fact stringlengths 6 14.3k | 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 | line_start int64 6 4.24k | line_end int64 7 4.25k | has_proof bool 2
classes | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
add_im_j : (a + b).im_j = a.im_j + b.im_j := rfl | add_im_j : (a + b).im_j = a.im_j + b.im_j | rfl | lemma | quaternion.add_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"
] | [] | 506 | 506 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_im_k : (a + b).im_k = a.im_k + b.im_k := rfl | add_im_k : (a + b).im_k = a.im_k + b.im_k | rfl | lemma | quaternion.add_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"
] | [] | 507 | 507 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_im : (a + b).im = a.im + b.im := ext _ _ (add_zero _).symm rfl rfl rfl | add_im : (a + b).im = a.im + b.im | ext _ _ (add_zero _).symm rfl rfl rfl | lemma | quaternion.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"
] | [] | 508 | 508 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add : ((x + y : R) : ℍ[R]) = x + y := quaternion_algebra.coe_add x y | coe_add : ((x + y : R) : ℍ[R]) = x + y | quaternion_algebra.coe_add x y | lemma | quaternion.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"
] | [
"quaternion_algebra.coe_add"
] | 509 | 509 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_re : (-a).re = -a.re := rfl | neg_re : (-a).re = -a.re | rfl | lemma | quaternion.neg_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"
] | [] | 511 | 511 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_im_i : (-a).im_i = -a.im_i := rfl | neg_im_i : (-a).im_i = -a.im_i | rfl | lemma | quaternion.neg_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"
] | [] | 512 | 512 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_im_j : (-a).im_j = -a.im_j := rfl | neg_im_j : (-a).im_j = -a.im_j | rfl | lemma | quaternion.neg_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"
] | [] | 513 | 513 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_im_k : (-a).im_k = -a.im_k := rfl | neg_im_k : (-a).im_k = -a.im_k | rfl | lemma | quaternion.neg_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"
] | [] | 514 | 514 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_im : (-a).im = -a.im := ext _ _ neg_zero.symm rfl rfl rfl | neg_im : (-a).im = -a.im | ext _ _ neg_zero.symm rfl rfl rfl | lemma | quaternion.neg_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"
] | [] | 515 | 515 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_neg : ((-x : R) : ℍ[R]) = -x := quaternion_algebra.coe_neg x | coe_neg : ((-x : R) : ℍ[R]) = -x | quaternion_algebra.coe_neg x | lemma | quaternion.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"
] | [
"quaternion_algebra.coe_neg"
] | 516 | 516 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_re : (a - b).re = a.re - b.re := rfl | sub_re : (a - b).re = a.re - b.re | rfl | lemma | quaternion.sub_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"
] | [] | 518 | 518 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_im_i : (a - b).im_i = a.im_i - b.im_i := rfl | sub_im_i : (a - b).im_i = a.im_i - b.im_i | rfl | lemma | quaternion.sub_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"
] | [] | 519 | 519 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_im_j : (a - b).im_j = a.im_j - b.im_j := rfl | sub_im_j : (a - b).im_j = a.im_j - b.im_j | rfl | lemma | quaternion.sub_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"
] | [] | 520 | 520 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_im_k : (a - b).im_k = a.im_k - b.im_k := rfl | sub_im_k : (a - b).im_k = a.im_k - b.im_k | rfl | lemma | quaternion.sub_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"
] | [] | 521 | 521 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_im : (a - b).im = a.im - b.im := ext _ _ (sub_zero _).symm rfl rfl rfl | sub_im : (a - b).im = a.im - b.im | ext _ _ (sub_zero _).symm rfl rfl rfl | lemma | quaternion.sub_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"
] | [] | 522 | 522 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sub : ((x - y : R) : ℍ[R]) = x - y := quaternion_algebra.coe_sub x y | coe_sub : ((x - y : R) : ℍ[R]) = x - y | quaternion_algebra.coe_sub x y | lemma | quaternion.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"
