fact
stringlengths 6
3.84k
| type
stringclasses 11
values | library
stringclasses 32
values | imports
listlengths 1
14
| filename
stringlengths 20
95
| symbolic_name
stringlengths 1
90
| docstring
stringlengths 7
20k
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|---|---|---|---|---|---|---|
@[ext]
ext : a.re = b.re → a.imI = b.imI → a.imJ = b.imJ → a.imK = b.imK → a = b :=
QuaternionAlgebra.ext
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
ext
| null |
im (x : ℍ[R]) : ℍ[R] := QuaternionAlgebra.im x
@[simp] theorem re_im : a.im.re = 0 := rfl
@[deprecated (since := "2025-08-31")] alias im_re := re_im
@[simp] theorem imI_im : a.im.imI = a.imI := rfl
@[deprecated (since := "2025-08-31")] alias im_imI := imI_im
@[simp] theorem imJ_im : a.im.imJ = a.imJ := rfl
@[deprecated (since := "2025-08-31")] alias im_imJ := imJ_im
@[simp] theorem imK_im : a.im.imK = a.imK := rfl
@[deprecated (since := "2025-08-31")] alias im_imK := imK_im
@[simp] theorem im_idem : a.im.im = a.im := rfl
@[simp] theorem re_add_im : ↑a.re + a.im = a := QuaternionAlgebra.re_add_im a
@[simp] theorem sub_im_self : a - a.im = a.re := QuaternionAlgebra.sub_im_self a
@[deprecated (since := "2025-08-31")] alias sub_self_im := sub_im_self
@[simp] theorem sub_re_self : a - ↑a.re = a.im := QuaternionAlgebra.sub_re_self a
@[deprecated (since := "2025-08-31")] alias sub_self_re := sub_re_self
@[simp, norm_cast]
|
def
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
im
|
The imaginary part of a quaternion.
|
re_coe : (x : ℍ[R]).re = x := rfl
@[deprecated (since := "2025-08-31")] alias coe_re := re_coe
@[simp, norm_cast]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
re_coe
| null |
imI_coe : (x : ℍ[R]).imI = 0 := rfl
@[deprecated (since := "2025-08-31")] alias coe_imI := imI_coe
@[simp, norm_cast]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
imI_coe
| null |
imJ_coe : (x : ℍ[R]).imJ = 0 := rfl
@[deprecated (since := "2025-08-31")] alias coe_imJ := imJ_coe
@[simp, norm_cast]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
imJ_coe
| null |
imK_coe : (x : ℍ[R]).imK = 0 := rfl
@[deprecated (since := "2025-08-31")] alias coe_imK := imK_coe
@[simp, norm_cast]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
imK_coe
| null |
im_coe : (x : ℍ[R]).im = 0 := rfl
@[deprecated (since := "2025-08-31")] alias coe_im := im_coe
@[scoped simp] theorem re_zero : (0 : ℍ[R]).re = 0 := rfl
@[deprecated (since := "2025-08-31")] alias zero_re := re_zero
@[scoped simp] theorem imI_zero : (0 : ℍ[R]).imI = 0 := rfl
@[deprecated (since := "2025-08-31")] alias zero_imI := imI_zero
@[scoped simp] theorem imJ_zero : (0 : ℍ[R]).imJ = 0 := rfl
@[deprecated (since := "2025-08-31")] alias zero_imJ := imJ_zero
@[scoped simp] theorem imK_zero : (0 : ℍ[R]).imK = 0 := rfl
@[deprecated (since := "2025-08-31")] alias zero_imK := imK_zero
@[scoped simp] theorem im_zero : (0 : ℍ[R]).im = 0 := rfl
@[deprecated (since := "2025-08-31")] alias zero_im := im_zero
@[simp, norm_cast]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
im_coe
| null |
coe_zero : ((0 : R) : ℍ[R]) = 0 := rfl
@[scoped simp] theorem re_one : (1 : ℍ[R]).re = 1 := rfl
@[deprecated (since := "2025-08-31")] alias one_re := re_one
@[scoped simp] theorem imI_one : (1 : ℍ[R]).imI = 0 := rfl
@[deprecated (since := "2025-08-31")] alias one_imI := imI_one
