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
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gcd_union (s₁ s₂ : multiset α) :
(s₁ ∪ s₂).gcd = gcd_monoid.gcd s₁.gcd s₂.gcd | by { rw [← gcd_dedup, dedup_ext.2, gcd_dedup, gcd_add], simp } | lemma | multiset.gcd_union | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_ndinsert (a : α) (s : multiset α) :
(ndinsert a s).gcd = gcd_monoid.gcd a s.gcd | by { rw [← gcd_dedup, dedup_ext.2, gcd_dedup, gcd_cons], simp } | lemma | multiset.gcd_ndinsert | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extract_gcd' (s t : multiset α) (hs : ∃ x, x ∈ s ∧ x ≠ (0 : α))
(ht : s = t.map ((*) s.gcd)) : t.gcd = 1 | ((@mul_right_eq_self₀ _ _ s.gcd _).1 $ by conv_lhs { rw [← normalize_gcd, ← gcd_map_mul, ← ht] })
.resolve_right $ by { contrapose! hs, exact s.gcd_eq_zero_iff.1 hs } | lemma | multiset.extract_gcd' | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"mul_right_eq_self₀",
"multiset",
"normalize_gcd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extract_gcd (s : multiset α) (hs : s ≠ 0) :
∃ t : multiset α, s = t.map ((*) s.gcd) ∧ t.gcd = 1 | begin
classical,
by_cases h : ∀ x ∈ s, x = (0 : α),
{ use replicate s.card 1,
rw [map_replicate, eq_replicate, mul_one, s.gcd_eq_zero_iff.2 h, ←nsmul_singleton, ←gcd_dedup],
rw [dedup_nsmul (card_pos.2 hs).ne', dedup_singleton, gcd_singleton],
exact ⟨⟨rfl, h⟩, normalize_one⟩ },
{ choose f hf using @... | lemma | multiset.extract_gcd | algebra.gcd_monoid | src/algebra/gcd_monoid/multiset.lean | [
"algebra.gcd_monoid.basic",
"data.multiset.finset_ops",
"data.multiset.fold"
] | [
"extract_gcd",
"mul_one",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc_left : (f x) ∘ (f y) = (f (f x y)) | by { ext z, rw [function.comp_apply, @is_associative.assoc _ f] } | lemma | comp_assoc_left | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"function.comp_apply"
] | Composing two associative operations of `f : α → α → α` on the left
is equal to an associative operation on the left. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_assoc_right : (λ z, f z x) ∘ (λ z, f z y) = (λ z, f z (f y x)) | by { ext z, rw [function.comp_apply, @is_associative.assoc _ f] } | lemma | comp_assoc_right | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"function.comp_apply"
] | Composing two associative operations of `f : α → α → α` on the right
is equal to an associative operation on the right. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_mul_left [semigroup α] (x y : α) :
((*) x) ∘ ((*) y) = ((*) (x * y)) | comp_assoc_left _ _ _ | lemma | comp_mul_left | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"comp_assoc_left",
"semigroup"
] | Composing two multiplications on the left by `y` then `x`
is equal to a multiplication on the left by `x * y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_mul_right [semigroup α] (x y : α) :
(* x) ∘ (* y) = (* (y * x)) | comp_assoc_right _ _ _ | lemma | comp_mul_right | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"comp_assoc_right",
"semigroup"
] | Composing two multiplications on the right by `y` and `x`
is equal to a multiplication on the right by `y * x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ite_mul_one {P : Prop} [decidable P] {a b : M} :
ite P (a * b) 1 = ite P a 1 * ite P b 1 | by { by_cases h : P; simp [h], } | lemma | ite_mul_one | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ite_one_mul {P : Prop} [decidable P] {a b : M} :
ite P 1 (a * b) = ite P 1 a * ite P 1 b | by { by_cases h : P; simp [h], } | lemma | ite_one_mul | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_one_iff_eq_one_of_mul_eq_one {a b : M} (h : a * b = 1) : a = 1 ↔ b = 1 | by split; { rintro rfl, simpa using h } | lemma | eq_one_iff_eq_one_of_mul_eq_one | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_mul_eq_id : ((*) (1 : M)) = id | funext one_mul | lemma | one_mul_eq_id | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_one_eq_id : (* (1 : M)) = id | funext mul_one | lemma | mul_one_eq_id | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_comm : ∀ a b c : G, a * (b * c) = b * (a * c) | left_comm has_mul.mul mul_comm mul_assoc | lemma | mul_left_comm | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"mul_assoc",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_comm : ∀ a b c : G, a * b * c = a * c * b | right_comm has_mul.mul mul_comm mul_assoc | lemma | mul_right_comm | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"mul_assoc",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mul_mul_comm (a b c d : G) : (a * b) * (c * d) = (a * c) * (b * d) | by simp only [mul_left_comm, mul_assoc] | theorem | mul_mul_mul_comm | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"mul_assoc",
"mul_left_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_rotate (a b c : G) : a * b * c = b * c * a | by simp only [mul_left_comm, mul_comm] | lemma | mul_rotate | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"mul_comm",