] | [
"quaternion_algebra.coe_sub"
] | 523 | 523 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_re :
(a * b).re = a.re * b.re - a.im_i * b.im_i - a.im_j * b.im_j - a.im_k * b.im_k :=
(quaternion_algebra.has_mul_mul_re a b).trans $
by simp only [one_mul, neg_mul, sub_eq_add_neg, neg_neg] | mul_re :
(a * b).re = a.re * b.re - a.im_i * b.im_i - a.im_j * b.im_j - a.im_k * b.im_k | (quaternion_algebra.has_mul_mul_re a b).trans $
by simp only [one_mul, neg_mul, sub_eq_add_neg, neg_neg] | lemma | quaternion.mul_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"
] | [
"neg_mul",
"one_mul"
] | 525 | 528 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_im_i :
(a * b).im_i = a.re * b.im_i + a.im_i * b.re + a.im_j * b.im_k - a.im_k * b.im_j :=
(quaternion_algebra.has_mul_mul_im_i a b).trans $
by simp only [one_mul, neg_mul, sub_eq_add_neg, neg_neg] | mul_im_i :
(a * b).im_i = a.re * b.im_i + a.im_i * b.re + a.im_j * b.im_k - a.im_k * b.im_j | (quaternion_algebra.has_mul_mul_im_i a b).trans $
by simp only [one_mul, neg_mul, sub_eq_add_neg, neg_neg] | lemma | quaternion.mul_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"
] | [
"neg_mul",
"one_mul"
] | 530 | 533 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_im_j :
(a * b).im_j = a.re * b.im_j - a.im_i * b.im_k + a.im_j * b.re + a.im_k * b.im_i :=
(quaternion_algebra.has_mul_mul_im_j a b).trans $
by simp only [one_mul, neg_mul, sub_eq_add_neg, neg_neg] | mul_im_j :
(a * b).im_j = a.re * b.im_j - a.im_i * b.im_k + a.im_j * b.re + a.im_k * b.im_i | (quaternion_algebra.has_mul_mul_im_j a b).trans $
by simp only [one_mul, neg_mul, sub_eq_add_neg, neg_neg] | lemma | quaternion.mul_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"
] | [
"neg_mul",
"one_mul"
] | 535 | 538 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_im_k :
(a * b).im_k = a.re * b.im_k + a.im_i * b.im_j - a.im_j * b.im_i + a.im_k * b.re :=
(quaternion_algebra.has_mul_mul_im_k a b).trans $
by simp only [one_mul, neg_mul, sub_eq_add_neg, neg_neg] | mul_im_k :
(a * b).im_k = a.re * b.im_k + a.im_i * b.im_j - a.im_j * b.im_i + a.im_k * b.re | (quaternion_algebra.has_mul_mul_im_k a b).trans $
by simp only [one_mul, neg_mul, sub_eq_add_neg, neg_neg] | lemma | quaternion.mul_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"
] | [
"neg_mul",
"one_mul"
] | 540 | 543 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul : ((x * y : R) : ℍ[R]) = x * y := quaternion_algebra.coe_mul x y | coe_mul : ((x * y : R) : ℍ[R]) = x * y | quaternion_algebra.coe_mul x y | lemma | quaternion.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"
] | [
"quaternion_algebra.coe_mul"
] | 545 | 545 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_pow (n : ℕ) : (↑(x ^ n) : ℍ[R]) = ↑x ^ n :=
quaternion_algebra.coe_pow x n | coe_pow (n : ℕ) : (↑(x ^ n) : ℍ[R]) = ↑x ^ n | quaternion_algebra.coe_pow x n | lemma | quaternion.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"
] | [
"quaternion_algebra.coe_pow"
] | 547 | 548 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_re (n : ℕ) : (n : ℍ[R]).re = n := rfl | nat_cast_re (n : ℕ) : (n : ℍ[R]).re = n | rfl | lemma | quaternion.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"
] | [] | 550 | 550 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_im_i (n : ℕ) : (n : ℍ[R]).im_i = 0 := rfl | nat_cast_im_i (n : ℕ) : (n : ℍ[R]).im_i = 0 | rfl | lemma | quaternion.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"
] | [] | 551 | 551 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_im_j (n : ℕ) : (n : ℍ[R]).im_j = 0 := rfl | nat_cast_im_j (n : ℕ) : (n : ℍ[R]).im_j = 0 | rfl | lemma | quaternion.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"
] | [] | 552 | 552 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_im_k (n : ℕ) : (n : ℍ[R]).im_k = 0 := rfl | nat_cast_im_k (n : ℕ) : (n : ℍ[R]).im_k = 0 | rfl | lemma | quaternion.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"
] | [] | 553 | 553 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_im (n : ℕ) : (n : ℍ[R]).im = 0 := rfl | nat_cast_im (n : ℕ) : (n : ℍ[R]).im = 0 | rfl | lemma | quaternion.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"
