_0.ip5LZLfGcNzFdsI | theorem toEven_comp_ofEven : (toEven Q).comp (ofEven Q) = AlgHom.id R _ | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m₁✝ m₂✝ : M × R
m₁ : M
r₁ : R
m₂ : M
r₂ : R
h1 : e0 Q * (v Q) m₁ * (e0 Q * (v Q) m₂) = (v Q) m₁ * (v Q) m₂
⊢ -(r₂ • e0 Q * (v Q) m₁) = (v Q) m₁ * r₂ • e0 Q | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | rw [mul_smul_comm, smul_mul_assoc, ← smul_neg, neg_e0_mul_v] | theorem toEven_comp_ofEven : (toEven Q).comp (ofEven Q) = AlgHom.id R _ :=
even.algHom_ext (Q' Q) <|
EvenHom.ext _ _ <|
LinearMap.ext fun m₁ =>
LinearMap.ext fun m₂ =>
Subtype.ext <|
let ⟨m₁, r₁⟩ := m₁
let ⟨m₂, r₂⟩ := m₂
calc
↑(toEven Q (of... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.188_0.ip5LZLfGcNzFdsI | theorem toEven_comp_ofEven : (toEven Q).comp (ofEven Q) = AlgHom.id R _ | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m₁✝ m₂✝ : M × R
m₁ : M
r₁ : R
m₂ : M
r₂ : R
h1 : e0 Q * (v Q) m₁ * (e0 Q * (v Q) m₂) = (v Q) m₁ * (v Q) m₂
h2 : -(r₂ • e0 Q * (v Q) m₁) = (v Q) m₁ * r₂ • e0 Q
⊢ e0 Q * (v Q) m₁ * (e0 Q * (v Q) m₂) + r₁ • e0 Q ... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | have h3 : -algebraMap R _ (r₁ * r₂) = r₁ • e0 Q * r₂ • e0 Q := by
rw [Algebra.algebraMap_eq_smul_one, smul_mul_smul, e0_mul_e0, smul_neg] | theorem toEven_comp_ofEven : (toEven Q).comp (ofEven Q) = AlgHom.id R _ :=
even.algHom_ext (Q' Q) <|
EvenHom.ext _ _ <|
LinearMap.ext fun m₁ =>
LinearMap.ext fun m₂ =>
Subtype.ext <|
let ⟨m₁, r₁⟩ := m₁
let ⟨m₂, r₂⟩ := m₂
calc
↑(toEven Q (of... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.188_0.ip5LZLfGcNzFdsI | theorem toEven_comp_ofEven : (toEven Q).comp (ofEven Q) = AlgHom.id R _ | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m₁✝ m₂✝ : M × R
m₁ : M
r₁ : R
m₂ : M
r₂ : R
h1 : e0 Q * (v Q) m₁ * (e0 Q * (v Q) m₂) = (v Q) m₁ * (v Q) m₂
h2 : -(r₂ • e0 Q * (v Q) m₁) = (v Q) m₁ * r₂ • e0 Q
⊢ -(algebraMap R (CliffordAlgebra (Q' Q))) (r₁ * r... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | rw [Algebra.algebraMap_eq_smul_one, smul_mul_smul, e0_mul_e0, smul_neg] | theorem toEven_comp_ofEven : (toEven Q).comp (ofEven Q) = AlgHom.id R _ :=
even.algHom_ext (Q' Q) <|
EvenHom.ext _ _ <|
LinearMap.ext fun m₁ =>
LinearMap.ext fun m₂ =>
Subtype.ext <|
let ⟨m₁, r₁⟩ := m₁
let ⟨m₂, r₂⟩ := m₂
calc
↑(toEven Q (of... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.188_0.ip5LZLfGcNzFdsI | theorem toEven_comp_ofEven : (toEven Q).comp (ofEven Q) = AlgHom.id R _ | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m₁✝ m₂✝ : M × R
m₁ : M
r₁ : R
m₂ : M
r₂ : R
h1 : e0 Q * (v Q) m₁ * (e0 Q * (v Q) m₂) = (v Q) m₁ * (v Q) m₂
h2 : -(r₂ • e0 Q * (v Q) m₁) = (v Q) m₁ * r₂ • e0 Q
h3 : -(algebraMap R (CliffordAlgebra (Q' Q))) (r₁ ... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | rw [sub_eq_add_neg, sub_eq_add_neg, h1, h2, h3] | theorem toEven_comp_ofEven : (toEven Q).comp (ofEven Q) = AlgHom.id R _ :=
even.algHom_ext (Q' Q) <|
EvenHom.ext _ _ <|
LinearMap.ext fun m₁ =>
LinearMap.ext fun m₂ =>
Subtype.ext <|
let ⟨m₁, r₁⟩ := m₁
let ⟨m₂, r₂⟩ := m₂
calc
↑(toEven Q (of... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.188_0.ip5LZLfGcNzFdsI | theorem toEven_comp_ofEven : (toEven Q).comp (ofEven Q) = AlgHom.id R _ | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m₁✝ m₂✝ : M × R
m₁ : M
r₁ : R
m₂ : M
r₂ : R
⊢ (v Q) m₁ * (v Q) m₂ + r₁ • e0 Q * (v Q) m₂ + (v Q) m₁ * r₂ • e0 Q + r₁ • e0 Q * r₂ • e0 Q =
(ι (Q' Q)) (m₁, r₁) * (ι (Q' Q)) (m₂, r₂) | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | rw [ι_eq_v_add_smul_e0, ι_eq_v_add_smul_e0, mul_add, add_mul, add_mul, add_assoc] | theorem toEven_comp_ofEven : (toEven Q).comp (ofEven Q) = AlgHom.id R _ :=
even.algHom_ext (Q' Q) <|
EvenHom.ext _ _ <|
LinearMap.ext fun m₁ =>
LinearMap.ext fun m₂ =>
Subtype.ext <|
let ⟨m₁, r₁⟩ := m₁
let ⟨m₂, r₂⟩ := m₂
calc
↑(toEven Q (of... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.188_0.ip5LZLfGcNzFdsI | theorem toEven_comp_ofEven : (toEven Q).comp (ofEven Q) = AlgHom.id R _ | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ (ofEven Q) ((toEven Q) ((ι Q) m)) =
(ofEven Q) { val := ↑((toEven Q) ((ι Q) m)), property := (_ : ↑((toEven Q) ((ι Q) m)) ∈ even (Q' Q)) } | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | rw [Subtype.coe_eta] | theorem ofEven_comp_toEven : (ofEven Q).comp (toEven Q) = AlgHom.id R _ :=
CliffordAlgebra.hom_ext <|
LinearMap.ext fun m =>
calc
ofEven Q (toEven Q (ι Q m)) = ofEven Q ⟨_, (toEven Q (ι Q m)).prop⟩ := by
| Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.221_0.ip5LZLfGcNzFdsI | theorem ofEven_comp_toEven : (ofEven Q).comp (toEven Q) = AlgHom.id R _ | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ (ofEven Q) { val := ↑((toEven Q) ((ι Q) m)), property := (_ : ↑((toEven Q) ((ι Q) m)) ∈ even (Q' Q)) } =
((ι Q) 0 + (algebraMap R (CliffordAlgebra Q)) 1) * ((ι Q) m - (algebraMap R (CliffordAlgebra... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | simp_rw [toEven_ι] | theorem ofEven_comp_toEven : (ofEven Q).comp (toEven Q) = AlgHom.id R _ :=
CliffordAlgebra.hom_ext <|
LinearMap.ext fun m =>
calc
ofEven Q (toEven Q (ι Q m)) = ofEven Q ⟨_, (toEven Q (ι Q m)).prop⟩ := by
rw [Subtype.coe_eta]
_ = (ι Q 0 + algebraMap R _ 1) * (ι Q m - algebraMap R _ ... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.221_0.ip5LZLfGcNzFdsI | theorem ofEven_comp_toEven : (ofEven Q).comp (toEven Q) = AlgHom.id R _ | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ (ofEven Q) { val := e0 Q * (v Q) m, property := (_ : (fun x => x ∈ even (Q' Q)) (e0 Q * (v Q) m)) } =
