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
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PreEnvelGroupRel.trans {R : Type u} [Rack R] {a b c : PreEnvelGroup R} :
PreEnvelGroupRel R a b → PreEnvelGroupRel R b c → PreEnvelGroupRel R a c | | ⟨rab⟩, ⟨rbc⟩ => (rab.trans rbc).rel | theorem | Rack.PreEnvelGroupRel.trans | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Rack"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PreEnvelGroup.setoid (R : Type*) [Rack R] : Setoid (PreEnvelGroup R) | where
r := PreEnvelGroupRel R
iseqv := by
constructor
· apply PreEnvelGroupRel.refl
· apply PreEnvelGroupRel.symm
· apply PreEnvelGroupRel.trans | instance | Rack.PreEnvelGroup.setoid | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Rack"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
EnvelGroup (R : Type*) [Rack R] | Quotient (PreEnvelGroup.setoid R) | def | Rack.EnvelGroup | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Rack"
] | The universal enveloping group for the rack R. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
EnvelGroup.inhabited (R : Type*) [Rack R] : Inhabited (EnvelGroup R) | ⟨1⟩ | instance | Rack.EnvelGroup.inhabited | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Rack"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toEnvelGroup (R : Type*) [Rack R] : R →◃ Quandle.Conj (EnvelGroup R) | where
toFun x := ⟦incl x⟧
map_act' := @fun x y => Quotient.sound (PreEnvelGroupRel'.act_incl x y).symm.rel | def | Rack.toEnvelGroup | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Quandle.Conj",
"Rack"
] | The canonical homomorphism from a rack to its enveloping group.
Satisfies universal properties given by `toEnvelGroup.map` and `toEnvelGroup.univ`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toEnvelGroup.mapAux {R : Type*} [Rack R] {G : Type*} [Group G] (f : R →◃ Quandle.Conj G) :
PreEnvelGroup R → G | | .unit => 1
| .incl x => f x
| .mul a b => toEnvelGroup.mapAux f a * toEnvelGroup.mapAux f b
| .inv a => (toEnvelGroup.mapAux f a)⁻¹ | def | Rack.toEnvelGroup.mapAux | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Group",
"Quandle.Conj",
"Rack"
] | The preliminary definition of the induced map from the enveloping group.
See `toEnvelGroup.map`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
well_def {R : Type*} [Rack R] {G : Type*} [Group G] (f : R →◃ Quandle.Conj G) :
∀ {a b : PreEnvelGroup R},
PreEnvelGroupRel' R a b → toEnvelGroup.mapAux f a = toEnvelGroup.mapAux f b | | _, _, PreEnvelGroupRel'.refl => rfl
| _, _, PreEnvelGroupRel'.symm h => (well_def f h).symm
| _, _, PreEnvelGroupRel'.trans hac hcb => Eq.trans (well_def f hac) (well_def f hcb)
| _, _, PreEnvelGroupRel'.congr_mul ha hb => by
simp [toEnvelGroup.mapAux, well_def f ha, well_def f hb]
| _, _, congr_inv ha =>... | theorem | Rack.toEnvelGroup.mapAux.well_def | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Group",
"Quandle.Conj",
"Rack",
"mul_assoc",
"symm"
] | Show that `toEnvelGroup.mapAux` sends equivalent expressions to equal terms. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toEnvelGroup.map {R : Type*} [Rack R] {G : Type*} [Group G] :
(R →◃ Quandle.Conj G) ≃ (EnvelGroup R →* G) | where
toFun f :=
{ toFun := fun x =>
Quotient.liftOn x (toEnvelGroup.mapAux f) fun _ _ ⟨hab⟩ =>
toEnvelGroup.mapAux.well_def f hab
map_one' := by
change Quotient.liftOn ⟦Rack.PreEnvelGroup.unit⟧ (toEnvelGroup.mapAux f) _ = 1
simp only [Quotient.lift_mk, mapAux]
map_mu... | def | Rack.toEnvelGroup.map | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Group",
"MonoidHom.coe_mk",
"MonoidHom.ext",
"OneHom.coe_mk",
"Quandle.Conj",
"Quandle.Conj.map",
"Quotient.lift_mk",
"Rack",
"map_inv"
] | Given a map from a rack to a group, lift it to being a map from the enveloping group.
