statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
mk_add_mk (a₁ a₂ a₃ a₄ b₁ b₂ b₃ b₄ : R) :
(mk a₁ a₂ a₃ a₄ : ℍ[R, c₁, c₂]) + mk b₁ b₂ b₃ b₄ = mk (a₁ + b₁) (a₂ + b₂) (a₃ + b₃) (a₄ + b₄) | rfl | lemma | quaternion_algebra.mk_add_mk | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add : ((x + y : R) : ℍ[R, c₁, c₂]) = x + y | by ext; simp | lemma | quaternion_algebra.coe_add | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_mk (a₁ a₂ a₃ a₄ : R) : -(mk a₁ a₂ a₃ a₄ : ℍ[R, c₁, c₂]) = ⟨-a₁, -a₂, -a₃, -a₄⟩ | rfl | lemma | quaternion_algebra.neg_mk | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_neg : ((-x : R) : ℍ[R, c₁, c₂]) = -x | by ext; simp | lemma | quaternion_algebra.coe_neg | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_sub_mk (a₁ a₂ a₃ a₄ b₁ b₂ b₃ b₄ : R) :
(mk a₁ a₂ a₃ a₄ : ℍ[R, c₁, c₂]) - mk b₁ b₂ b₃ b₄ = mk (a₁ - b₁) (a₂ - b₂) (a₃ - b₃) (a₄ - b₄) | rfl | lemma | quaternion_algebra.mk_sub_mk | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_im : (x : ℍ[R, c₁, c₂]).im = 0 | rfl | lemma | quaternion_algebra.coe_im | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
re_add_im : ↑a.re + a.im = a | ext _ _ (add_zero _) (zero_add _) (zero_add _) (zero_add _) | lemma | quaternion_algebra.re_add_im | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_self_im : a - a.im = a.re | ext _ _ (sub_zero _) (sub_self _) (sub_self _) (sub_self _) | lemma | quaternion_algebra.sub_self_im | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_self_re : a - a.re = a.im | ext _ _ (sub_self _) (sub_zero _) (sub_zero _) (sub_zero _) | lemma | quaternion_algebra.sub_self_re | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_mul_mk (a₁ a₂ a₃ a₄ b₁ b₂ b₃ b₄ : R) :
(mk a₁ a₂ a₃ a₄ : ℍ[R, c₁, c₂]) * mk b₁ b₂ b₃ b₄ =
⟨a₁ * b₁ + c₁ * a₂ * b₂ + c₂ * a₃ * b₃ - c₁ * c₂ * a₄ * b₄,
a₁ * b₂ + a₂ * b₁ - c₂ * a₃ * b₄ + c₂ * a₄ * b₃,
a₁ * b₃ + c₁ * a₂ * b₄ + a₃ * b₁ - c₁ * a₄ * b₂,
a₁ * b₄ + a₂ * b₃ - a₃ * b₂ + a₄ * b₁⟩ | rfl | lemma | quaternion_algebra.mk_mul_mk | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_re : (s • a).re = s • a.re | rfl | lemma | quaternion_algebra.smul_re | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_im_i : (s • a).im_i = s • a.im_i | rfl | lemma | quaternion_algebra.smul_im_i | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_im_j : (s • a).im_j = s • a.im_j | rfl | lemma | quaternion_algebra.smul_im_j | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_im_k : (s • a).im_k = s • a.im_k | rfl | lemma | quaternion_algebra.smul_im_k | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_mk (re im_i im_j im_k : R) :
s • (⟨re, im_i, im_j, im_k⟩ : ℍ[R, c₁, c₂]) = ⟨s • re, s • im_i, s • im_j, s • im_k⟩ | rfl | lemma | quaternion_algebra.smul_mk | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul [smul_zero_class S R] (s : S) (r : R) :
(↑(s • r) : ℍ[R, c₁, c₂]) = s • ↑r | ext _ _ rfl (smul_zero s).symm (smul_zero s).symm (smul_zero s).symm | lemma | quaternion_algebra.coe_smul | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"smul_zero",
"smul_zero_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_re (n : ℕ) : (n : ℍ[R, c₁, c₂]).re = n | rfl | lemma | quaternion_algebra.nat_cast_re | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_im_i (n : ℕ) : (n : ℍ[R, c₁, c₂]).im_i = 0 | rfl | lemma | quaternion_algebra.nat_cast_im_i | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_im_j (n : ℕ) : (n : ℍ[R, c₁, c₂]).im_j = 0 | rfl | lemma | quaternion_algebra.nat_cast_im_j | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_im_k (n : ℕ) : (n : ℍ[R, c₁, c₂]).im_k = 0 | rfl | lemma | quaternion_algebra.nat_cast_im_k | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_im (n : ℕ) : (n : ℍ[R, c₁, c₂]).im = 0 | rfl | lemma | quaternion_algebra.nat_cast_im | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_nat_cast (n : ℕ) : ↑(n : R) = (n : ℍ[R, c₁, c₂]) | rfl | lemma | quaternion_algebra.coe_nat_cast | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_cast_re (z : ℤ) : (z : ℍ[R, c₁, c₂]).re = z | rfl | lemma | quaternion_algebra.int_cast_re | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_cast_im_i (z : ℤ) : (z : ℍ[R, c₁, c₂]).im_i = 0 | rfl | lemma | quaternion_algebra.int_cast_im_i | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_cast_im_j (z : ℤ) : (z : ℍ[R, c₁, c₂]).im_j = 0 | rfl | lemma | quaternion_algebra.int_cast_im_j | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_cast_im_k (z : ℤ) : (z : ℍ[R, c₁, c₂]).im_k = 0 | rfl | lemma | quaternion_algebra.int_cast_im_k | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_cast_im (z : ℤ) : (z : ℍ[R, c₁, c₂]).im = 0 | rfl | lemma | quaternion_algebra.int_cast_im | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_int_cast (z : ℤ) : ↑(z : R) = (z : ℍ[R, c₁, c₂]) | rfl | lemma | quaternion_algebra.coe_int_cast | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul : ((x * y : R) : ℍ[R, c₁, c₂]) = x * y | by ext; simp | lemma | quaternion_algebra.coe_mul | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_eq (r : R) : algebra_map R ℍ[R,c₁,c₂] r = ⟨r, 0, 0, 0⟩ | rfl | lemma | quaternion_algebra.algebra_map_eq | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
re_lm : ℍ[R, c₁, c₂] →ₗ[R] R | { to_fun := re, map_add' := λ x y, rfl, map_smul' := λ r x, rfl } | def | quaternion_algebra.re_lm | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | `quaternion_algebra.re` as a `linear_map` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
im_i_lm : ℍ[R, c₁, c₂] →ₗ[R] R | { to_fun := im_i, map_add' := λ x y, rfl, map_smul' := λ r x, rfl } | def | quaternion_algebra.im_i_lm | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | `quaternion_algebra.im_i` as a `linear_map` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
im_j_lm : ℍ[R, c₁, c₂] →ₗ[R] R | { to_fun := im_j, map_add' := λ x y, rfl, map_smul' := λ r x, rfl } | def | quaternion_algebra.im_j_lm | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | `quaternion_algebra.im_j` as a `linear_map` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
im_k_lm : ℍ[R, c₁, c₂] →ₗ[R] R | { to_fun := im_k, map_add' := λ x y, rfl, map_smul' := λ r x, rfl } | def | quaternion_algebra.im_k_lm | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | `quaternion_algebra.im_k` as a `linear_map` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_equiv_tuple : ℍ[R,c₁,c₂] ≃ₗ[R] (fin 4 → R) | linear_equiv.symm -- proofs are not `rfl` in the forward direction
{ to_fun := (equiv_tuple c₁ c₂).symm,
inv_fun := equiv_tuple c₁ c₂,
map_add' := λ v₁ v₂, rfl,
map_smul' := λ v₁ v₂, rfl,
.. (equiv_tuple c₁ c₂).symm } | def | quaternion_algebra.linear_equiv_tuple | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"inv_fun",
"linear_equiv.symm"
] | `quaternion_algebra.equiv_tuple` as a linear equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_linear_equiv_tuple : ⇑(linear_equiv_tuple c₁ c₂) = equiv_tuple c₁ c₂ | rfl | lemma | quaternion_algebra.coe_linear_equiv_tuple | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_linear_equiv_tuple_symm :
⇑(linear_equiv_tuple c₁ c₂).symm = (equiv_tuple c₁ c₂).symm | rfl | lemma | quaternion_algebra.coe_linear_equiv_tuple_symm | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis_one_i_j_k : basis (fin 4) R ℍ[R, c₁, c₂] | basis.of_equiv_fun $ linear_equiv_tuple c₁ c₂ | def | quaternion_algebra.basis_one_i_j_k | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"basis",
"basis.of_equiv_fun"
] | `ℍ[R, c₁, c₂]` has a basis over `R` given by `1`, `i`, `j`, and `k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_basis_one_i_j_k_repr (q : ℍ[R, c₁, c₂]) :