((ι Q) 0 + (algebraMap R (CliffordAlgebra Q)) 1) * ((ι Q) m - (algebraMap R (CliffordAlgebra Q)... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | exact ofEven_ι Q _ _ | theorem ofEven_comp_toEven : (ofEven Q).comp (toEven Q) = AlgHom.id R _ :=
CliffordAlgebra.hom_ext <|
LinearMap.ext fun m =>
calc
ofEven Q (toEven Q (ι Q m)) = ofEven Q ⟨_, (toEven Q (ι Q m)).prop⟩ := by
rw [Subtype.coe_eta]
_ = (ι Q 0 + algebraMap R _ 1) * (ι Q m - algebraMap R _ ... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.221_0.ip5LZLfGcNzFdsI | theorem ofEven_comp_toEven : (ofEven Q).comp (toEven Q) = AlgHom.id R _ | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ ((ι Q) 0 + (algebraMap R (CliffordAlgebra Q)) 1) * ((ι Q) m - (algebraMap R (CliffordAlgebra Q)) 0) = (ι Q) m | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | rw [map_one, map_zero, map_zero, sub_zero, zero_add, one_mul] | theorem ofEven_comp_toEven : (ofEven Q).comp (toEven Q) = AlgHom.id R _ :=
CliffordAlgebra.hom_ext <|
LinearMap.ext fun m =>
calc
ofEven Q (toEven Q (ι Q m)) = ofEven Q ⟨_, (toEven Q (ι Q m)).prop⟩ := by
rw [Subtype.coe_eta]
_ = (ι Q 0 + algebraMap R _ 1) * (ι Q m - algebraMap R _ ... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.221_0.ip5LZLfGcNzFdsI | theorem ofEven_comp_toEven : (ofEven Q).comp (toEven Q) = AlgHom.id R _ | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x : CliffordAlgebra Q
⊢ ↑((toEven Q) (reverse (involute x))) = reverse ↑((toEven Q) x) | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | induction x using CliffordAlgebra.induction | set_option synthInstance.maxHeartbeats 30000 in
/-- The representation of the clifford conjugate (i.e. the reverse of the involute) in the even
subalgebra is just the reverse of the representation. -/
theorem coe_toEven_reverse_involute (x : CliffordAlgebra Q) :
↑(toEven Q (reverse (involute x))) =
reverse (Q... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.244_0.ip5LZLfGcNzFdsI | set_option synthInstance.maxHeartbeats 30000 in
/-- The representation of the clifford conjugate (i.e. the reverse of the involute) in the even
subalgebra is just the reverse of the representation. -/
theorem coe_toEven_reverse_involute (x : CliffordAlgebra Q) :
↑(toEven Q (reverse (involute x))) =
reverse (Q... | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
case h_grade0
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
r✝ : R
⊢ ↑((toEven Q) (reverse (involute ((algebraMap R (CliffordAlgebra Q)) r✝)))) =
reverse ↑((toEven Q) ((algebraMap R (CliffordAlgebra Q)) r✝))
case h_grade1
R : Type u_1
M : Type u_2
ins... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | case h_grade0 r => simp only [AlgHom.commutes, Subalgebra.coe_algebraMap, reverse.commutes] | set_option synthInstance.maxHeartbeats 30000 in
/-- The representation of the clifford conjugate (i.e. the reverse of the involute) in the even
subalgebra is just the reverse of the representation. -/
theorem coe_toEven_reverse_involute (x : CliffordAlgebra Q) :
↑(toEven Q (reverse (involute x))) =
reverse (Q... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.244_0.ip5LZLfGcNzFdsI | set_option synthInstance.maxHeartbeats 30000 in
/-- The representation of the clifford conjugate (i.e. the reverse of the involute) in the even
subalgebra is just the reverse of the representation. -/
theorem coe_toEven_reverse_involute (x : CliffordAlgebra Q) :
↑(toEven Q (reverse (involute x))) =
reverse (Q... | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
r : R
⊢ ↑((toEven Q) (reverse (involute ((algebraMap R (CliffordAlgebra Q)) r)))) =
reverse ↑((toEven Q) ((algebraMap R (CliffordAlgebra Q)) r)) | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | case h_grade0 r => simp only [AlgHom.commutes, Subalgebra.coe_algebraMap, reverse.commutes] | set_option synthInstance.maxHeartbeats 30000 in
/-- The representation of the clifford conjugate (i.e. the reverse of the involute) in the even
subalgebra is just the reverse of the representation. -/
theorem coe_toEven_reverse_involute (x : CliffordAlgebra Q) :
↑(toEven Q (reverse (involute x))) =
reverse (Q... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.244_0.ip5LZLfGcNzFdsI | set_option synthInstance.maxHeartbeats 30000 in
/-- The representation of the clifford conjugate (i.e. the reverse of the involute) in the even
subalgebra is just the reverse of the representation. -/
theorem coe_toEven_reverse_involute (x : CliffordAlgebra Q) :
↑(toEven Q (reverse (involute x))) =
reverse (Q... | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
r : R
⊢ ↑((toEven Q) (reverse (involute ((algebraMap R (CliffordAlgebra Q)) r)))) =
reverse ↑((toEven Q) ((algebraMap R (CliffordAlgebra Q)) r)) | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | simp only [AlgHom.commutes, Subalgebra.coe_algebraMap, reverse.commutes] | set_option synthInstance.maxHeartbeats 30000 in
/-- The representation of the clifford conjugate (i.e. the reverse of the involute) in the even
subalgebra is just the reverse of the representation. -/
theorem coe_toEven_reverse_involute (x : CliffordAlgebra Q) :
↑(toEven Q (reverse (involute x))) =
reverse (Q... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.244_0.ip5LZLfGcNzFdsI | set_option synthInstance.maxHeartbeats 30000 in