"mul_left_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_rotate' (a b c : G) : a * (b * c) = b * (c * a) | by simp only [mul_left_comm, mul_comm] | lemma | mul_rotate' | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"mul_comm",
"mul_left_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_add (a b : M) : bit0 (a + b) = bit0 a + bit0 b | add_add_add_comm _ _ _ _ | lemma | bit0_add | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit1_add [has_one M] (a b : M) : bit1 (a + b) = bit0 a + bit1 b | (congr_arg (+ (1 : M)) $ bit0_add a b : _).trans (add_assoc _ _ _) | lemma | bit1_add | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"bit0_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit1_add' [has_one M] (a b : M) : bit1 (a + b) = bit1 a + bit0 b | by rw [add_comm, bit1_add, add_comm] | lemma | bit1_add' | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"bit1_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_zero : bit0 (0 : M) = 0 | add_zero _ | lemma | bit0_zero | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit1_zero [has_one M] : bit1 (0 : M) = 1 | by rw [bit1, bit0_zero, zero_add] | lemma | bit1_zero | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"bit0_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_unique (hy : x * y = 1) (hz : x * z = 1) : y = z | left_inv_eq_right_inv (trans (mul_comm _ _) hy) hz | lemma | inv_unique | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"left_inv_eq_right_inv",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_eq_self : a * b = a ↔ b = 1 | calc a * b = a ↔ a * b = a * 1 : by rw mul_one
... ↔ b = 1 : mul_left_cancel_iff | lemma | mul_right_eq_self | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"mul_left_cancel_iff",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_eq_mul_right : a = a * b ↔ b = 1 | eq_comm.trans mul_right_eq_self | lemma | self_eq_mul_right | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"mul_right_eq_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_ne_self : a * b ≠ a ↔ b ≠ 1 | mul_right_eq_self.not | lemma | mul_right_ne_self | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_ne_mul_right : a ≠ a * b ↔ b ≠ 1 | self_eq_mul_right.not | lemma | self_ne_mul_right | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_eq_self : a * b = b ↔ a = 1 | calc a * b = b ↔ a * b = 1 * b : by rw one_mul
... ↔ a = 1 : mul_right_cancel_iff | lemma | mul_left_eq_self | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"mul_right_cancel_iff",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_eq_mul_left : b = a * b ↔ a = 1 | eq_comm.trans mul_left_eq_self | lemma | self_eq_mul_left | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"mul_left_eq_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_ne_self : a * b ≠ b ↔ a ≠ 1 | mul_left_eq_self.not | lemma | mul_left_ne_self | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_ne_mul_left : b ≠ a * b ↔ a ≠ 1 | self_eq_mul_left.not | lemma | self_ne_mul_left | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_involutive : function.involutive (has_inv.inv : G → G) | inv_inv | lemma | inv_involutive | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"inv_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_surjective : function.surjective (has_inv.inv : G → G) | inv_involutive.surjective | lemma | inv_surjective | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_injective : function.injective (has_inv.inv : G → G) | inv_involutive.injective | lemma | inv_injective | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_inj {a b : G} : a⁻¹ = b⁻¹ ↔ a = b | inv_injective.eq_iff | theorem | inv_inj | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_eq_iff_eq_inv : a⁻¹ = b ↔ a = b⁻¹ | ⟨λ h, h ▸ (inv_inv a).symm, λ h, h.symm ▸ inv_inv b⟩ | theorem | inv_eq_iff_eq_inv | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"inv_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_comp_inv : has_inv.inv ∘ has_inv.inv = @id G | inv_involutive.comp_self | lemma | inv_comp_inv | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_inverse_inv : left_inverse (λ a : G, a⁻¹) (λ a, a⁻¹) | inv_inv | lemma | left_inverse_inv | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"inv_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_inverse_inv : left_inverse (λ a : G, a⁻¹) (λ a, a⁻¹) | inv_inv | lemma | right_inverse_inv | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"inv_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_eq_one_div (x : G) :
x⁻¹ = 1 / x | by rw [div_eq_mul_inv, one_mul] | lemma | inv_eq_one_div | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_eq_mul_inv",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_one_div (x y : G) :
x * (1 / y) = x / y | by rw [div_eq_mul_inv, one_mul, div_eq_mul_inv] | lemma | mul_one_div | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_eq_mul_inv",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_assoc (a b c : G) : a * b / c = a * (b / c) | by rw [div_eq_mul_inv, div_eq_mul_inv, mul_assoc _ _ _] | lemma | mul_div_assoc | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_eq_mul_inv",
