statement stringlengths 1 2.93k | proof stringlengths 0 19.2k | type stringclasses 13
values | symbolic_name stringlengths 1 131 | library stringlengths 4 62 | filename stringlengths 20 95 | imports listlengths 0 10 | deps listlengths 0 64 | docstring stringlengths 0 4.95k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
imI_star : (star a).imI = -a.imI | rfl | theorem | QuaternionAlgebra.imI_star | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imJ_star : (star a).imJ = -a.imJ | rfl | theorem | QuaternionAlgebra.imJ_star | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imK_star : (star a).imK = -a.imK | rfl | theorem | QuaternionAlgebra.imK_star | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
im_star : (star a).im = -a.im | QuaternionAlgebra.ext neg_zero.symm rfl rfl rfl | theorem | QuaternionAlgebra.im_star | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
star_mk (a₁ a₂ a₃ a₄ : R) : star (mk a₁ a₂ a₃ a₄ : ℍ[R,c₁,c₂,c₃]) =
⟨a₁ + c₂ * a₂, -a₂, -a₃, -a₄⟩ | rfl | theorem | QuaternionAlgebra.star_mk | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
instStarRing : StarRing ℍ[R,c₁,c₂,c₃] | where
star_involutive x := by simp [Star.star]
star_add a b := by ext <;> simp [add_comm]; ring
star_mul a b := by ext <;> simp <;> ring | instance | QuaternionAlgebra.instStarRing | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"StarRing"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
self_add_star' : a + star a = ↑(2 * a.re + c₂ * a.imI) | by ext <;> simp [two_mul]; ring | theorem | QuaternionAlgebra.self_add_star' | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"two_mul"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
self_add_star : a + star a = 2 * a.re + c₂ * a.imI | by simp [self_add_star'] | theorem | QuaternionAlgebra.self_add_star | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
star_add_self' : star a + a = ↑(2 * a.re + c₂ * a.imI) | by rw [add_comm, self_add_star'] | theorem | QuaternionAlgebra.star_add_self' | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
star_add_self : star a + a = 2 * a.re + c₂ * a.imI | by rw [add_comm, self_add_star] | theorem | QuaternionAlgebra.star_add_self | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
star_eq_two_re_sub : star a = ↑(2 * a.re + c₂ * a.imI) - a | eq_sub_iff_add_eq.2 a.star_add_self' | theorem | QuaternionAlgebra.star_eq_two_re_sub | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comm (r : R) (x : ℍ[R,c₁,c₂,c₃]) : r * x = x * r | by
ext <;> simp [mul_comm] | lemma | QuaternionAlgebra.comm | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"comm",
"mul_comm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
star_coe : star (x : ℍ[R,c₁,c₂,c₃]) = x | by ext <;> simp | theorem | QuaternionAlgebra.star_coe | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
star_im : star a.im = -a.im + c₂ * a.imI | by ext <;> simp | theorem | QuaternionAlgebra.star_im | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
star_smul [Monoid S] [DistribMulAction S R] [SMulCommClass S R R]
(s : S) (a : ℍ[R,c₁,c₂,c₃]) :
star (s • a) = s • star a | QuaternionAlgebra.ext
(by simp [mul_smul_comm]) (smul_neg _ _).symm (smul_neg _ _).symm (smul_neg _ _).symm | theorem | QuaternionAlgebra.star_smul | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"DistribMulAction",
"Monoid",
"SMulCommClass",
"mul_smul_comm",
"smul_neg",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
star_smul' [Monoid S] [DistribMulAction S R] (s : S) (a : ℍ[R,c₁,0,c₃]) :
star (s • a) = s • star a | QuaternionAlgebra.ext (by simp) (smul_neg _ _).symm (smul_neg _ _).symm (smul_neg _ _).symm | theorem | QuaternionAlgebra.star_smul' | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"DistribMulAction",
"Monoid",
"smul_neg",
"symm"
] | A version of `star_smul` for the special case when `c₂ = 0`, without `SMulCommClass S R R`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
eq_re_of_eq_coe {a : ℍ[R,c₁,c₂,c₃]} {x : R} (h : a = x) : a = a.re | by rw [h, re_coe] | theorem | QuaternionAlgebra.eq_re_of_eq_coe | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eq_re_iff_mem_range_coe {a : ℍ[R,c₁,c₂,c₃]} :
a = a.re ↔ a ∈ Set.range (coe : R → ℍ[R,c₁,c₂,c₃]) | ⟨fun h => ⟨a.re, h.symm⟩, fun ⟨_, h⟩ => eq_re_of_eq_coe h.symm⟩ | theorem | QuaternionAlgebra.eq_re_iff_mem_range_coe | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"Set.range"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
star_eq_self {c₁ c₂ : R} {a : ℍ[R,c₁,c₂,c₃]} : star a = a ↔ a = a.re | by