/-- The representation of the clifford conjugate (i.e. the reverse of the involute) in the even
subalgebra is just the reverse of the representation. -/
theorem coe_toEven_reverse_involute (x : CliffordAlgebra Q) :
↑(toEven Q (reverse (involute x))) =
reverse (Q... | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
case h_grade1
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x✝ : M
⊢ ↑((toEven Q) (reverse (involute ((ι Q) x✝)))) = reverse ↑((toEven Q) ((ι Q) x✝))
case h_mul
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : Q... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | case h_grade1 m =>
-- porting note: added `letI`
letI : SubtractionMonoid (even (Q' Q)) := AddGroup.toSubtractionMonoid
simp only [involute_ι, Subalgebra.coe_neg, toEven_ι, reverse.map_mul, reverse_v, reverse_e0,
reverse_ι, neg_e0_mul_v, map_neg] | set_option synthInstance.maxHeartbeats 30000 in
/-- The representation of the clifford conjugate (i.e. the reverse of the involute) in the even
subalgebra is just the reverse of the representation. -/
theorem coe_toEven_reverse_involute (x : CliffordAlgebra Q) :
↑(toEven Q (reverse (involute x))) =
reverse (Q... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.244_0.ip5LZLfGcNzFdsI | set_option synthInstance.maxHeartbeats 30000 in
/-- The representation of the clifford conjugate (i.e. the reverse of the involute) in the even
subalgebra is just the reverse of the representation. -/
theorem coe_toEven_reverse_involute (x : CliffordAlgebra Q) :
↑(toEven Q (reverse (involute x))) =
reverse (Q... | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ ↑((toEven Q) (reverse (involute ((ι Q) m)))) = reverse ↑((toEven Q) ((ι Q) m)) | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | case h_grade1 m =>
-- porting note: added `letI`
letI : SubtractionMonoid (even (Q' Q)) := AddGroup.toSubtractionMonoid
simp only [involute_ι, Subalgebra.coe_neg, toEven_ι, reverse.map_mul, reverse_v, reverse_e0,
reverse_ι, neg_e0_mul_v, map_neg] | set_option synthInstance.maxHeartbeats 30000 in
/-- The representation of the clifford conjugate (i.e. the reverse of the involute) in the even
subalgebra is just the reverse of the representation. -/
theorem coe_toEven_reverse_involute (x : CliffordAlgebra Q) :
↑(toEven Q (reverse (involute x))) =
reverse (Q... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.244_0.ip5LZLfGcNzFdsI | set_option synthInstance.maxHeartbeats 30000 in
/-- The representation of the clifford conjugate (i.e. the reverse of the involute) in the even
subalgebra is just the reverse of the representation. -/
theorem coe_toEven_reverse_involute (x : CliffordAlgebra Q) :
↑(toEven Q (reverse (involute x))) =
reverse (Q... | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
⊢ ↑((toEven Q) (reverse (involute ((ι Q) m)))) = reverse ↑((toEven Q) ((ι Q) m)) | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | letI : SubtractionMonoid (even (Q' Q)) := AddGroup.toSubtractionMonoid | set_option synthInstance.maxHeartbeats 30000 in
/-- The representation of the clifford conjugate (i.e. the reverse of the involute) in the even
subalgebra is just the reverse of the representation. -/
theorem coe_toEven_reverse_involute (x : CliffordAlgebra Q) :
↑(toEven Q (reverse (involute x))) =
reverse (Q... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.244_0.ip5LZLfGcNzFdsI | set_option synthInstance.maxHeartbeats 30000 in
/-- The representation of the clifford conjugate (i.e. the reverse of the involute) in the even
subalgebra is just the reverse of the representation. -/
theorem coe_toEven_reverse_involute (x : CliffordAlgebra Q) :
↑(toEven Q (reverse (involute x))) =
reverse (Q... | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
m : M
this : SubtractionMonoid ↥(even (Q' Q)) := AddGroup.toSubtractionMonoid
⊢ ↑((toEven Q) (reverse (involute ((ι Q) m)))) = reverse ↑((toEven Q) ((ι Q) m)) | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | simp only [involute_ι, Subalgebra.coe_neg, toEven_ι, reverse.map_mul, reverse_v, reverse_e0,
reverse_ι, neg_e0_mul_v, map_neg] | set_option synthInstance.maxHeartbeats 30000 in
/-- The representation of the clifford conjugate (i.e. the reverse of the involute) in the even
subalgebra is just the reverse of the representation. -/
theorem coe_toEven_reverse_involute (x : CliffordAlgebra Q) :
↑(toEven Q (reverse (involute x))) =
reverse (Q... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.244_0.ip5LZLfGcNzFdsI | set_option synthInstance.maxHeartbeats 30000 in
/-- The representation of the clifford conjugate (i.e. the reverse of the involute) in the even
subalgebra is just the reverse of the representation. -/
theorem coe_toEven_reverse_involute (x : CliffordAlgebra Q) :
↑(toEven Q (reverse (involute x))) =
reverse (Q... | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
case h_mul
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
a✝² b✝ : CliffordAlgebra Q
a✝¹ : ↑((toEven Q) (reverse (involute a✝²))) = reverse ↑((toEven Q) a✝²)
a✝ : ↑((toEven Q) (reverse (involute b✝))) = reverse ↑((toEven Q) b✝)
⊢ ↑((toEven Q) (reverse (inv... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | case h_mul x y hx hy => simp only [map_mul, Subalgebra.coe_mul, reverse.map_mul, hx, hy] | set_option synthInstance.maxHeartbeats 30000 in
/-- The representation of the clifford conjugate (i.e. the reverse of the involute) in the even
subalgebra is just the reverse of the representation. -/