More precisely, the `EnvelGroup` functor is left adjoint to `Quandle.Conj`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toEnvelGroup.univ (R : Type*) [Rack R] (G : Type*) [Group G] (f : R →◃ Quandle.Conj G) :
(Quandle.Conj.map (toEnvelGroup.map f)).comp (toEnvelGroup R) = f | toEnvelGroup.map.symm_apply_apply f | theorem | Rack.toEnvelGroup.univ | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Group",
"Quandle.Conj",
"Quandle.Conj.map",
"Rack"
] | Given a homomorphism from a rack to a group, it factors through the enveloping group. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toEnvelGroup.univ_uniq (R : Type*) [Rack R] (G : Type*) [Group G]
(f : R →◃ Quandle.Conj G) (g : EnvelGroup R →* G)
(h : f = (Quandle.Conj.map g).comp (toEnvelGroup R)) : g = toEnvelGroup.map f | h.symm ▸ (toEnvelGroup.map.apply_symm_apply g).symm | theorem | Rack.toEnvelGroup.univ_uniq | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Group",
"Quandle.Conj",
"Quandle.Conj.map",
"Rack",
"symm"
] | The homomorphism `toEnvelGroup.map f` is the unique map that fits into the commutative
triangle in `toEnvelGroup.univ`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
envelAction {R : Type*} [Rack R] : EnvelGroup R →* R ≃ R | toEnvelGroup.map (toConj R) | def | Rack.envelAction | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Rack"
] | The induced group homomorphism from the enveloping group into bijections of the rack,
using `Rack.toConj`. Satisfies the property `envelAction_prop`.
This gives the rack `R` the structure of an augmented rack over `EnvelGroup R`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
envelAction_prop {R : Type*} [Rack R] (x y : R) :
envelAction (toEnvelGroup R x) y = x ◃ y | rfl | theorem | Rack.envelAction_prop | Algebra | Mathlib/Algebra/Quandle.lean | [] | [
"Rack"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
QuaternionAlgebra (R : Type*) (a b c : R) where
/-- Real part of a quaternion. -/
re : R
/-- First imaginary part (i) of a quaternion. -/
imI : R
/-- Second imaginary part (j) of a quaternion. -/
imJ : R
/-- Third imaginary part (k) of a quaternion. -/
imK : R
initialize_simps_projections QuaternionAlg... | structure | QuaternionAlgebra | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"Quaternion"
] | Quaternion algebra over a type with fixed coefficients where $i^2 = a + bi$ and $j^2 = c$,
denoted as `ℍ[R,a,b]`.
Implemented as a structure with four fields: `re`, `imI`, `imJ`, and `imK`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
equivProd {R : Type*} (c₁ c₂ c₃ : R) : ℍ[R,c₁,c₂,c₃] ≃ R × R × R × R | where
toFun a := ⟨a.1, a.2, a.3, a.4⟩
invFun a := ⟨a.1, a.2.1, a.2.2.1, a.2.2.2⟩ | def | QuaternionAlgebra.equivProd | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | The equivalence between a quaternion algebra over `R` and `R × R × R × R`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
equivTuple {R : Type*} (c₁ c₂ c₃ : R) : ℍ[R,c₁,c₂,c₃] ≃ (Fin 4 → R) | where
toFun a := ![a.1, a.2, a.3, a.4]
invFun a := ⟨a 0, a 1, a 2, a 3⟩
right_inv _ := by ext ⟨_, _ | _ | _ | _ | _ | ⟨⟩⟩ <;> rfl | def | QuaternionAlgebra.equivTuple | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | The equivalence between a quaternion algebra over `R` and `Fin 4 → R`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
equivTuple_apply {R : Type*} (c₁ c₂ c₃ : R) (x : ℍ[R,c₁,c₂,c₃]) :
equivTuple c₁ c₂ c₃ x = ![x.re, x.imI, x.imJ, x.imK] | rfl | theorem | QuaternionAlgebra.equivTuple_apply | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mk.eta {R : Type*} {c₁ c₂ c₃} (a : ℍ[R,c₁,c₂,c₃]) : mk a.1 a.2 a.3 a.4 = a | rfl | theorem | QuaternionAlgebra.mk.eta | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
im (x : ℍ[R,c₁,c₂,c₃]) : ℍ[R,c₁,c₂,c₃] | ⟨0, x.imI, x.imJ, x.imK⟩ | def | QuaternionAlgebra.im | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | The imaginary part of a quaternion.