simp_all [QuaternionAlgebra.ext_iff, neg_eq_iff_add_eq_zero, add_self_eq_zero] | theorem | QuaternionAlgebra.star_eq_self | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"add_self_eq_zero"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
star_eq_neg {c₁ : R} {a : ℍ[R,c₁,0,c₃]} : star a = -a ↔ a.re = 0 | by
simp [QuaternionAlgebra.ext_iff, eq_neg_iff_add_eq_zero] | theorem | QuaternionAlgebra.star_eq_neg | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
star_mul_eq_coe : star a * a = (star a * a).re | by ext <;> simp <;> ring | theorem | QuaternionAlgebra.star_mul_eq_coe | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mul_star_eq_coe : a * star a = (a * star a).re | by
rw [← star_comm_self']
exact a.star_mul_eq_coe | theorem | QuaternionAlgebra.mul_star_eq_coe | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"star_comm_self'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
starAe : ℍ[R,c₁,c₂,c₃] ≃ₐ[R] ℍ[R,c₁,c₂,c₃]ᵐᵒᵖ | { starAddEquiv.trans opAddEquiv with
toFun := op ∘ star
invFun := star ∘ unop
map_mul' := fun x y => by simp
commutes' := fun r => by simp } | def | QuaternionAlgebra.starAe | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | Quaternion conjugate as an `AlgEquiv` to the opposite ring. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_starAe : ⇑(starAe : ℍ[R,c₁,c₂,c₃] ≃ₐ[R] _) = op ∘ star | rfl | theorem | QuaternionAlgebra.coe_starAe | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Quaternion (R : Type*) [Zero R] [One R] [Neg R] | QuaternionAlgebra R (-1) (0) (-1)
@[inherit_doc]
scoped[Quaternion] notation "ℍ[" R "]" => Quaternion R | def | Quaternion | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"QuaternionAlgebra"
] | Space of quaternions over a type, denoted as `ℍ[R]`.
Implemented as a structure with four fields: `re`, `im_i`, `im_j`, and `im_k`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Quaternion.equivProd (R : Type*) [Zero R] [One R] [Neg R] : ℍ[R] ≃ R × R × R × R | QuaternionAlgebra.equivProd _ _ _ | def | Quaternion.equivProd | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"QuaternionAlgebra.equivProd"
] | The equivalence between the quaternions over `R` and `R × R × R × R`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Quaternion.equivTuple (R : Type*) [Zero R] [One R] [Neg R] : ℍ[R] ≃ (Fin 4 → R) | QuaternionAlgebra.equivTuple _ _ _ | def | Quaternion.equivTuple | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"QuaternionAlgebra.equivTuple"
] | The equivalence between the quaternions over `R` and `Fin 4 → R`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Quaternion.equivTuple_apply (R : Type*) [Zero R] [One R] [Neg R] (x : ℍ[R]) :
Quaternion.equivTuple R x = ![x.re, x.imI, x.imJ, x.imK] | rfl | theorem | Quaternion.equivTuple_apply | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"Quaternion.equivTuple"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe : R → ℍ[R] | QuaternionAlgebra.coe | def | Quaternion.coe | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"QuaternionAlgebra.coe"
] | Coercion `R → ℍ[R]`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
instRing : Ring ℍ[R] | inferInstanceAs <| Ring ℍ[R,-1,0,-1] | instance | Quaternion.instRing | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"Ring"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
algebra [CommSemiring S] [Algebra S R] : Algebra S ℍ[R] | inferInstanceAs <| Algebra S ℍ[R,-1,0,-1] | instance | Quaternion.algebra | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"Algebra",
"CommSemiring"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ext : a.re = b.re → a.imI = b.imI → a.imJ = b.imJ → a.imK = b.imK → a = b | QuaternionAlgebra.ext | theorem | Quaternion.ext | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
im (x : ℍ[R]) : ℍ[R] | QuaternionAlgebra.im x | def | Quaternion.im | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"QuaternionAlgebra.im"
] | The imaginary part of a quaternion. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
re_im : a.im.re = 0 | rfl | theorem | Quaternion.re_im | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imI_im : a.im.imI = a.imI | rfl | theorem | Quaternion.imI_im | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imJ_im : a.im.imJ = a.imJ | rfl | theorem | Quaternion.imJ_im | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imK_im : a.im.imK = a.imK | rfl | theorem | Quaternion.imK_im | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
im_idem : a.im.im = a.im | rfl | theorem | Quaternion.im_idem | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