@[scoped simp] theorem imJ_one : (1 : ℍ[R]).imJ = 0 := rfl
@[deprecated (since := "2025-08-31")] alias one_imJ := imJ_one
@[scoped simp] theorem imK_one : (1 : ℍ[R]).imK = 0 := rfl
@[deprecated (since := "2025-08-31")] alias one_imK := imK_one
@[scoped simp] theorem im_one : (1 : ℍ[R]).im = 0 := rfl
@[deprecated (since := "2025-08-31")] alias one_im := im_one
@[simp, norm_cast]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
coe_zero
| null |
coe_one : ((1 : R) : ℍ[R]) = 1 := rfl
@[simp] theorem re_add : (a + b).re = a.re + b.re := rfl
@[deprecated (since := "2025-08-31")] alias add_re := re_add
@[simp] theorem imI_add : (a + b).imI = a.imI + b.imI := rfl
@[deprecated (since := "2025-08-31")] alias add_imI := imI_add
@[simp] theorem imJ_add : (a + b).imJ = a.imJ + b.imJ := rfl
@[deprecated (since := "2025-08-31")] alias add_imJ := imJ_add
@[simp] theorem imK_add : (a + b).imK = a.imK + b.imK := rfl
@[deprecated (since := "2025-08-31")] alias add_imK := imK_add
@[simp] theorem im_add : (a + b).im = a.im + b.im := QuaternionAlgebra.im_add a b
@[deprecated (since := "2025-08-31")] alias add_im := im_add
@[simp, norm_cast]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
coe_one
| null |
coe_add : ((x + y : R) : ℍ[R]) = x + y :=
QuaternionAlgebra.coe_add x y
@[simp] theorem re_neg : (-a).re = -a.re := rfl
@[deprecated (since := "2025-08-31")] alias neg_re := re_neg
@[simp] theorem imI_neg : (-a).imI = -a.imI := rfl
@[deprecated (since := "2025-08-31")] alias neg_imI := imI_neg
@[simp] theorem imJ_neg : (-a).imJ = -a.imJ := rfl
@[deprecated (since := "2025-08-31")] alias neg_imJ := imJ_neg
@[simp] theorem imK_neg : (-a).imK = -a.imK := rfl
@[deprecated (since := "2025-08-31")] alias neg_imK := imK_neg
@[simp] theorem im_neg : (-a).im = -a.im := QuaternionAlgebra.im_neg a
@[deprecated (since := "2025-08-31")] alias neg_im := im_neg
@[simp, norm_cast]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
coe_add
| null |
coe_neg : ((-x : R) : ℍ[R]) = -x :=
QuaternionAlgebra.coe_neg x
@[simp] theorem re_sub : (a - b).re = a.re - b.re := rfl
@[deprecated (since := "2025-08-31")] alias sub_re := re_sub
@[simp] theorem imI_sub : (a - b).imI = a.imI - b.imI := rfl
@[deprecated (since := "2025-08-31")] alias sub_imI := imI_sub
@[simp] theorem imJ_sub : (a - b).imJ = a.imJ - b.imJ := rfl
@[deprecated (since := "2025-08-31")] alias sub_imJ := imJ_sub
@[simp] theorem imK_sub : (a - b).imK = a.imK - b.imK := rfl
@[deprecated (since := "2025-08-31")] alias sub_imK := imK_sub
@[simp] theorem im_sub : (a - b).im = a.im - b.im := QuaternionAlgebra.im_sub a b
@[deprecated (since := "2025-08-31")] alias sub_im := im_sub
@[simp, norm_cast]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
coe_neg
| null |
coe_sub : ((x - y : R) : ℍ[R]) = x - y :=
QuaternionAlgebra.coe_sub x y
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
coe_sub
| null |
re_mul : (a * b).re = a.re * b.re - a.imI * b.imI - a.imJ * b.imJ - a.imK * b.imK :=
(QuaternionAlgebra.re_mul a b).trans <| by simp [one_mul, neg_mul, sub_eq_add_neg, neg_neg]
@[deprecated (since := "2025-08-31")] alias mul_re := re_mul
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
re_mul
| null |
imI_mul : (a * b).imI = a.re * b.imI + a.imI * b.re + a.imJ * b.imK - a.imK * b.imJ :=