Note that unless `c₂ = 0`, this definition is not particularly well-behaved;
for instance, `QuaternionAlgebra.star_im` only says that the star of an imaginary quaternion
is imaginary under this condition. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
re_im : a.im.re = 0 | rfl | theorem | QuaternionAlgebra.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 | QuaternionAlgebra.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 | QuaternionAlgebra.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 | QuaternionAlgebra.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 | QuaternionAlgebra.im_idem | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe (x : R) : ℍ[R,c₁,c₂,c₃] | ⟨x, 0, 0, 0⟩ | def | QuaternionAlgebra.coe | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | Coercion `R → ℍ[R,c₁,c₂,c₃]`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
re_coe : (x : ℍ[R,c₁,c₂,c₃]).re = x | rfl | theorem | QuaternionAlgebra.re_coe | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imI_coe : (x : ℍ[R,c₁,c₂,c₃]).imI = 0 | rfl | theorem | QuaternionAlgebra.imI_coe | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imJ_coe : (x : ℍ[R,c₁,c₂,c₃]).imJ = 0 | rfl | theorem | QuaternionAlgebra.imJ_coe | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imK_coe : (x : ℍ[R,c₁,c₂,c₃]).imK = 0 | rfl | theorem | QuaternionAlgebra.imK_coe | 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,c₁,c₂,c₃]) | fun _ _ h => congr_arg re h | theorem | QuaternionAlgebra.coe_injective | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_inj {x y : R} : (x : ℍ[R,c₁,c₂,c₃]) = y ↔ x = y | coe_injective.eq_iff | theorem | QuaternionAlgebra.coe_inj | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
im_zero : (0 : ℍ[R,c₁,c₂,c₃]).im = 0 | rfl | theorem | QuaternionAlgebra.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,c₁,c₂,c₃]) = 0 | rfl | theorem | QuaternionAlgebra.coe_zero | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
im_one : (1 : ℍ[R,c₁,c₂,c₃]).im = 0 | rfl | theorem | QuaternionAlgebra.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,c₁,c₂,c₃]) = 1 | rfl | theorem | QuaternionAlgebra.coe_one | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mk_add_mk (a₁ a₂ a₃ a₄ b₁ b₂ b₃ b₄ : R) :
(mk a₁ a₂ a₃ a₄ : ℍ[R,c₁,c₂,c₃]) + mk b₁ b₂ b₃ b₄ =
mk (a₁ + b₁) (a₂ + b₂) (a₃ + b₃) (a₄ + b₄) | rfl | theorem | QuaternionAlgebra.mk_add_mk | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
addEquivTuple (c₁ c₂ c₃ : R) : ℍ[R,c₁,c₂,c₃] ≃+ (Fin 4 → R) | (equivTuple ..).addEquiv | def | QuaternionAlgebra.addEquivTuple | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | The additive equivalence between a quaternion algebra over `R` and `Fin 4 → R`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_addEquivTuple (c₁ c₂ c₃ : R) : ⇑(addEquivTuple c₁ c₂ c₃) = equivTuple c₁ c₂ c₃ | rfl | lemma | QuaternionAlgebra.coe_addEquivTuple | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_symm_addEquivTuple (c₁ c₂ c₃ : R) :
⇑(addEquivTuple c₁ c₂ c₃).symm = (equivTuple c₁ c₂ c₃).symm | rfl | lemma | QuaternionAlgebra.coe_symm_addEquivTuple | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