⇑((basis_one_i_j_k c₁ c₂).repr q) = ![q.re, q.im_i, q.im_j, q.im_k] | rfl | lemma | quaternion_algebra.coe_basis_one_i_j_k_repr | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rank_eq_four [strong_rank_condition R] : module.rank R ℍ[R, c₁, c₂] = 4 | by { rw [rank_eq_card_basis (basis_one_i_j_k c₁ c₂), fintype.card_fin], norm_num } | lemma | quaternion_algebra.rank_eq_four | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"fintype.card_fin",
"module.rank",
"rank_eq_card_basis",
"strong_rank_condition"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finrank_eq_four [strong_rank_condition R] : finite_dimensional.finrank R ℍ[R, c₁, c₂] = 4 | have cardinal.to_nat 4 = 4,
by rw [←cardinal.to_nat_cast 4, nat.cast_bit0, nat.cast_bit0, nat.cast_one],
by rw [finite_dimensional.finrank, rank_eq_four, this] | lemma | quaternion_algebra.finrank_eq_four | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"cardinal.to_nat",
"finite_dimensional.finrank",
"nat.cast_bit0",
"nat.cast_one",
"strong_rank_condition"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sub : ((x - y : R) : ℍ[R, c₁, c₂]) = x - y | (algebra_map R ℍ[R, c₁, c₂]).map_sub x y | lemma | quaternion_algebra.coe_sub | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_pow (n : ℕ) : (↑(x ^ n) : ℍ[R, c₁, c₂]) = ↑x ^ n | (algebra_map R ℍ[R, c₁, c₂]).map_pow x n | lemma | quaternion_algebra.coe_pow | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"algebra_map",
"map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_commutes : ↑r * a = a * r | algebra.commutes r a | lemma | quaternion_algebra.coe_commutes | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"algebra.commutes"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_commute : commute ↑r a | coe_commutes r a | lemma | quaternion_algebra.coe_commute | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul_eq_smul : ↑r * a = r • a | (algebra.smul_def r a).symm | lemma | quaternion_algebra.coe_mul_eq_smul | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"algebra.smul_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_coe_eq_smul : a * r = r • a | by rw [← coe_commutes, coe_mul_eq_smul] | lemma | quaternion_algebra.mul_coe_eq_smul | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_algebra_map : ⇑(algebra_map R ℍ[R, c₁, c₂]) = coe | rfl | lemma | quaternion_algebra.coe_algebra_map | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_coe : x • (y : ℍ[R, c₁, c₂]) = ↑(x * y) | by rw [coe_mul, coe_mul_eq_smul] | lemma | quaternion_algebra.smul_coe | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
re_star : (star a).re = a.re | rfl | lemma | quaternion_algebra.re_star | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
im_i_star : (star a).im_i = - a.im_i | rfl | lemma | quaternion_algebra.im_i_star | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
im_j_star : (star a).im_j = - a.im_j | rfl | lemma | quaternion_algebra.im_j_star | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
im_k_star : (star a).im_k = - a.im_k | rfl | lemma | quaternion_algebra.im_k_star | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
im_star : (star a).im = - a.im | ext _ _ neg_zero.symm rfl rfl rfl | lemma | quaternion_algebra.im_star | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_mk (a₁ a₂ a₃ a₄ : R) :
star (mk a₁ a₂ a₃ a₄ : ℍ[R, c₁, c₂]) = ⟨a₁, -a₂, -a₃, -a₄⟩ | rfl | lemma | quaternion_algebra.star_mk | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_add_star' : a + star a = ↑(2 * a.re) | by ext; simp [two_mul] | lemma | quaternion_algebra.self_add_star' | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_add_star : a + star a = 2 * a.re | by simp only [self_add_star', two_mul, coe_add] | lemma | quaternion_algebra.self_add_star | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_add_self' : star a + a = ↑(2 * a.re) | by rw [add_comm, self_add_star'] | lemma | quaternion_algebra.star_add_self' | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_add_self : star a + a = 2 * a.re | by rw [add_comm, self_add_star] | lemma | quaternion_algebra.star_add_self | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_eq_two_re_sub : star a = ↑(2 * a.re) - a | eq_sub_iff_add_eq.2 a.star_add_self' | lemma | quaternion_algebra.star_eq_two_re_sub | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_coe : star (x : ℍ[R, c₁, c₂]) = x | by ext; simp | lemma | quaternion_algebra.star_coe | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_im : star a.im = - a.im | im_star _ | lemma | quaternion_algebra.star_im | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_smul [monoid S] [distrib_mul_action S R] (s : S) (a : ℍ[R, c₁, c₂]) :
star (s • a) = s • star a | ext _ _ rfl (smul_neg _ _).symm (smul_neg _ _).symm (smul_neg _ _).symm | lemma | quaternion_algebra.star_smul | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"distrib_mul_action",
"monoid",
"smul_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_re_of_eq_coe {a : ℍ[R, c₁, c₂]} {x : R} (h : a = x) : a = a.re | by rw [h, coe_re] | lemma | quaternion_algebra.eq_re_of_eq_coe | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_re_iff_mem_range_coe {a : ℍ[R, c₁, c₂]} :
a = a.re ↔ a ∈ set.range (coe : R → ℍ[R, c₁, c₂]) | ⟨λ h, ⟨a.re, h.symm⟩, λ ⟨x, h⟩, eq_re_of_eq_coe h.symm⟩ | lemma | quaternion_algebra.eq_re_iff_mem_range_coe | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_eq_self {c₁ c₂ : R} {a : ℍ[R, c₁, c₂]} :
star a = a ↔ a = a.re | by simp [ext_iff, neg_eq_iff_add_eq_zero, add_self_eq_zero] | lemma | quaternion_algebra.star_eq_self | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"add_self_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_eq_neg {c₁ c₂ : R} {a : ℍ[R, c₁, c₂]} :
star a = -a ↔ a.re = 0 | by simp [ext_iff, eq_neg_iff_add_eq_zero] | lemma | quaternion_algebra.star_eq_neg | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_mul_eq_coe : star a * a = (star a * a).re | by ext; simp; ring_exp | lemma | quaternion_algebra.star_mul_eq_coe | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_star_eq_coe : a * star a = (a * star a).re | by { rw ←star_comm_self', exact a.star_mul_eq_coe } | lemma | quaternion_algebra.mul_star_eq_coe | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_ae : ℍ[R, c₁, c₂] ≃ₐ[R] (ℍ[R, c₁, c₂]ᵐᵒᵖ) | { to_fun := op ∘ star,
inv_fun := star ∘ unop,
map_mul' := λ x y, by simp,
commutes' := λ r, by simp,
.. star_add_equiv.trans op_add_equiv } | def | quaternion_algebra.star_ae | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"inv_fun"
] | Quaternion conjugate as an `alg_equiv` to the opposite ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_star_ae : ⇑(star_ae : ℍ[R, c₁, c₂] ≃ₐ[R] _) = op ∘ star | rfl | lemma | quaternion_algebra.coe_star_ae | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quaternion (R : Type*) [has_one R] [has_neg R] | quaternion_algebra R (-1) (-1) | def | quaternion | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"quaternion_algebra"
] | Space of quaternions over a type. Implemented as a structure with four fields:
`re`, `im_i`, `im_j`, and `im_k`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quaternion.equiv_prod (R : Type*) [has_one R] [has_neg R] : ℍ[R] ≃ R × R × R × R | quaternion_algebra.equiv_prod _ _ | def | quaternion.equiv_prod | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"quaternion_algebra.equiv_prod"
] | The equivalence between the quaternions over `R` and `R × R × R × R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quaternion.equiv_tuple (R : Type*) [has_one R] [has_neg R] : ℍ[R] ≃ (fin 4 → R) | quaternion_algebra.equiv_tuple _ _ | def | quaternion.equiv_tuple | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"quaternion_algebra.equiv_tuple"