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_assoc' (a b c : G) : a * (b / c) = (a * b) / c | (mul_div_assoc _ _ _).symm | lemma | mul_div_assoc' | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"mul_div_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div (a : G) : 1 / a = a⁻¹ | (inv_eq_one_div a).symm | lemma | one_div | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"inv_eq_one_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div (a b c : G) : a * (b / c) = a * b / c | by simp only [mul_assoc, div_eq_mul_inv] | lemma | mul_div | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_eq_mul_inv",
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_eq_mul_one_div (a b : G) : a / b = a * (1 / b) | by rw [div_eq_mul_inv, one_div] | lemma | div_eq_mul_one_div | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_eq_mul_inv",
"one_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_one (a : G) : a / 1 = a | by simp [div_eq_mul_inv] | lemma | div_one | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_one : (1 : G) / 1 = 1 | div_one _ | lemma | one_div_one | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_eq_of_mul_eq_one_left (h : a * b = 1) : b⁻¹ = a | by rw [←inv_eq_of_mul_eq_one_right h, inv_inv] | lemma | inv_eq_of_mul_eq_one_left | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"inv_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_inv_of_mul_eq_one_left (h : a * b = 1) : a = b⁻¹ | (inv_eq_of_mul_eq_one_left h).symm | lemma | eq_inv_of_mul_eq_one_left | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"inv_eq_of_mul_eq_one_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_inv_of_mul_eq_one_right (h : a * b = 1) : b = a⁻¹ | (inv_eq_of_mul_eq_one_right h).symm | lemma | eq_inv_of_mul_eq_one_right | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"inv_eq_of_mul_eq_one_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_one_div_of_mul_eq_one_left (h : b * a = 1) : b = 1 / a | by rw [eq_inv_of_mul_eq_one_left h, one_div] | lemma | eq_one_div_of_mul_eq_one_left | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"eq_inv_of_mul_eq_one_left",
"one_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_one_div_of_mul_eq_one_right (h : a * b = 1) : b = 1 / a | by rw [eq_inv_of_mul_eq_one_right h, one_div] | lemma | eq_one_div_of_mul_eq_one_right | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"eq_inv_of_mul_eq_one_right",
"one_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_div_eq_one (h : a / b = 1) : a = b | inv_injective $ inv_eq_of_mul_eq_one_right $ by rwa ←div_eq_mul_inv | lemma | eq_of_div_eq_one | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"inv_eq_of_mul_eq_one_right",
"inv_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_ne_one_of_ne : a ≠ b → a / b ≠ 1 | mt eq_of_div_eq_one | lemma | div_ne_one_of_ne | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"eq_of_div_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_mul_one_div_rev : (1 / a) * (1 / b) = 1 / (b * a) | by simp | lemma | one_div_mul_one_div_rev | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_div_left : a⁻¹ / b = (b * a)⁻¹ | by simp | lemma | inv_div_left | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_div : (a / b)⁻¹ = b / a | by simp | lemma | inv_div | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_div : 1 / (a / b) = b / a | by simp | lemma | one_div_div | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_one_div : 1 / (1 / a) = a | by simp | lemma | one_div_one_div | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
division_monoid.to_div_inv_one_monoid :
div_inv_one_monoid α | { inv_one := by simpa only [one_div, inv_inv] using (inv_div (1 : α) 1).symm,
..division_monoid.to_div_inv_monoid α } | instance | division_monoid.to_div_inv_one_monoid | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_inv_one_monoid",
"inv_div",
"inv_inv",
"inv_one",
"one_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_eq_one : a⁻¹ = 1 ↔ a = 1 | inv_injective.eq_iff' inv_one | lemma | inv_eq_one | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"inv_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_eq_inv : 1 = a⁻¹ ↔ a = 1 | eq_comm.trans inv_eq_one | lemma | one_eq_inv | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"inv_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_ne_one : a⁻¹ ≠ 1 ↔ a ≠ 1 | inv_eq_one.not | lemma | inv_ne_one | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_one_div_eq_one_div (h : 1 / a = 1 / b) : a = b | by rw [←one_div_one_div a, h, one_div_one_div] | lemma | eq_of_one_div_eq_one_div | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"one_div_one_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_div_eq_mul_div : a / (b / c) = a * c / b | by simp | lemma | div_div_eq_mul_div | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_inv_eq_mul : a / b⁻¹ = a * b | by simp | lemma | div_inv_eq_mul | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_mul_eq_div_div_swap : a / (b * c) = a / c / b | by simp only [mul_assoc, mul_inv_rev, div_eq_mul_inv] | lemma | div_mul_eq_div_div_swap | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_eq_mul_inv",