addEquivProd (c₁ c₂ c₃ : R) : ℍ[R,c₁,c₂,c₃] ≃+ R × R × R × R | (equivProd ..).addEquiv | def | QuaternionAlgebra.addEquivProd | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | The additive equivalence between a quaternion algebra over `R` and `R × R × R × R`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_addEquivProd (c₁ c₂ c₃ : R) : ⇑(addEquivProd c₁ c₂ c₃) = equivProd c₁ c₂ c₃ | rfl | lemma | QuaternionAlgebra.coe_addEquivProd | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_symm_addEquivProd (c₁ c₂ c₃ : R) :
⇑(addEquivProd c₁ c₂ c₃).symm = (equivProd c₁ c₂ c₃).symm | rfl | lemma | QuaternionAlgebra.coe_symm_addEquivProd | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
im_add : (a + b).im = a.im + b.im | QuaternionAlgebra.ext (zero_add _).symm rfl rfl rfl | theorem | QuaternionAlgebra.im_add | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_add : ((x + y : R) : ℍ[R,c₁,c₂,c₃]) = x + y | by ext <;> simp | theorem | QuaternionAlgebra.coe_add | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
neg_mk (a₁ a₂ a₃ a₄ : R) : -(mk a₁ a₂ a₃ a₄ : ℍ[R,c₁,c₂,c₃]) = ⟨-a₁, -a₂, -a₃, -a₄⟩ | rfl | theorem | QuaternionAlgebra.neg_mk | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
im_neg : (-a).im = -a.im | QuaternionAlgebra.ext neg_zero.symm rfl rfl rfl | theorem | QuaternionAlgebra.im_neg | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_neg : ((-x : R) : ℍ[R,c₁,c₂,c₃]) = -x | by ext <;> simp | theorem | QuaternionAlgebra.coe_neg | 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.ext (sub_zero _).symm rfl rfl rfl | theorem | QuaternionAlgebra.im_sub | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mk_sub_mk (a₁ a₂ a₃ a₄ b₁ b₂ b₃ b₄ : R) :
(mk a₁ a₂ a₃ a₄ : ℍ[R,c₁,c₂,c₃]) - mk b₁ b₂ b₃ b₄ =
mk (a₁ - b₁) (a₂ - b₂) (a₃ - b₃) (a₄ - b₄) | rfl | theorem | QuaternionAlgebra.mk_sub_mk | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
im_coe : (x : ℍ[R,c₁,c₂,c₃]).im = 0 | rfl | theorem | QuaternionAlgebra.im_coe | 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.ext (add_zero _) (zero_add _) (zero_add _) (zero_add _) | theorem | QuaternionAlgebra.re_add_im | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
sub_im_self : a - a.im = a.re | QuaternionAlgebra.ext (sub_zero _) (sub_self _) (sub_self _) (sub_self _) | theorem | QuaternionAlgebra.sub_im_self | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
sub_re_self : a - a.re = a.im | QuaternionAlgebra.ext (sub_self _) (sub_zero _) (sub_zero _) (sub_zero _) | theorem | QuaternionAlgebra.sub_re_self | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mk_mul_mk (a₁ a₂ a₃ a₄ b₁ b₂ b₃ b₄ : R) :
(mk a₁ a₂ a₃ a₄ : ℍ[R,c₁,c₂,c₃]) * mk b₁ b₂ b₃ b₄ =
mk
(a₁ * b₁ + c₁ * a₂ * b₂ + c₃ * a₃ * b₃ + c₂ * c₃ * a₃ * b₄ - c₁ * c₃ * a₄ * b₄)
(a₁ * b₂ + a₂ * b₁ + c₂ * a₂ * b₂ - c₃ * a₃ * b₄ + c₃ * a₄ * b₃)
(a₁ * b₃ + c₁ * a₂ * b₄ + a₃ * b₁ + c₂ * a₃ * b₂ - c... | rfl | theorem | QuaternionAlgebra.mk_mul_mk | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
im_smul {S} [CommRing R] [SMulZeroClass S R] (s : S) : (s • a).im = s • a.im | QuaternionAlgebra.ext (smul_zero s).symm rfl rfl rfl | theorem | QuaternionAlgebra.im_smul | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"CommRing",
"SMulZeroClass",
"smul_zero",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
smul_mk (re im_i im_j im_k : R) :