] | [] | 554 | 554 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_nat_cast (n : ℕ) : ↑(n : R) = (n : ℍ[R]) := rfl | coe_nat_cast (n : ℕ) : ↑(n : R) = (n : ℍ[R]) | rfl | lemma | quaternion.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"
] | [] | 555 | 555 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_cast_re (z : ℤ) : (z : ℍ[R]).re = z := rfl | int_cast_re (z : ℤ) : (z : ℍ[R]).re = z | rfl | lemma | quaternion.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"
] | [] | 557 | 557 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_cast_im_i (z : ℤ) : (z : ℍ[R]).im_i = 0 := rfl | int_cast_im_i (z : ℤ) : (z : ℍ[R]).im_i = 0 | rfl | lemma | quaternion.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"
] | [] | 558 | 558 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_cast_im_j (z : ℤ) : (z : ℍ[R]).im_j = 0 := rfl | int_cast_im_j (z : ℤ) : (z : ℍ[R]).im_j = 0 | rfl | lemma | quaternion.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"
] | [] | 559 | 559 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_cast_im_k (z : ℤ) : (z : ℍ[R]).im_k = 0 := rfl | int_cast_im_k (z : ℤ) : (z : ℍ[R]).im_k = 0 | rfl | lemma | quaternion.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"
] | [] | 560 | 560 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_cast_im (z : ℤ) : (z : ℍ[R]).im = 0 := rfl | int_cast_im (z : ℤ) : (z : ℍ[R]).im = 0 | rfl | lemma | quaternion.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"
] | [] | 561 | 561 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_int_cast (z : ℤ) : ↑(z : R) = (z : ℍ[R]) := rfl | coe_int_cast (z : ℤ) : ↑(z : R) = (z : ℍ[R]) | rfl | lemma | quaternion.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"
] | [] | 562 | 562 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_injective : function.injective (coe : R → ℍ[R]) := quaternion_algebra.coe_injective | coe_injective : function.injective (coe : R → ℍ[R]) | quaternion_algebra.coe_injective | lemma | quaternion.coe_injective | 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.coe_injective"
] | 564 | 564 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inj {x y : R} : (x : ℍ[R]) = y ↔ x = y := coe_injective.eq_iff | coe_inj {x y : R} : (x : ℍ[R]) = y ↔ x = y | coe_injective.eq_iff | lemma | quaternion.coe_inj | 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"
] | [] | 566 | 566 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_re [has_smul S R] (s : S) : (s • a).re = s • a.re := rfl | smul_re [has_smul S R] (s : S) : (s • a).re = s • a.re | rfl | lemma | quaternion.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"
] | [
"has_smul"
] | 568 | 568 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_im_i [has_smul S R] (s : S) : (s • a).im_i = s • a.im_i := rfl | smul_im_i [has_smul S R] (s : S) : (s • a).im_i = s • a.im_i | rfl | lemma | quaternion.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"
] | [
"has_smul"
] | 569 | 569 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_im_j [has_smul S R] (s : S) : (s • a).im_j = s • a.im_j := rfl | smul_im_j [has_smul S R] (s : S) : (s • a).im_j = s • a.im_j | rfl | lemma | quaternion.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"
] | [
"has_smul"
] | 570 | 570 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_im_k [has_smul S R] (s : S) : (s • a).im_k = s • a.im_k := rfl | smul_im_k [has_smul S R] (s : S) : (s • a).im_k = s • a.im_k | rfl | lemma | quaternion.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"
] | [
"has_smul"
] | 571 | 571 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_im [smul_zero_class S R] (s : S) : (s • a).im = s • a.im :=
ext _ _ (smul_zero _).symm rfl rfl rfl | smul_im [smul_zero_class S R] (s : S) : (s • a).im = s • a.im | ext _ _ (smul_zero _).symm rfl rfl rfl | lemma | quaternion.smul_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"
] | [
"smul_zero",
"smul_zero_class"
] | 572 | 573 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul [smul_zero_class S R] (s : S) (r : R) :
(↑(s • r) : ℍ[R]) = s • ↑r :=
quaternion_algebra.coe_smul _ _ | coe_smul [smul_zero_class S R] (s : S) (r : R) :