"mul_assoc",
"mul_inv_rev"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_neg [subtraction_monoid α] (a : α) : bit0 (-a) = -bit0 a | (neg_add_rev _ _).symm | lemma | bit0_neg | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"subtraction_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv : (a * b)⁻¹ = a⁻¹ * b⁻¹ | by simp | lemma | mul_inv | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_div' : (a / b)⁻¹ = a⁻¹ / b⁻¹ | by simp | lemma | inv_div' | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_eq_inv_mul : a / b = b⁻¹ * a | by simp | lemma | div_eq_inv_mul | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_eq_div : a⁻¹ * b = b / a | by simp | lemma | inv_mul_eq_div | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul' : (a * b)⁻¹ = a⁻¹ / b | by simp | lemma | inv_mul' | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_div_inv : (a⁻¹ / b⁻¹) = b / a | by simp | lemma | inv_div_inv | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_inv_div_inv : (a⁻¹ / b⁻¹)⁻¹ = a / b | by simp | lemma | inv_inv_div_inv | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_mul_one_div : (1 / a) * (1 / b) = 1 / (a * b) | by simp | lemma | one_div_mul_one_div | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_right_comm : a / b / c = a / c / b | by simp | lemma | div_right_comm | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_div : a / b / c = a / (b * c) | by simp | lemma | div_div | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_mul : a / b * c = a / (b / c) | by simp | lemma | div_mul | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_left_comm : a * (b / c) = b * (a / c) | by simp | lemma | mul_div_left_comm | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_right_comm : a * b / c = a / c * b | by simp | lemma | mul_div_right_comm | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_mul_eq_div_div : a / (b * c) = a / b / c | by simp | lemma | div_mul_eq_div_div | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_mul_eq_mul_div : a / b * c = a * c / b | by simp | lemma | div_mul_eq_mul_div | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_comm_div : a / b * c = a * (c / b) | by simp | lemma | mul_comm_div | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_mul_comm : a / b * c = c / b * a | by simp | lemma | div_mul_comm | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_mul_eq_div_mul_one_div : a / (b * c) = (a / b) * (1 / c) | by simp | lemma | div_mul_eq_div_mul_one_div | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_div_div_eq : a / b / (c / d) = a * d / (b * c) | by simp | lemma | div_div_div_eq | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_div_div_comm : a / b / (c / d) = a / c / (b / d) | by simp | lemma | div_div_div_comm | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_mul_div_comm : a / b * (c / d) = a * c / (b * d) | by simp | lemma | div_mul_div_comm | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_mul_comm : a * b / (c * d) = a / c * (b / d) | by simp | lemma | mul_div_mul_comm | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_eq_inv_self : a / b = b⁻¹ ↔ a = 1 | by rw [div_eq_mul_inv, mul_left_eq_self] | theorem | div_eq_inv_self | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_eq_mul_inv",
"mul_left_eq_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_surjective (a : G) : function.surjective ((*) a) | λ x, ⟨a⁻¹ * x, mul_inv_cancel_left a x⟩ | theorem | mul_left_surjective | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"mul_inv_cancel_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_surjective (a : G) : function.surjective (λ x, x * a) | λ x, ⟨x * a⁻¹, inv_mul_cancel_right x a⟩ | theorem | mul_right_surjective | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"inv_mul_cancel_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_mul_inv_of_mul_eq (h : a * c = b) : a = b * c⁻¹ | by simp [h.symm] | lemma | eq_mul_inv_of_mul_eq | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_inv_mul_of_mul_eq (h : b * a = c) : a = b⁻¹ * c | by simp [h.symm] | lemma | eq_inv_mul_of_mul_eq | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_eq_of_eq_mul (h : b = a * c) : a⁻¹ * b = c | by simp [h] | lemma | inv_mul_eq_of_eq_mul | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_eq_of_eq_mul (h : a = c * b) : a * b⁻¹ = c | by simp [h] | lemma | mul_inv_eq_of_eq_mul | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_mul_of_mul_inv_eq (h : a * c⁻¹ = b) : a = b * c | by simp [h.symm] | lemma | eq_mul_of_mul_inv_eq | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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