s • (⟨re, im_i, im_j, im_k⟩ : ℍ[R,c₁,c₂,c₃]) = ⟨s • re, s • im_i, s • im_j, s • im_k⟩ | rfl | theorem | QuaternionAlgebra.smul_mk | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_smul [Zero R] [SMulZeroClass S R] (s : S) (r : R) :
(↑(s • r) : ℍ[R,c₁,c₂,c₃]) = s • (r : ℍ[R,c₁,c₂,c₃]) | QuaternionAlgebra.ext rfl (smul_zero _).symm (smul_zero _).symm (smul_zero _).symm | theorem | QuaternionAlgebra.coe_smul | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"SMulZeroClass",
"smul_zero",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
re_natCast (n : ℕ) : (n : ℍ[R,c₁,c₂,c₃]).re = n | rfl | theorem | QuaternionAlgebra.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,c₁,c₂,c₃]).imI = 0 | rfl | theorem | QuaternionAlgebra.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,c₁,c₂,c₃]).imJ = 0 | rfl | theorem | QuaternionAlgebra.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,c₁,c₂,c₃]).imK = 0 | rfl | theorem | QuaternionAlgebra.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,c₁,c₂,c₃]).im = 0 | rfl | theorem | QuaternionAlgebra.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,c₁,c₂,c₃]) | rfl | theorem | QuaternionAlgebra.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,c₁,c₂,c₃]).re = z | rfl | theorem | QuaternionAlgebra.re_intCast | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
re_ofNat (n : ℕ) [n.AtLeastTwo] : (ofNat(n) : ℍ[R,c₁,c₂,c₃]).re = ofNat(n) | rfl | theorem | QuaternionAlgebra.re_ofNat | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imI_ofNat (n : ℕ) [n.AtLeastTwo] : (ofNat(n) : ℍ[R,c₁,c₂,c₃]).imI = 0 | rfl | theorem | QuaternionAlgebra.imI_ofNat | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imJ_ofNat (n : ℕ) [n.AtLeastTwo] : (ofNat(n) : ℍ[R,c₁,c₂,c₃]).imJ = 0 | rfl | theorem | QuaternionAlgebra.imJ_ofNat | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imK_ofNat (n : ℕ) [n.AtLeastTwo] : (ofNat(n) : ℍ[R,c₁,c₂,c₃]).imK = 0 | rfl | theorem | QuaternionAlgebra.imK_ofNat | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
im_ofNat (n : ℕ) [n.AtLeastTwo] : (ofNat(n) : ℍ[R,c₁,c₂,c₃]).im = 0 | rfl | theorem | QuaternionAlgebra.im_ofNat | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
imI_intCast (z : ℤ) : (z : ℍ[R,c₁,c₂,c₃]).imI = 0 | rfl | theorem | QuaternionAlgebra.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,c₁,c₂,c₃]).imJ = 0 | rfl | theorem | QuaternionAlgebra.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,c₁,c₂,c₃]).imK = 0 | rfl | theorem | QuaternionAlgebra.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,c₁,c₂,c₃]).im = 0 | rfl | theorem | QuaternionAlgebra.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,c₁,c₂,c₃]) | rfl | theorem | QuaternionAlgebra.coe_intCast | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
instRing : Ring ℍ[R,c₁,c₂,c₃] | where
__ := (inferInstance : AddCommGroupWithOne ℍ[R,c₁,c₂,c₃])
left_distrib _ _ _ := by ext <;> simp <;> ring
right_distrib _ _ _ := by ext <;> simp <;> ring
zero_mul _ := by ext <;> simp
mul_zero _ := by ext <;> simp
mul_assoc _ _ _ := by ext <;> simp <;> ring
one_mul _ := by ext <;> simp
mul_one _ :=... | instance | QuaternionAlgebra.instRing | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"AddCommGroupWithOne",
"Ring",
"left_distrib",
"mul_assoc",
"mul_one",
"one_mul",