theorem coe_toEven_reverse_involute (x : CliffordAlgebra Q) :
↑(toEven Q (reverse (involute x))) =
reverse (Q... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.244_0.ip5LZLfGcNzFdsI | set_option synthInstance.maxHeartbeats 30000 in
/-- The representation of the clifford conjugate (i.e. the reverse of the involute) in the even
subalgebra is just the reverse of the representation. -/
theorem coe_toEven_reverse_involute (x : CliffordAlgebra Q) :
↑(toEven Q (reverse (involute x))) =
reverse (Q... | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x y : CliffordAlgebra Q
hx : ↑((toEven Q) (reverse (involute x))) = reverse ↑((toEven Q) x)
hy : ↑((toEven Q) (reverse (involute y))) = reverse ↑((toEven Q) y)
⊢ ↑((toEven Q) (reverse (involute (x * y)))) = re... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | case h_mul x y hx hy => simp only [map_mul, Subalgebra.coe_mul, reverse.map_mul, hx, hy] | set_option synthInstance.maxHeartbeats 30000 in
/-- The representation of the clifford conjugate (i.e. the reverse of the involute) in the even
subalgebra is just the reverse of the representation. -/
theorem coe_toEven_reverse_involute (x : CliffordAlgebra Q) :
↑(toEven Q (reverse (involute x))) =
reverse (Q... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.244_0.ip5LZLfGcNzFdsI | set_option synthInstance.maxHeartbeats 30000 in
/-- The representation of the clifford conjugate (i.e. the reverse of the involute) in the even
subalgebra is just the reverse of the representation. -/
theorem coe_toEven_reverse_involute (x : CliffordAlgebra Q) :
↑(toEven Q (reverse (involute x))) =
reverse (Q... | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x y : CliffordAlgebra Q
hx : ↑((toEven Q) (reverse (involute x))) = reverse ↑((toEven Q) x)
hy : ↑((toEven Q) (reverse (involute y))) = reverse ↑((toEven Q) y)
⊢ ↑((toEven Q) (reverse (involute (x * y)))) = re... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | simp only [map_mul, Subalgebra.coe_mul, reverse.map_mul, hx, hy] | set_option synthInstance.maxHeartbeats 30000 in
/-- The representation of the clifford conjugate (i.e. the reverse of the involute) in the even
subalgebra is just the reverse of the representation. -/
theorem coe_toEven_reverse_involute (x : CliffordAlgebra Q) :
↑(toEven Q (reverse (involute x))) =
reverse (Q... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.244_0.ip5LZLfGcNzFdsI | set_option synthInstance.maxHeartbeats 30000 in
/-- The representation of the clifford conjugate (i.e. the reverse of the involute) in the even
subalgebra is just the reverse of the representation. -/
theorem coe_toEven_reverse_involute (x : CliffordAlgebra Q) :
↑(toEven Q (reverse (involute x))) =
reverse (Q... | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
case h_add
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
a✝² b✝ : CliffordAlgebra Q
a✝¹ : ↑((toEven Q) (reverse (involute a✝²))) = reverse ↑((toEven Q) a✝²)
a✝ : ↑((toEven Q) (reverse (involute b✝))) = reverse ↑((toEven Q) b✝)
⊢ ↑((toEven Q) (reverse (inv... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | case h_add x y hx hy => simp only [map_add, Subalgebra.coe_add, hx, hy] | set_option synthInstance.maxHeartbeats 30000 in
/-- The representation of the clifford conjugate (i.e. the reverse of the involute) in the even
subalgebra is just the reverse of the representation. -/
theorem coe_toEven_reverse_involute (x : CliffordAlgebra Q) :
↑(toEven Q (reverse (involute x))) =
reverse (Q... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.244_0.ip5LZLfGcNzFdsI | set_option synthInstance.maxHeartbeats 30000 in
/-- The representation of the clifford conjugate (i.e. the reverse of the involute) in the even
subalgebra is just the reverse of the representation. -/
theorem coe_toEven_reverse_involute (x : CliffordAlgebra Q) :
↑(toEven Q (reverse (involute x))) =
reverse (Q... | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x y : CliffordAlgebra Q
hx : ↑((toEven Q) (reverse (involute x))) = reverse ↑((toEven Q) x)
hy : ↑((toEven Q) (reverse (involute y))) = reverse ↑((toEven Q) y)
⊢ ↑((toEven Q) (reverse (involute (x + y)))) = re... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | case h_add x y hx hy => simp only [map_add, Subalgebra.coe_add, hx, hy] | set_option synthInstance.maxHeartbeats 30000 in
/-- The representation of the clifford conjugate (i.e. the reverse of the involute) in the even
subalgebra is just the reverse of the representation. -/
theorem coe_toEven_reverse_involute (x : CliffordAlgebra Q) :
↑(toEven Q (reverse (involute x))) =
reverse (Q... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.244_0.ip5LZLfGcNzFdsI | set_option synthInstance.maxHeartbeats 30000 in
/-- The representation of the clifford conjugate (i.e. the reverse of the involute) in the even
subalgebra is just the reverse of the representation. -/
theorem coe_toEven_reverse_involute (x : CliffordAlgebra Q) :
↑(toEven Q (reverse (involute x))) =
reverse (Q... | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x y : CliffordAlgebra Q
hx : ↑((toEven Q) (reverse (involute x))) = reverse ↑((toEven Q) x)
hy : ↑((toEven Q) (reverse (involute y))) = reverse ↑((toEven Q) y)
⊢ ↑((toEven Q) (reverse (involute (x + y)))) = re... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | simp only [map_add, Subalgebra.coe_add, hx, hy] | set_option synthInstance.maxHeartbeats 30000 in