(↑(s • r) : ℍ[R]) = s • ↑r | quaternion_algebra.coe_smul _ _ | lemma | quaternion.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"
] | [
"quaternion_algebra.coe_smul",
"smul_zero_class"
] | 575 | 577 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_commutes : ↑r * a = a * r := quaternion_algebra.coe_commutes r a | coe_commutes : ↑r * a = a * r | quaternion_algebra.coe_commutes r a | lemma | quaternion.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"
] | [
"quaternion_algebra.coe_commutes"
] | 579 | 579 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_commute : commute ↑r a := quaternion_algebra.coe_commute r a | coe_commute : commute ↑r a | quaternion_algebra.coe_commute r a | lemma | quaternion.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",
"quaternion_algebra.coe_commute"
] | 581 | 581 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul_eq_smul : ↑r * a = r • a := quaternion_algebra.coe_mul_eq_smul r a | coe_mul_eq_smul : ↑r * a = r • a | quaternion_algebra.coe_mul_eq_smul r a | lemma | quaternion.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"
] | [
"quaternion_algebra.coe_mul_eq_smul"
] | 583 | 583 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_coe_eq_smul : a * r = r • a := quaternion_algebra.mul_coe_eq_smul r a | mul_coe_eq_smul : a * r = r • a | quaternion_algebra.mul_coe_eq_smul r a | lemma | quaternion.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"
] | [
"quaternion_algebra.mul_coe_eq_smul"
] | 585 | 585 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_def : ⇑(algebra_map R ℍ[R]) = coe := rfl | algebra_map_def : ⇑(algebra_map R ℍ[R]) = coe | rfl | lemma | quaternion.algebra_map_def | 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"
] | 587 | 587 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_coe : x • (y : ℍ[R]) = ↑(x * y) := quaternion_algebra.smul_coe x y | smul_coe : x • (y : ℍ[R]) = ↑(x * y) | quaternion_algebra.smul_coe x y | lemma | quaternion.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"
] | [
"quaternion_algebra.smul_coe"
] | 589 | 589 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rank_eq_four [strong_rank_condition R] : module.rank R ℍ[R] = 4 :=
quaternion_algebra.rank_eq_four _ _ | rank_eq_four [strong_rank_condition R] : module.rank R ℍ[R] = 4 | quaternion_algebra.rank_eq_four _ _ | lemma | quaternion.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"
] | [
"module.rank",
"quaternion_algebra.rank_eq_four",
"strong_rank_condition"
] | 594 | 595 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finrank_eq_four [strong_rank_condition R] : finite_dimensional.finrank R ℍ[R] = 4 :=
quaternion_algebra.finrank_eq_four _ _ | finrank_eq_four [strong_rank_condition R] : finite_dimensional.finrank R ℍ[R] = 4 | quaternion_algebra.finrank_eq_four _ _ | lemma | quaternion.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"
] | [
"finite_dimensional.finrank",
"quaternion_algebra.finrank_eq_four",
"strong_rank_condition"
] | 597 | 598 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_re : (star a).re = a.re := rfl | star_re : (star a).re = a.re | rfl | lemma | quaternion.star_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"
] | [] | 600 | 600 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_im_i : (star a).im_i = - a.im_i := rfl | star_im_i : (star a).im_i = - a.im_i | rfl | lemma | quaternion.star_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"
] | [] | 601 | 601 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_im_j : (star a).im_j = - a.im_j := rfl | star_im_j : (star a).im_j = - a.im_j | rfl | lemma | quaternion.star_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"
] | [] | 602 | 602 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_im_k : (star a).im_k = - a.im_k := rfl | star_im_k : (star a).im_k = - a.im_k | rfl | lemma | quaternion.star_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"
] | [] | 603 | 603 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_im : (star a).im = - a.im := a.im_star | star_im : (star a).im = - a.im | a.im_star | lemma | quaternion.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"