re_add_im : ↑a.re + a.im = a | QuaternionAlgebra.re_add_im a | theorem | Quaternion.re_add_im | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"QuaternionAlgebra.re_add_im"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
sub_im_self : a - a.im = a.re | QuaternionAlgebra.sub_im_self a | theorem | Quaternion.sub_im_self | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"QuaternionAlgebra.sub_im_self"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
sub_re_self : a - ↑a.re = a.im | QuaternionAlgebra.sub_re_self a | theorem | Quaternion.sub_re_self | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"QuaternionAlgebra.sub_re_self"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
re_coe : (x : ℍ[R]).re = x | rfl | theorem | Quaternion.re_coe | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imI_coe : (x : ℍ[R]).imI = 0 | rfl | theorem | Quaternion.imI_coe | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imJ_coe : (x : ℍ[R]).imJ = 0 | rfl | theorem | Quaternion.imJ_coe | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imK_coe : (x : ℍ[R]).imK = 0 | rfl | theorem | Quaternion.imK_coe | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
im_coe : (x : ℍ[R]).im = 0 | rfl | theorem | Quaternion.im_coe | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
re_zero : (0 : ℍ[R]).re = 0 | rfl | theorem | Quaternion.re_zero | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imI_zero : (0 : ℍ[R]).imI = 0 | rfl | theorem | Quaternion.imI_zero | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imJ_zero : (0 : ℍ[R]).imJ = 0 | rfl | theorem | Quaternion.imJ_zero | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imK_zero : (0 : ℍ[R]).imK = 0 | rfl | theorem | Quaternion.imK_zero | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
im_zero : (0 : ℍ[R]).im = 0 | rfl | theorem | Quaternion.im_zero | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_zero : ((0 : R) : ℍ[R]) = 0 | rfl | theorem | Quaternion.coe_zero | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
re_one : (1 : ℍ[R]).re = 1 | rfl | theorem | Quaternion.re_one | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imI_one : (1 : ℍ[R]).imI = 0 | rfl | theorem | Quaternion.imI_one | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imJ_one : (1 : ℍ[R]).imJ = 0 | rfl | theorem | Quaternion.imJ_one | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imK_one : (1 : ℍ[R]).imK = 0 | rfl | theorem | Quaternion.imK_one | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
im_one : (1 : ℍ[R]).im = 0 | rfl | theorem | Quaternion.im_one | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_one : ((1 : R) : ℍ[R]) = 1 | rfl | theorem | Quaternion.coe_one | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
re_add : (a + b).re = a.re + b.re | rfl | theorem | Quaternion.re_add | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imI_add : (a + b).imI = a.imI + b.imI | rfl | theorem | Quaternion.imI_add | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imJ_add : (a + b).imJ = a.imJ + b.imJ | rfl | theorem | Quaternion.imJ_add | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imK_add : (a + b).imK = a.imK + b.imK | rfl | theorem | Quaternion.imK_add | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
im_add : (a + b).im = a.im + b.im | QuaternionAlgebra.im_add a b | theorem | Quaternion.im_add | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"QuaternionAlgebra.im_add"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_add : ((x + y : R) : ℍ[R]) = x + y | QuaternionAlgebra.coe_add x y | theorem | Quaternion.coe_add | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"QuaternionAlgebra.coe_add"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
re_neg : (-a).re = -a.re | rfl | theorem | Quaternion.re_neg | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imI_neg : (-a).imI = -a.imI | rfl | theorem | Quaternion.imI_neg | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imJ_neg : (-a).imJ = -a.imJ | rfl | theorem | Quaternion.imJ_neg | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imK_neg : (-a).imK = -a.imK | rfl | theorem | Quaternion.imK_neg | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
im_neg : (-a).im = -a.im | QuaternionAlgebra.im_neg a | theorem | Quaternion.im_neg | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"QuaternionAlgebra.im_neg"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_neg : ((-x : R) : ℍ[R]) = -x | QuaternionAlgebra.coe_neg x | theorem | Quaternion.coe_neg | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"QuaternionAlgebra.coe_neg"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