/-- The representation of the clifford conjugate (i.e. the reverse of the involute) in the even
subalgebra is just the reverse of the representation. -/
theorem coe_toEven_reverse_involute (x : CliffordAlgebra Q) :
↑(toEven Q (reverse (involute x))) =
reverse (Q... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.244_0.ip5LZLfGcNzFdsI | set_option synthInstance.maxHeartbeats 30000 in
/-- The representation of the clifford conjugate (i.e. the reverse of the involute) in the even
subalgebra is just the reverse of the representation. -/
theorem coe_toEven_reverse_involute (x : CliffordAlgebra Q) :
↑(toEven Q (reverse (involute x))) =
reverse (Q... | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q Q' : QuadraticForm R M
h : Q' = -Q
this✝ : AddCommGroup ↥(even Q') := AddSubgroupClass.toAddCommGroup (even Q')
this : HasDistribNeg ↥(even Q') := NonUnitalNonAssocRing.toHasDistribNeg
m : M
⊢ ((-(even.ι Q').bilin) m) m = (algebr... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | simp_rw [LinearMap.neg_apply, EvenHom.contract, h, QuadraticForm.neg_apply, map_neg,
neg_neg] | set_option synthInstance.maxHeartbeats 30000 in
/-- One direction of `CliffordAlgebra.evenEquivEvenNeg` -/
def evenToNeg (Q' : QuadraticForm R M) (h : Q' = -Q) :
CliffordAlgebra.even Q →ₐ[R] CliffordAlgebra.even Q' :=
even.lift Q <|
-- porting note: added `letI`s
letI : AddCommGroup (even Q') := AddSubgro... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.263_0.ip5LZLfGcNzFdsI | set_option synthInstance.maxHeartbeats 30000 in
/-- One direction of `CliffordAlgebra.evenEquivEvenNeg` -/
def evenToNeg (Q' : QuadraticForm R M) (h : Q' = -Q) :
CliffordAlgebra.even Q →ₐ[R] CliffordAlgebra.even Q' | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q Q' : QuadraticForm R M
h : Q' = -Q
this✝ : AddCommGroup ↥(even Q') := AddSubgroupClass.toAddCommGroup (even Q')
this : HasDistribNeg ↥(even Q') := NonUnitalNonAssocRing.toHasDistribNeg
m₁ m₂ m₃ : M
⊢ ((-(even.ι Q').bilin) m₁) m₂ ... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | simp_rw [LinearMap.neg_apply, neg_mul_neg, EvenHom.contract_mid, h,
QuadraticForm.neg_apply, smul_neg, neg_smul] | set_option synthInstance.maxHeartbeats 30000 in
/-- One direction of `CliffordAlgebra.evenEquivEvenNeg` -/
def evenToNeg (Q' : QuadraticForm R M) (h : Q' = -Q) :
CliffordAlgebra.even Q →ₐ[R] CliffordAlgebra.even Q' :=
even.lift Q <|
-- porting note: added `letI`s
letI : AddCommGroup (even Q') := AddSubgro... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.263_0.ip5LZLfGcNzFdsI | set_option synthInstance.maxHeartbeats 30000 in
/-- One direction of `CliffordAlgebra.evenEquivEvenNeg` -/
def evenToNeg (Q' : QuadraticForm R M) (h : Q' = -Q) :
CliffordAlgebra.even Q →ₐ[R] CliffordAlgebra.even Q' | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q Q' : QuadraticForm R M
h : Q' = -Q
h' : Q = -Q'
⊢ AlgHom.comp (evenToNeg Q' Q h') (evenToNeg Q Q' h) = AlgHom.id R ↥(even Q) | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | ext m₁ m₂ : 4 | set_option synthInstance.maxHeartbeats 100000 in
theorem evenToNeg_comp_evenToNeg (Q' : QuadraticForm R M) (h : Q' = -Q) (h' : Q = -Q') :
(evenToNeg Q' Q h').comp (evenToNeg Q Q' h) = AlgHom.id R _ := by
| Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.287_0.ip5LZLfGcNzFdsI | set_option synthInstance.maxHeartbeats 100000 in
theorem evenToNeg_comp_evenToNeg (Q' : QuadraticForm R M) (h : Q' = -Q) (h' : Q = -Q') :
(evenToNeg Q' Q h').comp (evenToNeg Q Q' h) = AlgHom.id R _ | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
case h.bilin.h.h
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q Q' : QuadraticForm R M
h : Q' = -Q
h' : Q = -Q'
m₁ m₂ : M
⊢ ((EvenHom.compr₂ (even.ι Q) (AlgHom.comp (evenToNeg Q' Q h') (evenToNeg Q Q' h))).bilin m₁) m₂ =
((EvenHom.compr₂ (even.ι Q) (AlgHom.id R ↥(even Q))... | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | dsimp only [EvenHom.compr₂_bilin, LinearMap.compr₂_apply, AlgHom.toLinearMap_apply,
AlgHom.comp_apply, AlgHom.id_apply] | set_option synthInstance.maxHeartbeats 100000 in
theorem evenToNeg_comp_evenToNeg (Q' : QuadraticForm R M) (h : Q' = -Q) (h' : Q = -Q') :
(evenToNeg Q' Q h').comp (evenToNeg Q Q' h) = AlgHom.id R _ := by
ext m₁ m₂ : 4
| Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.287_0.ip5LZLfGcNzFdsI | set_option synthInstance.maxHeartbeats 100000 in
theorem evenToNeg_comp_evenToNeg (Q' : QuadraticForm R M) (h : Q' = -Q) (h' : Q = -Q') :
(evenToNeg Q' Q h').comp (evenToNeg Q Q' h) = AlgHom.id R _ | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
case h.bilin.h.h
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q Q' : QuadraticForm R M
h : Q' = -Q
h' : Q = -Q'
m₁ m₂ : M
⊢ (evenToNeg Q' Q h') ((evenToNeg Q Q' h) (((even.ι Q).bilin m₁) m₂)) = ((even.ι Q).bilin m₁) m₂ | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | rw [evenToNeg_ι, map_neg, evenToNeg_ι, neg_neg] | set_option synthInstance.maxHeartbeats 100000 in
theorem evenToNeg_comp_evenToNeg (Q' : QuadraticForm R M) (h : Q' = -Q) (h' : Q = -Q') :
(evenToNeg Q' Q h').comp (evenToNeg Q Q' h) = AlgHom.id R _ := by
ext m₁ m₂ : 4
dsimp only [EvenHom.compr₂_bilin, LinearMap.compr₂_apply, AlgHom.toLinearMap_apply,
AlgHom... | Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv.287_0.ip5LZLfGcNzFdsI | set_option synthInstance.maxHeartbeats 100000 in
theorem evenToNeg_comp_evenToNeg (Q' : QuadraticForm R M) (h : Q' = -Q) (h' : Q = -Q') :