] | [] | 604 | 604 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_add_star' : a + star a = ↑(2 * a.re) := a.self_add_star' | self_add_star' : a + star a = ↑(2 * a.re) | a.self_add_star' | lemma | quaternion.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"
] | [] | 606 | 606 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_add_star : a + star a = 2 * a.re := a.self_add_star | self_add_star : a + star a = 2 * a.re | a.self_add_star | lemma | quaternion.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"
] | [] | 608 | 608 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_add_self' : star a + a = ↑(2 * a.re) := a.star_add_self' | star_add_self' : star a + a = ↑(2 * a.re) | a.star_add_self' | lemma | quaternion.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"
] | [] | 610 | 610 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_add_self : star a + a = 2 * a.re := a.star_add_self | star_add_self : star a + a = 2 * a.re | a.star_add_self | lemma | quaternion.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"
] | [] | 612 | 612 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_eq_two_re_sub : star a = ↑(2 * a.re) - a := a.star_eq_two_re_sub | star_eq_two_re_sub : star a = ↑(2 * a.re) - a | a.star_eq_two_re_sub | lemma | quaternion.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"
] | [] | 614 | 614 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_coe : star (x : ℍ[R]) = x := quaternion_algebra.star_coe x | star_coe : star (x : ℍ[R]) = x | quaternion_algebra.star_coe x | lemma | quaternion.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"
] | [
"quaternion_algebra.star_coe"
] | 616 | 616 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
im_star : star a.im = - a.im := quaternion_algebra.im_star _ | im_star : star a.im = - a.im | quaternion_algebra.im_star _ | lemma | quaternion.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"
] | [
"quaternion_algebra.im_star"
] | 617 | 617 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_smul [monoid S] [distrib_mul_action S R] (s : S) (a : ℍ[R]) :
star (s • a) = s • star a := quaternion_algebra.star_smul _ _ | star_smul [monoid S] [distrib_mul_action S R] (s : S) (a : ℍ[R]) :
star (s • a) = s • star a | quaternion_algebra.star_smul _ _ | lemma | quaternion.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",
"quaternion_algebra.star_smul"
] | 619 | 620 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_re_of_eq_coe {a : ℍ[R]} {x : R} (h : a = x) : a = a.re :=
quaternion_algebra.eq_re_of_eq_coe h | eq_re_of_eq_coe {a : ℍ[R]} {x : R} (h : a = x) : a = a.re | quaternion_algebra.eq_re_of_eq_coe h | lemma | quaternion.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"
] | [
"quaternion_algebra.eq_re_of_eq_coe"
] | 622 | 623 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_re_iff_mem_range_coe {a : ℍ[R]} : a = a.re ↔ a ∈ set.range (coe : R → ℍ[R]) :=
quaternion_algebra.eq_re_iff_mem_range_coe | eq_re_iff_mem_range_coe {a : ℍ[R]} : a = a.re ↔ a ∈ set.range (coe : R → ℍ[R]) | quaternion_algebra.eq_re_iff_mem_range_coe | lemma | quaternion.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"
] | [
"quaternion_algebra.eq_re_iff_mem_range_coe",
"set.range"
] | 625 | 626 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_eq_self {a : ℍ[R]} : star a = a ↔ a = a.re := quaternion_algebra.star_eq_self | star_eq_self {a : ℍ[R]} : star a = a ↔ a = a.re | quaternion_algebra.star_eq_self | lemma | quaternion.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"
] | [
"quaternion_algebra.star_eq_self"
] | 631 | 631 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_eq_neg {a : ℍ[R]} : star a = -a ↔ a.re = 0 := quaternion_algebra.star_eq_neg | star_eq_neg {a : ℍ[R]} : star a = -a ↔ a.re = 0 | quaternion_algebra.star_eq_neg | lemma | quaternion.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"
] | [
"quaternion_algebra.star_eq_neg"
] | 633 | 633 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_mul_eq_coe : star a * a = (star a * a).re := a.star_mul_eq_coe | star_mul_eq_coe : star a * a = (star a * a).re | a.star_mul_eq_coe | lemma | quaternion.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"