(QuaternionAlgebra.imI_mul a b).trans <| by ring
@[deprecated (since := "2025-08-31")] alias mul_imI := imI_mul
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
imI_mul
| null |
imJ_mul : (a * b).imJ = a.re * b.imJ - a.imI * b.imK + a.imJ * b.re + a.imK * b.imI :=
(QuaternionAlgebra.imJ_mul a b).trans <| by ring
@[deprecated (since := "2025-08-31")] alias mul_imJ := imJ_mul
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
imJ_mul
| null |
imK_mul : (a * b).imK = a.re * b.imK + a.imI * b.imJ - a.imJ * b.imI + a.imK * b.re :=
(QuaternionAlgebra.imK_mul a b).trans <| by ring
@[deprecated (since := "2025-08-31")] alias mul_imK := imK_mul
@[simp, norm_cast]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
imK_mul
| null |
coe_mul : ((x * y : R) : ℍ[R]) = x * y := QuaternionAlgebra.coe_mul x y
@[norm_cast, simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
coe_mul
| null |
coe_pow (n : ℕ) : (↑(x ^ n) : ℍ[R]) = (x : ℍ[R]) ^ n :=
QuaternionAlgebra.coe_pow x n
@[simp, norm_cast]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
coe_pow
| null |
re_natCast (n : ℕ) : (n : ℍ[R]).re = n := rfl
@[deprecated (since := "2025-08-31")] alias natCast_re := re_natCast
@[simp, norm_cast]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
re_natCast
| null |
imI_natCast (n : ℕ) : (n : ℍ[R]).imI = 0 := rfl
@[deprecated (since := "2025-08-31")] alias natCast_imI := imI_natCast
@[simp, norm_cast]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
imI_natCast
| null |
imJ_natCast (n : ℕ) : (n : ℍ[R]).imJ = 0 := rfl
@[deprecated (since := "2025-08-31")] alias natCast_imJ := imJ_natCast
@[simp, norm_cast]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
imJ_natCast
| null |
imK_natCast (n : ℕ) : (n : ℍ[R]).imK = 0 := rfl
@[deprecated (since := "2025-08-31")] alias natCast_imK := imK_natCast
@[simp, norm_cast]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
imK_natCast
| null |
im_natCast (n : ℕ) : (n : ℍ[R]).im = 0 := rfl
@[deprecated (since := "2025-08-31")] alias natCast_im := im_natCast
@[norm_cast]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
im_natCast
| null |
coe_natCast (n : ℕ) : ↑(n : R) = (n : ℍ[R]) := rfl
@[simp, norm_cast]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
coe_natCast
| null |
re_intCast (z : ℤ) : (z : ℍ[R]).re = z := rfl
@[deprecated (since := "2025-08-31")] alias intCast_re := re_intCast
@[simp, norm_cast]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
re_intCast
| null |
imI_intCast (z : ℤ) : (z : ℍ[R]).imI = 0 := rfl
@[deprecated (since := "2025-08-31")] alias intCast_imI := imI_intCast
@[simp, norm_cast]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
imI_intCast
| null |
imJ_intCast (z : ℤ) : (z : ℍ[R]).imJ = 0 := rfl
@[deprecated (since := "2025-08-31")] alias intCast_imJ := imJ_intCast
@[simp, norm_cast]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
imJ_intCast
| null |
imK_intCast (z : ℤ) : (z : ℍ[R]).imK = 0 := rfl
@[deprecated (since := "2025-08-31")] alias intCast_imK := imK_intCast
@[simp, norm_cast]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
imK_intCast
| null |
im_intCast (z : ℤ) : (z : ℍ[R]).im = 0 := rfl
@[deprecated (since := "2025-08-31")] alias intCast_im := im_intCast
@[norm_cast]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
im_intCast
| null |
coe_intCast (z : ℤ) : ↑(z : R) = (z : ℍ[R]) := rfl