(evenToNeg Q' Q h').comp (evenToNeg Q Q' h) = AlgHom.id R _ | Mathlib_LinearAlgebra_CliffordAlgebra_EvenEquiv |
ι : Type uι
R : Type uR
A : Type uA
M₁ : Type uM₁
M₂ : Type uM₂
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring A
inst✝⁸ : AddCommMonoid M₁
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : Algebra R A
inst✝⁵ : Module R M₁
inst✝⁴ : Module A M₁
inst✝³ : SMulCommClass R A M₁
inst✝² : SMulCommClass A R M₁
inst✝¹ : IsScalarTower R A M₁
in... | /-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.BilinearForm.Properties
import Mathlib.LinearAlgebra.TensorProduct
import Mathlib.LinearAlgebra.TensorProduct.Tower
#align_import linear_algeb... | rw [isSymm_iff_flip R] | attribute [ext] TensorProduct.ext in
/-- A tensor product of symmetric bilinear forms is symmetric. -/
lemma IsSymm.tmul {B₁ : BilinForm A M₁} {B₂ : BilinForm R M₂}
(hB₁ : B₁.IsSymm) (hB₂ : B₂.IsSymm) : (B₁.tmul B₂).IsSymm := by
| Mathlib.LinearAlgebra.BilinearForm.TensorProduct.72_0.6CsxQfmKdckKOA2 | attribute [ext] TensorProduct.ext in
/-- A tensor product of symmetric bilinear forms is symmetric. -/
lemma IsSymm.tmul {B₁ : BilinForm A M₁} {B₂ : BilinForm R M₂}
(hB₁ : B₁.IsSymm) (hB₂ : B₂.IsSymm) : (B₁.tmul B₂).IsSymm | Mathlib_LinearAlgebra_BilinearForm_TensorProduct |
ι : Type uι
R : Type uR
A : Type uA
M₁ : Type uM₁
M₂ : Type uM₂
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring A
inst✝⁸ : AddCommMonoid M₁
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : Algebra R A
inst✝⁵ : Module R M₁
inst✝⁴ : Module A M₁
inst✝³ : SMulCommClass R A M₁
inst✝² : SMulCommClass A R M₁
inst✝¹ : IsScalarTower R A M₁
in... | /-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.BilinearForm.Properties
import Mathlib.LinearAlgebra.TensorProduct
import Mathlib.LinearAlgebra.TensorProduct.Tower
#align_import linear_algeb... | apply toLin.injective | attribute [ext] TensorProduct.ext in
/-- A tensor product of symmetric bilinear forms is symmetric. -/
lemma IsSymm.tmul {B₁ : BilinForm A M₁} {B₂ : BilinForm R M₂}
(hB₁ : B₁.IsSymm) (hB₂ : B₂.IsSymm) : (B₁.tmul B₂).IsSymm := by
rw [isSymm_iff_flip R]
| Mathlib.LinearAlgebra.BilinearForm.TensorProduct.72_0.6CsxQfmKdckKOA2 | attribute [ext] TensorProduct.ext in
/-- A tensor product of symmetric bilinear forms is symmetric. -/
lemma IsSymm.tmul {B₁ : BilinForm A M₁} {B₂ : BilinForm R M₂}
(hB₁ : B₁.IsSymm) (hB₂ : B₂.IsSymm) : (B₁.tmul B₂).IsSymm | Mathlib_LinearAlgebra_BilinearForm_TensorProduct |
case a
ι : Type uι
R : Type uR
A : Type uA
M₁ : Type uM₁
M₂ : Type uM₂
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring A
inst✝⁸ : AddCommMonoid M₁
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : Algebra R A
inst✝⁵ : Module R M₁
inst✝⁴ : Module A M₁
inst✝³ : SMulCommClass R A M₁
inst✝² : SMulCommClass A R M₁
inst✝¹ : IsScalarTower R ... | /-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.BilinearForm.Properties
import Mathlib.LinearAlgebra.TensorProduct
import Mathlib.LinearAlgebra.TensorProduct.Tower
#align_import linear_algeb... | ext x₁ x₂ y₁ y₂ | attribute [ext] TensorProduct.ext in
/-- A tensor product of symmetric bilinear forms is symmetric. -/
lemma IsSymm.tmul {B₁ : BilinForm A M₁} {B₂ : BilinForm R M₂}
(hB₁ : B₁.IsSymm) (hB₂ : B₂.IsSymm) : (B₁.tmul B₂).IsSymm := by
rw [isSymm_iff_flip R]
apply toLin.injective
| Mathlib.LinearAlgebra.BilinearForm.TensorProduct.72_0.6CsxQfmKdckKOA2 | attribute [ext] TensorProduct.ext in
/-- A tensor product of symmetric bilinear forms is symmetric. -/
lemma IsSymm.tmul {B₁ : BilinForm A M₁} {B₂ : BilinForm R M₂}
(hB₁ : B₁.IsSymm) (hB₂ : B₂.IsSymm) : (B₁.tmul B₂).IsSymm | Mathlib_LinearAlgebra_BilinearForm_TensorProduct |
case a.a.h.h.a.h.h
ι : Type uι
R : Type uR
A : Type uA
M₁ : Type uM₁
M₂ : Type uM₂
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring A
inst✝⁸ : AddCommMonoid M₁
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : Algebra R A
inst✝⁵ : Module R M₁
inst✝⁴ : Module A M₁
inst✝³ : SMulCommClass R A M₁
inst✝² : SMulCommClass A R M₁
inst✝¹ : IsSc... | /-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.BilinearForm.Properties
import Mathlib.LinearAlgebra.TensorProduct
import Mathlib.LinearAlgebra.TensorProduct.Tower
#align_import linear_algeb... | exact (congr_arg₂ (HSMul.hSMul) (hB₂ x₂ y₂) (hB₁ x₁ y₁)).symm | attribute [ext] TensorProduct.ext in
/-- A tensor product of symmetric bilinear forms is symmetric. -/
lemma IsSymm.tmul {B₁ : BilinForm A M₁} {B₂ : BilinForm R M₂}
(hB₁ : B₁.IsSymm) (hB₂ : B₂.IsSymm) : (B₁.tmul B₂).IsSymm := by
rw [isSymm_iff_flip R]
apply toLin.injective
ext x₁ x₂ y₁ y₂
| Mathlib.LinearAlgebra.BilinearForm.TensorProduct.72_0.6CsxQfmKdckKOA2 | attribute [ext] TensorProduct.ext in
/-- A tensor product of symmetric bilinear forms is symmetric. -/
lemma IsSymm.tmul {B₁ : BilinForm A M₁} {B₂ : BilinForm R M₂}
(hB₁ : B₁.IsSymm) (hB₂ : B₂.IsSymm) : (B₁.tmul B₂).IsSymm | Mathlib_LinearAlgebra_BilinearForm_TensorProduct |
ι : Type uι
R : Type uR
A : Type uA
M₁ : Type uM₁
M₂ : Type uM₂
inst✝⁹ : CommRing R
inst✝⁸ : AddCommGroup M₁
inst✝⁷ : AddCommGroup M₂
inst✝⁶ : Module R M₁
inst✝⁵ : Module R M₂
inst✝⁴ : Module.Free R M₁
inst✝³ : Module.Finite R M₁
inst✝² : Module.Free R M₂
inst✝¹ : Module.Finite R M₂
inst✝ : Nontrivial R
⊢ ↑(tensorDistr... | /-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.BilinearForm.Properties
import Mathlib.LinearAlgebra.TensorProduct
import Mathlib.LinearAlgebra.TensorProduct.Tower
#align_import linear_algeb... | ext B₁ B₂ : 3 | variable (R M₁ M₂) in