] | [] | 637 | 637 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_star_eq_coe : a * star a = (a * star a).re := a.mul_star_eq_coe | mul_star_eq_coe : a * star a = (a * star a).re | a.mul_star_eq_coe | lemma | quaternion.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"
] | [] | 639 | 639 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_ae : ℍ[R] ≃ₐ[R] (ℍ[R]ᵐᵒᵖ) := quaternion_algebra.star_ae | star_ae : ℍ[R] ≃ₐ[R] (ℍ[R]ᵐᵒᵖ) | quaternion_algebra.star_ae | def | quaternion.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"
] | [
"quaternion_algebra.star_ae"
] | Quaternion conjugate as an `alg_equiv` to the opposite ring. | 644 | 644 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_star_ae : ⇑(star_ae : ℍ[R] ≃ₐ[R] ℍ[R]ᵐᵒᵖ) = op ∘ star := rfl | coe_star_ae : ⇑(star_ae : ℍ[R] ≃ₐ[R] ℍ[R]ᵐᵒᵖ) = op ∘ star | rfl | lemma | quaternion.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"
] | [] | 646 | 646 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sq : ℍ[R] →*₀ R :=
{ to_fun := λ a, (a * star a).re,
map_zero' := by rw [star_zero, zero_mul, zero_re],
map_one' := by rw [star_one, one_mul, one_re],
map_mul' := λ x y, coe_injective $ by conv_lhs { rw [← mul_star_eq_coe, star_mul, mul_assoc,
← mul_assoc y, y.mul_star_eq_coe, coe_commutes, ← mul_assoc, ... | norm_sq : ℍ[R] →*₀ R | { to_fun := λ a, (a * star a).re,
map_zero' := by rw [star_zero, zero_mul, zero_re],
map_one' := by rw [star_one, one_mul, one_re],
map_mul' := λ x y, coe_injective $ by conv_lhs { rw [← mul_star_eq_coe, star_mul, mul_assoc,
← mul_assoc y, y.mul_star_eq_coe, coe_commutes, ← mul_assoc, x.mul_star_eq_coe, ← coe... | def | quaternion.norm_sq | 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"
] | [
"mul_assoc",
"one_mul",
"star_one",
"star_zero",
"zero_mul"
] | Square of the norm. | 649 | 654 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_sq_def : norm_sq a = (a * star a).re := rfl | norm_sq_def : norm_sq a = (a * star a).re | rfl | lemma | quaternion.norm_sq_def | 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"
] | [] | 656 | 656 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sq_def' : norm_sq a = a.1^2 + a.2^2 + a.3^2 + a.4^2 :=
by simp only [norm_sq_def, sq, mul_neg, sub_neg_eq_add,
mul_re, star_re, star_im_i, star_im_j, star_im_k] | norm_sq_def' : norm_sq a = a.1^2 + a.2^2 + a.3^2 + a.4^2 | by simp only [norm_sq_def, sq, mul_neg, sub_neg_eq_add,
mul_re, star_re, star_im_i, star_im_j, star_im_k] | lemma | quaternion.norm_sq_def' | 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"
] | [
"mul_neg"
] | 658 | 660 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sq_coe : norm_sq (x : ℍ[R]) = x^2 :=
by rw [norm_sq_def, star_coe, ← coe_mul, coe_re, sq] | norm_sq_coe : norm_sq (x : ℍ[R]) = x^2 | by rw [norm_sq_def, star_coe, ← coe_mul, coe_re, sq] | lemma | quaternion.norm_sq_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"
] | [] | 662 | 663 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sq_star : norm_sq (star a) = norm_sq a := by simp [norm_sq_def'] | norm_sq_star : norm_sq (star a) = norm_sq a | by simp [norm_sq_def'] | lemma | quaternion.norm_sq_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"
] | [] | 665 | 665 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sq_nat_cast (n : ℕ) : norm_sq (n : ℍ[R]) = n^2 :=
by rw [←coe_nat_cast, norm_sq_coe] | norm_sq_nat_cast (n : ℕ) : norm_sq (n : ℍ[R]) = n^2 | by rw [←coe_nat_cast, norm_sq_coe] | lemma | quaternion.norm_sq_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"
] | [] | 667 | 668 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sq_int_cast (z : ℤ) : norm_sq (z : ℍ[R]) = z^2 :=
by rw [←coe_int_cast, norm_sq_coe] | norm_sq_int_cast (z : ℤ) : norm_sq (z : ℍ[R]) = z^2 | by rw [←coe_int_cast, norm_sq_coe] | lemma | quaternion.norm_sq_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"
] | [] | 670 | 671 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sq_neg : norm_sq (-a) = norm_sq a :=
by simp only [norm_sq_def, star_neg, neg_mul_neg] | norm_sq_neg : norm_sq (-a) = norm_sq a | by simp only [norm_sq_def, star_neg, neg_mul_neg] | lemma | quaternion.norm_sq_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"
] | [
"neg_mul_neg",
"star_neg"