"right_distrib"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_mul : ((x * y : R) : ℍ[R,c₁,c₂,c₃]) = x * y | by ext <;> simp | theorem | QuaternionAlgebra.coe_mul | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_ofNat {n : ℕ} [n.AtLeastTwo] :
((ofNat(n) : R) : ℍ[R,c₁,c₂,c₃]) = (ofNat(n) : ℍ[R,c₁,c₂,c₃]) | rfl | lemma | QuaternionAlgebra.coe_ofNat | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
algebraMap_eq (r : R) : algebraMap R ℍ[R,c₁,c₂,c₃] r = ⟨r, 0, 0, 0⟩ | rfl | theorem | QuaternionAlgebra.algebraMap_eq | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
algebraMap_injective : (algebraMap R ℍ[R,c₁,c₂,c₃] : _ → _).Injective | fun _ _ ↦ by simp [algebraMap_eq] | theorem | QuaternionAlgebra.algebraMap_injective | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
reₗ : ℍ[R,c₁,c₂,c₃] →ₗ[R] R | where
toFun := re
map_add' _ _ := rfl
map_smul' _ _ := rfl | def | QuaternionAlgebra.reₗ | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | `QuaternionAlgebra.re` as a `LinearMap` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
imIₗ : ℍ[R,c₁,c₂,c₃] →ₗ[R] R | where
toFun := imI
map_add' _ _ := rfl
map_smul' _ _ := rfl | def | QuaternionAlgebra.imIₗ | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | `QuaternionAlgebra.imI` as a `LinearMap` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
imJₗ : ℍ[R,c₁,c₂,c₃] →ₗ[R] R | where
toFun := imJ
map_add' _ _ := rfl
map_smul' _ _ := rfl | def | QuaternionAlgebra.imJₗ | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | `QuaternionAlgebra.imJ` as a `LinearMap` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
imKₗ : ℍ[R,c₁,c₂,c₃] →ₗ[R] R | where
toFun := imK
map_add' _ _ := rfl
map_smul' _ _ := rfl | def | QuaternionAlgebra.imKₗ | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | `QuaternionAlgebra.imK` as a `LinearMap` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
linearEquivTuple : ℍ[R,c₁,c₂,c₃] ≃ₗ[R] Fin 4 → R | (equivTuple ..).linearEquiv _ | def | QuaternionAlgebra.linearEquivTuple | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | `QuaternionAlgebra.equivTuple` as a linear equivalence. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_linearEquivTuple :
⇑(linearEquivTuple c₁ c₂ c₃) = equivTuple c₁ c₂ c₃ | rfl | theorem | QuaternionAlgebra.coe_linearEquivTuple | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_linearEquivTuple_symm :
⇑(linearEquivTuple c₁ c₂ c₃).symm = (equivTuple c₁ c₂ c₃).symm | rfl | theorem | QuaternionAlgebra.coe_linearEquivTuple_symm | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
basisOneIJK : Basis (Fin 4) R ℍ[R,c₁,c₂,c₃] | .ofEquivFun <| linearEquivTuple c₁ c₂ c₃ | def | QuaternionAlgebra.basisOneIJK | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | `ℍ[R,c₁,c₂,c₃]` has a basis over `R` given by `1`, `i`, `j`, and `k`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_basisOneIJK_repr (q : ℍ[R,c₁,c₂,c₃]) :
((basisOneIJK c₁ c₂ c₃).repr q) = ![q.re, q.imI, q.imJ, q.imK] | rfl | theorem | QuaternionAlgebra.coe_basisOneIJK_repr | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
rank_eq_four [StrongRankCondition R] : Module.rank R ℍ[R,c₁,c₂,c₃] = 4 | by
rw [rank_eq_card_basis (basisOneIJK c₁ c₂ c₃), Fintype.card_fin]