-- TODO: make this `rfl`
@[simp]
theorem tensorDistribEquiv_toLinearMap :
(tensorDistribEquiv R (M₁ := M₁) (M₂ := M₂)).toLinearMap = tensorDistrib R R := by
| Mathlib.LinearAlgebra.BilinearForm.TensorProduct.134_0.6CsxQfmKdckKOA2 | variable (R M₁ M₂) in
-- TODO: make this `rfl`
@[simp]
theorem tensorDistribEquiv_toLinearMap :
(tensorDistribEquiv R (M₁ | Mathlib_LinearAlgebra_BilinearForm_TensorProduct |
case a.h.h
ι : Type uι
R : Type uR
A : Type uA
M₁ : Type uM₁
M₂ : Type uM₂
inst✝⁹ : CommRing R
inst✝⁸ : AddCommGroup M₁
inst✝⁷ : AddCommGroup M₂
inst✝⁶ : Module R M₁
inst✝⁵ : Module R M₂
inst✝⁴ : Module.Free R M₁
inst✝³ : Module.Finite R M₁
inst✝² : Module.Free R M₂
inst✝¹ : Module.Finite R M₂
inst✝ : Nontrivial R
B₁ :... | /-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.BilinearForm.Properties
import Mathlib.LinearAlgebra.TensorProduct
import Mathlib.LinearAlgebra.TensorProduct.Tower
#align_import linear_algeb... | apply toLin.injective | variable (R M₁ M₂) in
-- TODO: make this `rfl`
@[simp]
theorem tensorDistribEquiv_toLinearMap :
(tensorDistribEquiv R (M₁ := M₁) (M₂ := M₂)).toLinearMap = tensorDistrib R R := by
ext B₁ B₂ : 3
| Mathlib.LinearAlgebra.BilinearForm.TensorProduct.134_0.6CsxQfmKdckKOA2 | variable (R M₁ M₂) in
-- TODO: make this `rfl`
@[simp]
theorem tensorDistribEquiv_toLinearMap :
(tensorDistribEquiv R (M₁ | Mathlib_LinearAlgebra_BilinearForm_TensorProduct |
case a.h.h.a
ι : Type uι
R : Type uR
A : Type uA
M₁ : Type uM₁
M₂ : Type uM₂
inst✝⁹ : CommRing R
inst✝⁸ : AddCommGroup M₁
inst✝⁷ : AddCommGroup M₂
inst✝⁶ : Module R M₁
inst✝⁵ : Module R M₂
inst✝⁴ : Module.Free R M₁
inst✝³ : Module.Finite R M₁
inst✝² : Module.Free R M₂
inst✝¹ : Module.Finite R M₂
inst✝ : Nontrivial R
B₁... | /-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.BilinearForm.Properties
import Mathlib.LinearAlgebra.TensorProduct
import Mathlib.LinearAlgebra.TensorProduct.Tower
#align_import linear_algeb... | ext | variable (R M₁ M₂) in
-- TODO: make this `rfl`
@[simp]
theorem tensorDistribEquiv_toLinearMap :
(tensorDistribEquiv R (M₁ := M₁) (M₂ := M₂)).toLinearMap = tensorDistrib R R := by
ext B₁ B₂ : 3
apply toLin.injective
| Mathlib.LinearAlgebra.BilinearForm.TensorProduct.134_0.6CsxQfmKdckKOA2 | variable (R M₁ M₂) in
-- TODO: make this `rfl`
@[simp]
theorem tensorDistribEquiv_toLinearMap :
(tensorDistribEquiv R (M₁ | Mathlib_LinearAlgebra_BilinearForm_TensorProduct |
case a.h.h.a.a.h.h.a.h.h
ι : Type uι
R : Type uR
A : Type uA
M₁ : Type uM₁
M₂ : Type uM₂
inst✝⁹ : CommRing R
inst✝⁸ : AddCommGroup M₁
inst✝⁷ : AddCommGroup M₂
inst✝⁶ : Module R M₁
inst✝⁵ : Module R M₂
inst✝⁴ : Module.Free R M₁
inst✝³ : Module.Finite R M₁
inst✝² : Module.Free R M₂
inst✝¹ : Module.Finite R M₂
inst✝ : Non... | /-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.BilinearForm.Properties
import Mathlib.LinearAlgebra.TensorProduct
import Mathlib.LinearAlgebra.TensorProduct.Tower
#align_import linear_algeb... | exact mul_comm _ _ | variable (R M₁ M₂) in
-- TODO: make this `rfl`
@[simp]
theorem tensorDistribEquiv_toLinearMap :
(tensorDistribEquiv R (M₁ := M₁) (M₂ := M₂)).toLinearMap = tensorDistrib R R := by
ext B₁ B₂ : 3
apply toLin.injective
ext
| Mathlib.LinearAlgebra.BilinearForm.TensorProduct.134_0.6CsxQfmKdckKOA2 | variable (R M₁ M₂) in
-- TODO: make this `rfl`
@[simp]
theorem tensorDistribEquiv_toLinearMap :
(tensorDistribEquiv R (M₁ | Mathlib_LinearAlgebra_BilinearForm_TensorProduct |
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : MetricSpace β
inst✝ : NormedAddCommGroup E
f : C(α, β)
⊢ (mkOfCompact f).toContinuousMap = f | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | ext | /-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
| Mathlib.Topology.ContinuousFunction.Compact.46_0.Mig2jTVnn2FLKEB | /-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config | Mathlib_Topology_ContinuousFunction_Compact |
case h
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : MetricSpace β
inst✝ : NormedAddCommGroup E
f : C(α, β)
a✝ : α
⊢ (mkOfCompact f).toContinuousMap a✝ = f a✝ | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | rfl | /-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
| Mathlib.Topology.ContinuousFunction.Compact.46_0.Mig2jTVnn2FLKEB | /-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config | Mathlib_Topology_ContinuousFunction_Compact |
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : MetricSpace β
inst✝ : NormedAddCommGroup E
f : α →ᵇ β
⊢ mkOfCompact f.toContinuousMap = f | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | ext | /-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
| Mathlib.Topology.ContinuousFunction.Compact.46_0.Mig2jTVnn2FLKEB | /-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config | Mathlib_Topology_ContinuousFunction_Compact |
case h
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : MetricSpace β
inst✝ : NormedAddCommGroup E
f : α →ᵇ β
x✝ : α
⊢ (mkOfCompact f.toContinuousMap) x✝ = f x✝ | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | rfl | /-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config := .asFn)]
def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
⟨mkOfCompact, BoundedContinuousFunction.toContinuousMap, fun f => by
ext
rfl, fun f => by
ext
| Mathlib.Topology.ContinuousFunction.Compact.46_0.Mig2jTVnn2FLKEB | /-- When `α` is compact, the bounded continuous maps `α →ᵇ β` are
equivalent to `C(α, β)`.