] | The equivalence between the quaternions over `R` and `fin 4 → R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quaternion.equiv_tuple_apply (R : Type*) [has_one R] [has_neg R] (x : ℍ[R]) :
quaternion.equiv_tuple R x = ![x.re, x.im_i, x.im_j, x.im_k] | rfl | lemma | quaternion.equiv_tuple_apply | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [
"quaternion.equiv_tuple"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext : a.re = b.re → a.im_i = b.im_i → a.im_j = b.im_j → a.im_k = b.im_k → a = b | quaternion_algebra.ext a b | lemma | quaternion.ext | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {a b : ℍ[R]} :
a = b ↔ a.re = b.re ∧ a.im_i = b.im_i ∧ a.im_j = b.im_j ∧ a.im_k = b.im_k | quaternion_algebra.ext_iff a b | lemma | quaternion.ext_iff | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
im (x : ℍ[R]) : ℍ[R] | x.im | def | quaternion.im | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | The imaginary part of a quaternion. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
re_add_im : ↑a.re + a.im = a | a.re_add_im | lemma | quaternion.re_add_im | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_self_im : a - a.im = a.re | a.sub_self_im | lemma | quaternion.sub_self_im | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_self_re : a - a.re = a.im | a.sub_self_re | lemma | quaternion.sub_self_re | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_re : (x : ℍ[R]).re = x | rfl | lemma | quaternion.coe_re | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_im_i : (x : ℍ[R]).im_i = 0 | rfl | lemma | quaternion.coe_im_i | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_im_j : (x : ℍ[R]).im_j = 0 | rfl | lemma | quaternion.coe_im_j | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_im_k : (x : ℍ[R]).im_k = 0 | rfl | lemma | quaternion.coe_im_k | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_im : (x : ℍ[R]).im = 0 | rfl | lemma | quaternion.coe_im | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_re : (0 : ℍ[R]).re = 0 | rfl | lemma | quaternion.zero_re | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_im_i : (0 : ℍ[R]).im_i = 0 | rfl | lemma | quaternion.zero_im_i | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_im_j : (0 : ℍ[R]).im_j = 0 | rfl | lemma | quaternion.zero_im_j | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_im_k : (0 : ℍ[R]).im_k = 0 | rfl | lemma | quaternion.zero_im_k | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_im : (0 : ℍ[R]).im = 0 | rfl | lemma | quaternion.zero_im | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero : ((0 : R) : ℍ[R]) = 0 | rfl | lemma | quaternion.coe_zero | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_re : (1 : ℍ[R]).re = 1 | rfl | lemma | quaternion.one_re | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_im_i : (1 : ℍ[R]).im_i = 0 | rfl | lemma | quaternion.one_im_i | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_im_j : (1 : ℍ[R]).im_j = 0 | rfl | lemma | quaternion.one_im_j | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_im_k : (1 : ℍ[R]).im_k = 0 | rfl | lemma | quaternion.one_im_k | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_im : (1 : ℍ[R]).im = 0 | rfl | lemma | quaternion.one_im | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ((1 : R) : ℍ[R]) = 1 | rfl | lemma | quaternion.coe_one | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_re : (a + b).re = a.re + b.re | rfl | lemma | quaternion.add_re | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_im_i : (a + b).im_i = a.im_i + b.im_i | rfl | lemma | quaternion.add_im_i | algebra | src/algebra/quaternion.lean | [
"algebra.algebra.equiv",
"linear_algebra.finrank",
"linear_algebra.free_module.basic",
"linear_algebra.free_module.finite.basic",
"set_theory.cardinal.ordinal",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.