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
coe_intCast
| null |
coe_injective : Function.Injective (coe : R → ℍ[R]) :=
QuaternionAlgebra.coe_injective
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
coe_injective
| null |
coe_inj {x y : R} : (x : ℍ[R]) = y ↔ x = y :=
coe_injective.eq_iff
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
coe_inj
| null |
re_smul [SMul S R] (s : S) : (s • a).re = s • a.re :=
rfl
@[deprecated (since := "2025-08-31")] alias smul_re := re_smul
@[simp] theorem imI_smul [SMul S R] (s : S) : (s • a).imI = s • a.imI := rfl
@[deprecated (since := "2025-08-31")] alias smul_imI := imI_smul
@[simp] theorem imJ_smul [SMul S R] (s : S) : (s • a).imJ = s • a.imJ := rfl
@[deprecated (since := "2025-08-31")] alias smul_imJ := imJ_smul
@[simp] theorem imK_smul [SMul S R] (s : S) : (s • a).imK = s • a.imK := rfl
@[deprecated (since := "2025-08-31")] alias smul_imK := imK_smul
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
re_smul
| null |
im_smul [SMulZeroClass S R] (s : S) : (s • a).im = s • a.im :=
QuaternionAlgebra.im_smul a s
@[deprecated (since := "2025-08-31")] alias smul_im := im_smul
@[simp, norm_cast]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
im_smul
| null |
coe_smul [SMulZeroClass S R] (s : S) (r : R) : (↑(s • r) : ℍ[R]) = s • (r : ℍ[R]) :=
QuaternionAlgebra.coe_smul _ _
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
coe_smul
| null |
coe_commutes : ↑r * a = a * r :=
QuaternionAlgebra.coe_commutes r a
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
coe_commutes
| null |
coe_commute : Commute (↑r) a :=
QuaternionAlgebra.coe_commute r a
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
coe_commute
| null |
coe_mul_eq_smul : ↑r * a = r • a :=
QuaternionAlgebra.coe_mul_eq_smul r a
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
coe_mul_eq_smul
| null |
mul_coe_eq_smul : a * r = r • a :=
QuaternionAlgebra.mul_coe_eq_smul r a
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
mul_coe_eq_smul
| null |
algebraMap_def : ⇑(algebraMap R ℍ[R]) = coe :=
rfl
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
algebraMap_def
| null |
algebraMap_injective : (algebraMap R ℍ[R] : _ → _).Injective :=
QuaternionAlgebra.algebraMap_injective
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
algebraMap_injective
| null |
smul_coe : x • (y : ℍ[R]) = ↑(x * y) :=
QuaternionAlgebra.smul_coe x y
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
smul_coe
| null |
rank_eq_four [StrongRankCondition R] : Module.rank R ℍ[R] = 4 :=
QuaternionAlgebra.rank_eq_four _ _ _
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
rank_eq_four
| null |
finrank_eq_four [StrongRankCondition R] : Module.finrank R ℍ[R] = 4 :=
QuaternionAlgebra.finrank_eq_four _ _ _
@[simp] theorem re_star : (star a).re = a.re := by
rw [QuaternionAlgebra.re_star, zero_mul, add_zero]
@[deprecated (since := "2025-08-31")] alias star_re := re_star
@[simp] theorem imI_star : (star a).imI = -a.imI := rfl
@[deprecated (since := "2025-08-31")] alias star_imI := imI_star
@[simp] theorem imJ_star : (star a).imJ = -a.imJ := rfl
@[deprecated (since := "2025-08-31")] alias star_imJ := imJ_star
@[simp] theorem imK_star : (star a).imK = -a.imK := rfl
@[deprecated (since := "2025-08-31")] alias star_imK := imK_star