-/
@[simps (config | Mathlib_Topology_ContinuousFunction_Compact |
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : MetricSpace β
inst✝ : NormedAddCommGroup E
⊢ ∀ (s : Set (C(α, β) × C(α, β))),
s ∈ uniformity C(α, β) ↔
∃ t ∈ uniformity (α →ᵇ β),
∀ (x y : C(α, β)), ((equivBoundedOfCompact α β) x, (equivBoundedOfCompac... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff] | theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
| Mathlib.Topology.ContinuousFunction.Compact.58_0.Mig2jTVnn2FLKEB | theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) | Mathlib_Topology_ContinuousFunction_Compact |
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : MetricSpace β
inst✝ : NormedAddCommGroup E
⊢ ∀ (s : Set (C(α, β) × C(α, β))),
(∃ i,
(IsCompact i.1 ∧ ∃ i_1, 0 < i_1 ∧ {p | dist p.1 p.2 ≤ i_1} ⊆ i.2) ∧
{fg | ∀ x ∈ i.1, (fg.1 x, fg.2 x) ∈ i.2} ⊆ s) ... | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | exact fun s =>
⟨fun ⟨⟨a, b⟩, ⟨_, ⟨ε, hε, hb⟩⟩, hs⟩ =>
⟨{ p | ∀ x, (p.1 x, p.2 x) ∈ b }, ⟨ε, hε, fun _ h x => hb ((dist_le hε.le).mp h x)⟩,
fun f g h => hs fun x _ => h x⟩,
fun ⟨_, ⟨ε, hε, ht⟩, hs⟩ =>
⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ ... | theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
UniformInducing.mk'
(by
simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
| Mathlib.Topology.ContinuousFunction.Compact.58_0.Mig2jTVnn2FLKEB | theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) | Mathlib_Topology_ContinuousFunction_Compact |
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : MetricSpace β
inst✝ : NormedAddCommGroup E
f g : C(α, β)
C : ℝ
x : α
⊢ dist (f x) (g x) ≤ dist f g | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | simp only [← dist_mkOfCompact, dist_coe_le_dist, ← mkOfCompact_apply] | /-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g := by
| Mathlib.Topology.ContinuousFunction.Compact.129_0.Mig2jTVnn2FLKEB | /-- The pointwise distance is controlled by the distance between functions, by definition. -/
theorem dist_apply_le_dist (x : α) : dist (f x) (g x) ≤ dist f g | Mathlib_Topology_ContinuousFunction_Compact |
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : MetricSpace β
inst✝ : NormedAddCommGroup E
f g : C(α, β)
C : ℝ
C0 : 0 ≤ C
⊢ dist f g ≤ C ↔ ∀ (x : α), dist (f x) (g x) ≤ C | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply] | /-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by
| Mathlib.Topology.ContinuousFunction.Compact.134_0.Mig2jTVnn2FLKEB | /-- The distance between two functions is controlled by the supremum of the pointwise distances. -/
theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C | Mathlib_Topology_ContinuousFunction_Compact |
α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁴ : TopologicalSpace α
inst✝³ : CompactSpace α
inst✝² : MetricSpace β
inst✝¹ : NormedAddCommGroup E
f g : C(α, β)
C : ℝ
inst✝ : Nonempty α
⊢ dist f g ≤ C ↔ ∀ (x : α), dist (f x) (g x) ≤ C | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le_iff_of_nonempty,
mkOfCompact_apply] | theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C := by
| Mathlib.Topology.ContinuousFunction.Compact.139_0.Mig2jTVnn2FLKEB | theorem dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C | Mathlib_Topology_ContinuousFunction_Compact |
α : Type u_1
β : Type u_2
E : Type u_3
inst✝⁴ : TopologicalSpace α
inst✝³ : CompactSpace α
inst✝² : MetricSpace β
inst✝¹ : NormedAddCommGroup E
f g : C(α, β)
C : ℝ
inst✝ : Nonempty α
⊢ dist f g < C ↔ ∀ (x : α), dist (f x) (g x) < C | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply] | theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
| Mathlib.Topology.ContinuousFunction.Compact.144_0.Mig2jTVnn2FLKEB | theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C | Mathlib_Topology_ContinuousFunction_Compact |
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : MetricSpace β
inst✝ : NormedAddCommGroup E
f g : C(α, β)
C : ℝ
C0 : 0 < C
⊢ dist f g < C ↔ ∀ (x : α), dist (f x) (g x) < C | /-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0] | theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
| Mathlib.Topology.ContinuousFunction.Compact.152_0.Mig2jTVnn2FLKEB | theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C | Mathlib_Topology_ContinuousFunction_Compact |
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