] | 673 | 674 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_mul_star : a * star a = norm_sq a := by rw [mul_star_eq_coe, norm_sq_def] | self_mul_star : a * star a = norm_sq a | by rw [mul_star_eq_coe, norm_sq_def] | lemma | quaternion.self_mul_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"
] | [] | 676 | 676 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_mul_self : star a * a = norm_sq a := by rw [star_comm_self', self_mul_star] | star_mul_self : star a * a = norm_sq a | by rw [star_comm_self', self_mul_star] | lemma | quaternion.star_mul_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"
] | [
"star_comm_self'"
] | 678 | 678 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
im_sq : a.im^2 = -norm_sq a.im :=
by simp_rw [sq, ←star_mul_self, im_star, neg_mul, neg_neg] | im_sq : a.im^2 = -norm_sq a.im | by simp_rw [sq, ←star_mul_self, im_star, neg_mul, neg_neg] | lemma | quaternion.im_sq | 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"
] | [
"neg_mul"
] | 680 | 681 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_norm_sq_add :
(norm_sq (a + b) : ℍ[R]) = norm_sq a + a * star b + b * star a + norm_sq b :=
by simp [← self_mul_star, mul_add, add_mul, add_assoc] | coe_norm_sq_add :
(norm_sq (a + b) : ℍ[R]) = norm_sq a + a * star b + b * star a + norm_sq b | by simp [← self_mul_star, mul_add, add_mul, add_assoc] | lemma | quaternion.coe_norm_sq_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"
] | [] | 683 | 685 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sq_smul (r : R) (q : ℍ[R]) : norm_sq (r • q) = r^2 * norm_sq q :=
by simp_rw [norm_sq_def, star_smul, smul_mul_smul, smul_re, sq, smul_eq_mul] | norm_sq_smul (r : R) (q : ℍ[R]) : norm_sq (r • q) = r^2 * norm_sq q | by simp_rw [norm_sq_def, star_smul, smul_mul_smul, smul_re, sq, smul_eq_mul] | lemma | quaternion.norm_sq_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_eq_mul",
"smul_mul_smul"
] | 687 | 688 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sq_add (a b : ℍ[R]) : norm_sq (a + b) = norm_sq a + norm_sq b + 2 * (a * star b).re :=
calc norm_sq (a + b) = (norm_sq a + (a * star b).re) + ((b * star a).re + norm_sq b)
: by simp_rw [norm_sq_def, star_add, add_mul, mul_add, add_re]
... = norm_sq a + norm_sq b + ((a * star b... | norm_sq_add (a b : ℍ[R]) : norm_sq (a + b) = norm_sq a + norm_sq b + 2 * (a * star b).re | calc norm_sq (a + b) = (norm_sq a + (a * star b).re) + ((b * star a).re + norm_sq b)
: by simp_rw [norm_sq_def, star_add, add_mul, mul_add, add_re]
... = norm_sq a + norm_sq b + ((a * star b).re + (b * star a).re) : by abel
... = norm_sq a + norm_sq b + 2 * (a * st... | lemma | quaternion.norm_sq_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"
] | [] | 690 | 695 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sq_eq_zero : norm_sq a = 0 ↔ a = 0 :=
begin
refine ⟨λ h, _, λ h, h.symm ▸ norm_sq.map_zero⟩,
rw [norm_sq_def', add_eq_zero_iff', add_eq_zero_iff', add_eq_zero_iff'] at h,
exact ext a 0 (pow_eq_zero h.1.1.1) (pow_eq_zero h.1.1.2) (pow_eq_zero h.1.2) (pow_eq_zero h.2),
all_goals { apply_rules [sq_nonneg, add... | norm_sq_eq_zero : norm_sq a = 0 ↔ a = 0 | begin
refine ⟨λ h, _, λ h, h.symm ▸ norm_sq.map_zero⟩,
rw [norm_sq_def', add_eq_zero_iff', add_eq_zero_iff', add_eq_zero_iff'] at h,
exact ext a 0 (pow_eq_zero h.1.1.1) (pow_eq_zero h.1.1.2) (pow_eq_zero h.1.2) (pow_eq_zero h.2),
all_goals { apply_rules [sq_nonneg, add_nonneg] }
end | lemma | quaternion.norm_sq_eq_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"
] | [
"pow_eq_zero",
"sq_nonneg"
] | 707 | 713 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sq_ne_zero : norm_sq a ≠ 0 ↔ a ≠ 0 := not_congr norm_sq_eq_zero | norm_sq_ne_zero : norm_sq a ≠ 0 ↔ a ≠ 0 | not_congr norm_sq_eq_zero | lemma | quaternion.norm_sq_ne_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"
] | [] | 715 | 715 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sq_nonneg : 0 ≤ norm_sq a :=
by { rw norm_sq_def', apply_rules [sq_nonneg, add_nonneg] } | norm_sq_nonneg : 0 ≤ norm_sq a | by { rw norm_sq_def', apply_rules [sq_nonneg, add_nonneg] } | lemma | quaternion.norm_sq_nonneg | 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"
] | [
"sq_nonneg"