@[simp] theorem im_star : (star a).im = -a.im := QuaternionAlgebra.im_star a
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
finrank_eq_four
| null |
self_add_star' : a + star a = ↑(2 * a.re) := by
simpa using QuaternionAlgebra.self_add_star' a
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
self_add_star'
| null |
self_add_star : a + star a = 2 * a.re := by
simpa using QuaternionAlgebra.self_add_star a
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
self_add_star
| null |
star_add_self' : star a + a = ↑(2 * a.re) := by
simpa using QuaternionAlgebra.star_add_self' a
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
star_add_self'
| null |
star_add_self : star a + a = 2 * a.re := by
simpa using QuaternionAlgebra.star_add_self a
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
star_add_self
| null |
star_eq_two_re_sub : star a = ↑(2 * a.re) - a := by
simpa using QuaternionAlgebra.star_eq_two_re_sub a
@[simp, norm_cast]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
star_eq_two_re_sub
| null |
star_coe : star (x : ℍ[R]) = x :=
QuaternionAlgebra.star_coe x
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
star_coe
| null |
star_im : star a.im = -a.im := by ext <;> simp
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
star_im
| null |
star_smul [Monoid S] [DistribMulAction S R] (s : S) (a : ℍ[R]) :
star (s • a) = s • star a := QuaternionAlgebra.star_smul' s a
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
star_smul
| null |
eq_re_of_eq_coe {a : ℍ[R]} {x : R} (h : a = x) : a = a.re :=
QuaternionAlgebra.eq_re_of_eq_coe h
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
eq_re_of_eq_coe
| null |
eq_re_iff_mem_range_coe {a : ℍ[R]} : a = a.re ↔ a ∈ Set.range (coe : R → ℍ[R]) :=
QuaternionAlgebra.eq_re_iff_mem_range_coe
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
eq_re_iff_mem_range_coe
| null |
@[simp]
star_eq_self {a : ℍ[R]} : star a = a ↔ a = a.re :=
QuaternionAlgebra.star_eq_self
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
star_eq_self
| null |
star_eq_neg {a : ℍ[R]} : star a = -a ↔ a.re = 0 :=
QuaternionAlgebra.star_eq_neg
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
star_eq_neg
| null |
star_mul_eq_coe : star a * a = (star a * a).re :=
QuaternionAlgebra.star_mul_eq_coe a
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
star_mul_eq_coe
| null |
mul_star_eq_coe : a * star a = (a * star a).re :=
QuaternionAlgebra.mul_star_eq_coe a
open MulOpposite
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
mul_star_eq_coe
| null |
starAe : ℍ[R] ≃ₐ[R] ℍ[R]ᵐᵒᵖ :=
QuaternionAlgebra.starAe
@[simp]
|
def
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
starAe
|
Quaternion conjugate as an `AlgEquiv` to the opposite ring.
|
coe_starAe : ⇑(starAe : ℍ[R] ≃ₐ[R] ℍ[R]ᵐᵒᵖ) = op ∘ star :=
rfl
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
coe_starAe
| null |
normSq : ℍ[R] →*₀ R where
toFun a := (a * star a).re
map_zero' := by simp only [star_zero, zero_mul, re_zero]
map_one' := by simp only [star_one, one_mul, re_one]
map_mul' x y := coe_injective <| by
conv_lhs => rw [← mul_star_eq_coe, star_mul, mul_assoc, ← mul_assoc y, y.mul_star_eq_coe,
coe_commutes, ← mul_assoc, x.mul_star_eq_coe, ← coe_mul]
|
def
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
normSq
|
Square of the norm.