norm_num | theorem | QuaternionAlgebra.rank_eq_four | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"Fintype.card_fin",
"StrongRankCondition",
"rank_eq_card_basis"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
finrank_eq_four [StrongRankCondition R] : Module.finrank R ℍ[R,c₁,c₂,c₃] = 4 | by
rw [Module.finrank, rank_eq_four, Cardinal.toNat_ofNat] | theorem | QuaternionAlgebra.finrank_eq_four | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"Cardinal.toNat_ofNat",
"Module.finrank",
"StrongRankCondition"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
swapEquiv : ℍ[R,c₁,0,c₃] ≃ₐ[R] ℍ[R,c₃,0,c₁] | where
toFun t := ⟨t.1, t.3, t.2, -t.4⟩
invFun t := ⟨t.1, t.3, t.2, -t.4⟩
left_inv _ := by simp
right_inv _ := by simp
map_mul' _ _ := by ext <;> simp <;> ring
map_add' _ _ := by ext <;> simp [add_comm]
commutes' _ := by simp [algebraMap_eq] | def | QuaternionAlgebra.swapEquiv | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | There is a natural equivalence when swapping the first and third coefficients of a
quaternion algebra if `c₂` is 0. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_sub : ((x - y : R) : ℍ[R,c₁,c₂,c₃]) = x - y | (algebraMap R ℍ[R,c₁,c₂,c₃]).map_sub x y | theorem | QuaternionAlgebra.coe_sub | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_pow (n : ℕ) : (↑(x ^ n) : ℍ[R,c₁,c₂,c₃]) = (x : ℍ[R,c₁,c₂,c₃]) ^ n | (algebraMap R ℍ[R,c₁,c₂,c₃]).map_pow x n | theorem | QuaternionAlgebra.coe_pow | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"map_pow"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_commutes : ↑r * a = a * r | Algebra.commutes r a | theorem | QuaternionAlgebra.coe_commutes | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"Algebra.commutes"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_commute : Commute (↑r) a | coe_commutes r a | theorem | QuaternionAlgebra.coe_commute | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"Commute"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_mul_eq_smul : ↑r * a = r • a | (Algebra.smul_def r a).symm | theorem | QuaternionAlgebra.coe_mul_eq_smul | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"Algebra.smul_def",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mul_coe_eq_smul : a * r = r • a | by rw [← coe_commutes, coe_mul_eq_smul] | theorem | QuaternionAlgebra.mul_coe_eq_smul | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_algebraMap : ⇑(algebraMap R ℍ[R,c₁,c₂,c₃]) = coe | rfl | theorem | QuaternionAlgebra.coe_algebraMap | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
smul_coe : x • (y : ℍ[R,c₁,c₂,c₃]) = ↑(x * y) | by rw [coe_mul, coe_mul_eq_smul] | theorem | QuaternionAlgebra.smul_coe | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
instStarQuaternionAlgebra : Star ℍ[R,c₁,c₂,c₃] | where star a :=
⟨a.1 + c₂ * a.2, -a.2, -a.3, -a.4⟩ | instance | QuaternionAlgebra.instStarQuaternionAlgebra | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [
"Star"
] | Quaternion conjugate. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
re_star : (star a).re = a.re + c₂ * a.imI | rfl | theorem | QuaternionAlgebra.re_star | Algebra | Mathlib/Algebra/Quaternion.lean | [
"Mathlib.Algebra.Module.Torsion.Prod"
] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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