] | 717 | 718 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sq_le_zero : norm_sq a ≤ 0 ↔ a = 0 :=
by simpa only [le_antisymm_iff, norm_sq_nonneg, and_true] using @norm_sq_eq_zero _ _ a | norm_sq_le_zero : norm_sq a ≤ 0 ↔ a = 0 | by simpa only [le_antisymm_iff, norm_sq_nonneg, and_true] using @norm_sq_eq_zero _ _ a | lemma | quaternion.norm_sq_le_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"
] | [] | 720 | 721 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sq_eq_norm_sq : a^2 = norm_sq a ↔ a = a.re :=
begin
simp_rw [←star_eq_self],
obtain rfl | hq0 := eq_or_ne a 0,
{ simp },
{ rw [←star_mul_self, sq, mul_left_inj' hq0, eq_comm] }
end | sq_eq_norm_sq : a^2 = norm_sq a ↔ a = a.re | begin
simp_rw [←star_eq_self],
obtain rfl | hq0 := eq_or_ne a 0,
{ simp },
{ rw [←star_mul_self, sq, mul_left_inj' hq0, eq_comm] }
end | lemma | quaternion.sq_eq_norm_sq | 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"
] | [
"eq_or_ne",
"mul_left_inj'"
] | 735 | 741 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sq_eq_neg_norm_sq : a^2 = -norm_sq a ↔ a.re = 0 :=
begin
simp_rw [←star_eq_neg],
obtain rfl | hq0 := eq_or_ne a 0,
{ simp },
rw [←star_mul_self, ←mul_neg, ←neg_sq, sq, mul_left_inj' (neg_ne_zero.mpr hq0), eq_comm],
end | sq_eq_neg_norm_sq : a^2 = -norm_sq a ↔ a.re = 0 | begin
simp_rw [←star_eq_neg],
obtain rfl | hq0 := eq_or_ne a 0,
{ simp },
rw [←star_mul_self, ←mul_neg, ←neg_sq, sq, mul_left_inj' (neg_ne_zero.mpr hq0), eq_comm],
end | lemma | quaternion.sq_eq_neg_norm_sq | 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"
] | [
"eq_or_ne",
"mul_left_inj'"
] | 743 | 749 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inv (x : R) : ((x⁻¹ : R) : ℍ[R]) = x⁻¹ :=
map_inv₀ (algebra_map R ℍ[R]) _ | coe_inv (x : R) : ((x⁻¹ : R) : ℍ[R]) = x⁻¹ | map_inv₀ (algebra_map R ℍ[R]) _ | lemma | quaternion.coe_inv | 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_inv₀"
] | 767 | 768 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_div (x y : R) : ((x / y : R) : ℍ[R]) = x / y :=
map_div₀ (algebra_map R ℍ[R]) x y | coe_div (x y : R) : ((x / y : R) : ℍ[R]) = x / y | map_div₀ (algebra_map R ℍ[R]) x y | lemma | quaternion.coe_div | 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_div₀"
] | 770 | 771 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zpow (x : R) (z : ℤ) : ((x ^ z : R) : ℍ[R]) = x ^ z :=
map_zpow₀ (algebra_map R ℍ[R]) x z | coe_zpow (x : R) (z : ℤ) : ((x ^ z : R) : ℍ[R]) = x ^ z | map_zpow₀ (algebra_map R ℍ[R]) x z | lemma | quaternion.coe_zpow | 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_zpow₀"
] | 773 | 774 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rat_cast_re (q : ℚ) : (q : ℍ[R]).re = q := rfl | rat_cast_re (q : ℚ) : (q : ℍ[R]).re = q | rfl | lemma | quaternion.rat_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"
] | [] | 787 | 787 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rat_cast_im_i (q : ℚ) : (q : ℍ[R]).im_i = 0 := rfl | rat_cast_im_i (q : ℚ) : (q : ℍ[R]).im_i = 0 | rfl | lemma | quaternion.rat_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"
] | [] | 788 | 788 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rat_cast_im_j (q : ℚ) : (q : ℍ[R]).im_j = 0 := rfl | rat_cast_im_j (q : ℚ) : (q : ℍ[R]).im_j = 0 | rfl | lemma | quaternion.rat_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"
] | [] | 789 | 789 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rat_cast_im_k (q : ℚ) : (q : ℍ[R]).im_k = 0 := rfl | rat_cast_im_k (q : ℚ) : (q : ℍ[R]).im_k = 0 | rfl | lemma | quaternion.rat_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"
] | [] | 790 | 790 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rat_cast_im (q : ℚ) : (q : ℍ[R]).im = 0 := rfl | rat_cast_im (q : ℚ) : (q : ℍ[R]).im = 0 | rfl | lemma | quaternion.rat_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"
] | [] | 791 | 791 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_rat_cast (q : ℚ) : ↑(q : R) = (q : ℍ[R]) := rfl | coe_rat_cast (q : ℚ) : ↑(q : R) = (q : ℍ[R]) | rfl | lemma | quaternion.coe_rat_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"
] | [] | 792 | 792 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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