|
normSq_def : normSq a = (a * star a).re := rfl
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
normSq_def
| null |
normSq_def' : normSq a = a.1 ^ 2 + a.2 ^ 2 + a.3 ^ 2 + a.4 ^ 2 := by
simp only [normSq_def, sq, mul_neg, sub_neg_eq_add, re_mul, re_star, imI_star, imJ_star,
imK_star]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
normSq_def'
| null |
normSq_coe : normSq (x : ℍ[R]) = x ^ 2 := by
rw [normSq_def, star_coe, ← coe_mul, re_coe, sq]
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
normSq_coe
| null |
normSq_star : normSq (star a) = normSq a := by simp [normSq_def']
@[norm_cast]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
normSq_star
| null |
normSq_natCast (n : ℕ) : normSq (n : ℍ[R]) = (n : R) ^ 2 := by
rw [← coe_natCast, normSq_coe]
@[norm_cast]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
normSq_natCast
| null |
normSq_intCast (z : ℤ) : normSq (z : ℍ[R]) = (z : R) ^ 2 := by
rw [← coe_intCast, normSq_coe]
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
normSq_intCast
| null |
normSq_neg : normSq (-a) = normSq a := by simp only [normSq_def, star_neg, neg_mul_neg]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
normSq_neg
| null |
self_mul_star : a * star a = normSq a := by rw [mul_star_eq_coe, normSq_def]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
self_mul_star
| null |
star_mul_self : star a * a = normSq a := by rw [star_comm_self, self_mul_star]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
star_mul_self
| null |
im_sq : a.im ^ 2 = -normSq a.im := by
simp_rw [sq, ← star_mul_self, star_im, neg_mul, neg_neg]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
im_sq
| null |
coe_normSq_add : normSq (a + b) = normSq a + a * star b + b * star a + normSq b := by
simp only [star_add, ← self_mul_star, mul_add, add_mul, add_assoc, add_left_comm]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
coe_normSq_add
| null |
normSq_smul (r : R) (q : ℍ[R]) : normSq (r • q) = r ^ 2 * normSq q := by
simp only [normSq_def', re_smul, imI_smul, imJ_smul, imK_smul, mul_pow, mul_add, smul_eq_mul]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
normSq_smul
| null |
normSq_add (a b : ℍ[R]) : normSq (a + b) = normSq a + normSq b + 2 * (a * star b).re :=
calc
normSq (a + b) = normSq a + (a * star b).re + ((b * star a).re + normSq b) := by
simp_rw [normSq_def, star_add, add_mul, mul_add, re_add]
_ = normSq a + normSq b + ((a * star b).re + (b * star a).re) := by abel
_ = normSq a + normSq b + 2 * (a * star b).re := by
rw [← re_add, ← star_mul_star a b, self_add_star', re_coe]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
normSq_add
| null |
@[simp]
normSq_eq_zero : normSq a = 0 ↔ a = 0 := by
refine ⟨fun h => ?_, fun h => h.symm ▸ normSq.map_zero⟩
rw [normSq_def', add_eq_zero_iff_of_nonneg, add_eq_zero_iff_of_nonneg, add_eq_zero_iff_of_nonneg]
at h
· exact ext a 0 (pow_eq_zero h.1.1.1) (pow_eq_zero h.1.1.2) (pow_eq_zero h.1.2) (pow_eq_zero h.2)
all_goals apply_rules [sq_nonneg, add_nonneg]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
normSq_eq_zero
| null |
normSq_ne_zero : normSq a ≠ 0 ↔ a ≠ 0 := normSq_eq_zero.not
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
normSq_ne_zero
| null |
normSq_nonneg : 0 ≤ normSq a := by
rw [normSq_def']
apply_rules [sq_nonneg, add_nonneg]
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
normSq_nonneg
| null |
normSq_le_zero : normSq a ≤ 0 ↔ a = 0 :=
normSq_nonneg.ge_iff_eq'.trans normSq_eq_zero
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
normSq_le_zero
| null |
instNontrivial : Nontrivial ℍ[R] where
exists_pair_ne := ⟨0, 1, mt (congr_arg QuaternionAlgebra.re) zero_ne_one⟩
|
instance