re_sub : (a - b).re = a.re - b.re | rfl | theorem | Quaternion.re_sub | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imI_sub : (a - b).imI = a.imI - b.imI | rfl | theorem | Quaternion.imI_sub | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imJ_sub : (a - b).imJ = a.imJ - b.imJ | rfl | theorem | Quaternion.imJ_sub | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imK_sub : (a - b).imK = a.imK - b.imK | rfl | theorem | Quaternion.imK_sub | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
im_sub : (a - b).im = a.im - b.im | QuaternionAlgebra.im_sub a b | theorem | Quaternion.im_sub | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"QuaternionAlgebra.im_sub"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_sub : ((x - y : R) : ℍ[R]) = x - y | QuaternionAlgebra.coe_sub x y | theorem | Quaternion.coe_sub | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"QuaternionAlgebra.coe_sub"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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] | theorem | Quaternion.re_mul | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"neg_mul",
"one_mul",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | theorem | Quaternion.imI_mul | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | theorem | Quaternion.imJ_mul | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | theorem | Quaternion.imK_mul | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_mul : ((x * y : R) : ℍ[R]) = x * y | QuaternionAlgebra.coe_mul x y | theorem | Quaternion.coe_mul | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"QuaternionAlgebra.coe_mul"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_pow (n : ℕ) : (↑(x ^ n) : ℍ[R]) = (x : ℍ[R]) ^ n | QuaternionAlgebra.coe_pow x n | theorem | Quaternion.coe_pow | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"QuaternionAlgebra.coe_pow"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
re_natCast (n : ℕ) : (n : ℍ[R]).re = n | rfl | theorem | Quaternion.re_natCast | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imI_natCast (n : ℕ) : (n : ℍ[R]).imI = 0 | rfl | theorem | Quaternion.imI_natCast | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imJ_natCast (n : ℕ) : (n : ℍ[R]).imJ = 0 | rfl | theorem | Quaternion.imJ_natCast | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imK_natCast (n : ℕ) : (n : ℍ[R]).imK = 0 | rfl | theorem | Quaternion.imK_natCast | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
im_natCast (n : ℕ) : (n : ℍ[R]).im = 0 | rfl | theorem | Quaternion.im_natCast | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_natCast (n : ℕ) : ↑(n : R) = (n : ℍ[R]) | rfl | theorem | Quaternion.coe_natCast | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
re_intCast (z : ℤ) : (z : ℍ[R]).re = z | rfl | theorem | Quaternion.re_intCast | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imI_intCast (z : ℤ) : (z : ℍ[R]).imI = 0 | rfl | theorem | Quaternion.imI_intCast | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imJ_intCast (z : ℤ) : (z : ℍ[R]).imJ = 0 | rfl | theorem | Quaternion.imJ_intCast | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imK_intCast (z : ℤ) : (z : ℍ[R]).imK = 0 | rfl | theorem | Quaternion.imK_intCast | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
im_intCast (z : ℤ) : (z : ℍ[R]).im = 0 | rfl | theorem | Quaternion.im_intCast | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_intCast (z : ℤ) : ↑(z : R) = (z : ℍ[R]) | rfl | theorem | Quaternion.coe_intCast | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_injective : Function.Injective (coe : R → ℍ[R]) | QuaternionAlgebra.coe_injective | theorem | Quaternion.coe_injective | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"QuaternionAlgebra.coe_injective"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_inj {x y : R} : (x : ℍ[R]) = y ↔ x = y | coe_injective.eq_iff | theorem | Quaternion.coe_inj | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
re_smul [SMul S R] (s : S) : (s • a).re = s • a.re | rfl | theorem | Quaternion.re_smul | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imI_smul [SMul S R] (s : S) : (s • a).imI = s • a.imI | rfl | theorem | Quaternion.imI_smul | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imJ_smul [SMul S R] (s : S) : (s • a).imJ = s • a.imJ | rfl | theorem | Quaternion.imJ_smul | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imK_smul [SMul S R] (s : S) : (s • a).imK = s • a.imK | rfl | theorem | Quaternion.imK_smul | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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