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
instNontrivial
| null |
sq_eq_normSq : a ^ 2 = normSq a ↔ a = a.re := by
rw [← star_eq_self, ← star_mul_self, sq, mul_eq_mul_right_iff, eq_comm]
exact or_iff_left_of_imp fun ha ↦ ha.symm ▸ star_zero _
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
sq_eq_normSq
| null |
sq_eq_neg_normSq : a ^ 2 = -normSq a ↔ a.re = 0 := by
simp_rw [← star_eq_neg]
obtain rfl | hq0 := eq_or_ne a 0
· simp
· rw [← star_mul_self, ← mul_neg, ← neg_sq, sq, mul_left_inj' (neg_ne_zero.mpr hq0), eq_comm]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
sq_eq_neg_normSq
| null |
instNNRatCast : NNRatCast ℍ[R] where nnratCast q := (q : R)
|
instance
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
instNNRatCast
| null |
instRatCast : RatCast ℍ[R] where ratCast q := (q : R)
@[simp, norm_cast] lemma re_nnratCast (q : ℚ≥0) : (q : ℍ[R]).re = q := rfl
@[simp, norm_cast] lemma im_nnratCast (q : ℚ≥0) : (q : ℍ[R]).im = 0 := rfl
@[simp, norm_cast] lemma imI_nnratCast (q : ℚ≥0) : (q : ℍ[R]).imI = 0 := rfl
@[simp, norm_cast] lemma imJ_nnratCast (q : ℚ≥0) : (q : ℍ[R]).imJ = 0 := rfl
@[simp, norm_cast] lemma imK_nnratCast (q : ℚ≥0) : (q : ℍ[R]).imK = 0 := rfl
@[simp, norm_cast] lemma re_ratCast (q : ℚ) : (q : ℍ[R]).re = q := rfl
@[simp, norm_cast] lemma im_ratCast (q : ℚ) : (q : ℍ[R]).im = 0 := rfl
@[simp, norm_cast] lemma imI_ratCast (q : ℚ) : (q : ℍ[R]).imI = 0 := rfl
@[simp, norm_cast] lemma imJ_ratCast (q : ℚ) : (q : ℍ[R]).imJ = 0 := rfl
@[simp, norm_cast] lemma imK_ratCast (q : ℚ) : (q : ℍ[R]).imK = 0 := rfl
@[deprecated (since := "2025-08-31")] alias ratCast_re := re_ratCast
@[deprecated (since := "2025-08-31")] alias ratCast_im := im_ratCast
@[deprecated (since := "2025-08-31")] alias ratCast_imI := imI_ratCast
@[deprecated (since := "2025-08-31")] alias ratCast_imJ := imJ_ratCast
@[deprecated (since := "2025-08-31")] alias ratCast_imK := imK_ratCast
@[norm_cast] lemma coe_nnratCast (q : ℚ≥0) : ↑(q : R) = (q : ℍ[R]) := rfl
@[norm_cast] lemma coe_ratCast (q : ℚ) : ↑(q : R) = (q : ℍ[R]) := rfl
variable [LinearOrder R] [IsStrictOrderedRing R] (a b : ℍ[R])
@[simps -isSimp]
|
instance
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
instRatCast
| null |
instInv : Inv ℍ[R] :=
⟨fun a => (normSq a)⁻¹ • star a⟩
|
instance
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
instInv
| null |
instGroupWithZero : GroupWithZero ℍ[R] :=
{ Quaternion.instNontrivial with
inv := Inv.inv
inv_zero := by rw [inv_def, star_zero, smul_zero]
mul_inv_cancel := fun a ha => by
rw [inv_def, Algebra.mul_smul_comm (normSq a)⁻¹ a (star a), self_mul_star, smul_coe,
inv_mul_cancel₀ (normSq_ne_zero.2 ha), coe_one] }
@[norm_cast, simp]
|
instance
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
instGroupWithZero
| null |
coe_inv (x : R) : ((x⁻¹ : R) : ℍ[R]) = (↑x)⁻¹ :=
map_inv₀ (algebraMap R ℍ[R]) _
@[norm_cast, simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
coe_inv
| null |
coe_div (x y : R) : ((x / y : R) : ℍ[R]) = x / y :=
map_div₀ (algebraMap R ℍ[R]) x y
@[norm_cast, simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
coe_div
| null |
coe_zpow (x : R) (z : ℤ) : ((x ^ z : R) : ℍ[R]) = (x : ℍ[R]) ^ z :=
map_zpow₀ (algebraMap R ℍ[R]) x z
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
coe_zpow
| null |
instDivisionRing : DivisionRing ℍ[R] where
__ := Quaternion.instRing
__ := Quaternion.instGroupWithZero
nnqsmul := (· • ·)
qsmul := (· • ·)
nnratCast_def _ := by rw [← coe_nnratCast, NNRat.cast_def, coe_div, coe_natCast, coe_natCast]
ratCast_def _ := by rw [← coe_ratCast, Rat.cast_def, coe_div, coe_intCast, coe_natCast]
nnqsmul_def _ _ := by rw [← coe_nnratCast, coe_mul_eq_smul]; ext <;> exact NNRat.smul_def ..
qsmul_def _ _ := by rw [← coe_ratCast, coe_mul_eq_smul]; ext <;> exact Rat.smul_def ..
|
instance
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
instDivisionRing
| null |
normSq_inv : normSq a⁻¹ = (normSq a)⁻¹ :=
map_inv₀ normSq _
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
normSq_inv
| null |
normSq_div : normSq (a / b) = normSq a / normSq b :=
map_div₀ normSq a b
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
normSq_div
| null |
normSq_zpow (z : ℤ) : normSq (a ^ z) = normSq a ^ z :=
map_zpow₀ normSq a z
@[norm_cast]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
normSq_zpow
| null |
normSq_ratCast (q : ℚ) : normSq (q : ℍ[R]) = (q : ℍ[R]) ^ 2 := by
rw [← coe_ratCast, normSq_coe, coe_pow]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
normSq_ratCast
| null |
private pow_four [Infinite R] : #R ^ 4 = #R :=
power_nat_eq (aleph0_le_mk R) <| by decide
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
pow_four
| null |
mk_quaternionAlgebra : #(ℍ[R,c₁,c₂,c₃]) = #R ^ 4 := by
rw [mk_congr (QuaternionAlgebra.equivProd c₁ c₂ c₃)]
simp only [mk_prod, lift_id]
ring
@[simp]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
mk_quaternionAlgebra
|
The cardinality of a quaternion algebra, as a type.
|
mk_quaternionAlgebra_of_infinite [Infinite R] : #(ℍ[R,c₁,c₂,c₃]) = #R := by
rw [mk_quaternionAlgebra, pow_four]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
mk_quaternionAlgebra_of_infinite
| null |
mk_univ_quaternionAlgebra : #(Set.univ : Set ℍ[R,c₁,c₂,c₃]) = #R ^ 4 := by
rw [mk_univ, mk_quaternionAlgebra]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
mk_univ_quaternionAlgebra
|
The cardinality of a quaternion algebra, as a set.
|
mk_univ_quaternionAlgebra_of_infinite [Infinite R] :
#(Set.univ : Set ℍ[R,c₁,c₂,c₃]) = #R := by rw [mk_univ_quaternionAlgebra, pow_four]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
mk_univ_quaternionAlgebra_of_infinite
| null |
@[simp]
mk_quaternion : #(ℍ[R]) = #R ^ 4 :=
mk_quaternionAlgebra _ _ _
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
mk_quaternion
|
Show the quaternion ⟨w, x, y, z⟩ as a string "{ re := w, imI := x, imJ := y, imK := z }".
For the typical case of quaternions over ℝ, each component will show as a Cauchy sequence due to
the way Real numbers are represented.
-/
instance [Repr R] {a b c : R} : Repr ℍ[R, a, b, c] where
reprPrec q _ :=
s!"\{ re := {repr q.re}, imI := {repr q.imI}, imJ := {repr q.imJ}, imK := {repr q.imK} }"
end QuaternionAlgebra
section Quaternion
variable (R : Type*) [Zero R] [One R] [Neg R]
/-- The cardinality of the quaternions, as a type.
|
mk_quaternion_of_infinite [Infinite R] : #(ℍ[R]) = #R :=
mk_quaternionAlgebra_of_infinite _ _ _
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic"
] |
Mathlib/Algebra/Quaternion.lean
|
mk_quaternion_of_infinite
| null |
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