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
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|---|---|---|---|---|---|---|---|---|---|---|
eq_mul_of_inv_mul_eq (h : b⁻¹ * a = c) : a = b * c | by simp [h.symm, mul_inv_cancel_left] | lemma | eq_mul_of_inv_mul_eq | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"mul_inv_cancel_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_of_eq_inv_mul (h : b = a⁻¹ * c) : a * b = c | by rw [h, mul_inv_cancel_left] | lemma | mul_eq_of_eq_inv_mul | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"mul_inv_cancel_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_of_eq_mul_inv (h : a = c * b⁻¹) : a * b = c | by simp [h] | lemma | mul_eq_of_eq_mul_inv | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_one_iff_eq_inv : a * b = 1 ↔ a = b⁻¹ | ⟨eq_inv_of_mul_eq_one_left, λ h, by rw [h, mul_left_inv]⟩ | theorem | mul_eq_one_iff_eq_inv | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"mul_left_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_one_iff_inv_eq : a * b = 1 ↔ a⁻¹ = b | by rw [mul_eq_one_iff_eq_inv, inv_eq_iff_eq_inv] | theorem | mul_eq_one_iff_inv_eq | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"inv_eq_iff_eq_inv",
"mul_eq_one_iff_eq_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_inv_iff_mul_eq_one : a = b⁻¹ ↔ a * b = 1 | mul_eq_one_iff_eq_inv.symm | theorem | eq_inv_iff_mul_eq_one | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_eq_iff_mul_eq_one : a⁻¹ = b ↔ a * b = 1 | mul_eq_one_iff_inv_eq.symm | theorem | inv_eq_iff_mul_eq_one | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_mul_inv_iff_mul_eq : a = b * c⁻¹ ↔ a * c = b | ⟨λ h, by rw [h, inv_mul_cancel_right], λ h, by rw [← h, mul_inv_cancel_right]⟩ | theorem | eq_mul_inv_iff_mul_eq | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"inv_mul_cancel_right",
"mul_inv_cancel_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_inv_mul_iff_mul_eq : a = b⁻¹ * c ↔ b * a = c | ⟨λ h, by rw [h, mul_inv_cancel_left], λ h, by rw [← h, inv_mul_cancel_left]⟩ | theorem | eq_inv_mul_iff_mul_eq | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"inv_mul_cancel_left",
"mul_inv_cancel_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_eq_iff_eq_mul : a⁻¹ * b = c ↔ b = a * c | ⟨λ h, by rw [← h, mul_inv_cancel_left], λ h, by rw [h, inv_mul_cancel_left]⟩ | theorem | inv_mul_eq_iff_eq_mul | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"inv_mul_cancel_left",
"mul_inv_cancel_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_eq_iff_eq_mul : a * b⁻¹ = c ↔ a = c * b | ⟨λ h, by rw [← h, inv_mul_cancel_right], λ h, by rw [h, mul_inv_cancel_right]⟩ | theorem | mul_inv_eq_iff_eq_mul | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"inv_mul_cancel_right",
"mul_inv_cancel_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_eq_one : a * b⁻¹ = 1 ↔ a = b | by rw [mul_eq_one_iff_eq_inv, inv_inv] | theorem | mul_inv_eq_one | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"inv_inv",
"mul_eq_one_iff_eq_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_eq_one : a⁻¹ * b = 1 ↔ a = b | by rw [mul_eq_one_iff_eq_inv, inv_inj] | theorem | inv_mul_eq_one | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"inv_inj",
"mul_eq_one_iff_eq_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_left_injective : function.injective (λ a, a / b) | by simpa only [div_eq_mul_inv] using λ a a' h, mul_left_injective (b⁻¹) h | lemma | div_left_injective | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_eq_mul_inv",
"mul_left_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_right_injective : function.injective (λ a, b / a) | by simpa only [div_eq_mul_inv] using λ a a' h, inv_injective (mul_right_injective b h) | lemma | div_right_injective | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_eq_mul_inv",
"inv_injective",
"mul_right_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_mul_cancel' (a b : G) : a / b * b = a | by rw [div_eq_mul_inv, inv_mul_cancel_right a b] | lemma | div_mul_cancel' | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_eq_mul_inv",
"inv_mul_cancel_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_self' (a : G) : a / a = 1 | by rw [div_eq_mul_inv, mul_right_inv a] | lemma | div_self' | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_eq_mul_inv",
"mul_right_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_cancel'' (a b : G) : a * b / b = a | by rw [div_eq_mul_inv, mul_inv_cancel_right a b] | lemma | mul_div_cancel'' | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_eq_mul_inv",
"mul_inv_cancel_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_mul_cancel''' (a b : G) : a / (b * a) = b⁻¹ | by rw [←inv_div, mul_div_cancel''] | lemma | div_mul_cancel''' | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"mul_div_cancel''"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_mul_right_eq_div (a b c : G) : (a * c) / (b * c) = a / b | by rw [div_mul_eq_div_div_swap]; simp only [mul_left_inj, eq_self_iff_true, mul_div_cancel''] | lemma | mul_div_mul_right_eq_div | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_mul_eq_div_div_swap",
"mul_div_cancel''",
"mul_left_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_div_of_mul_eq' (h : a * c = b) : a = b / c | by simp [← h] | lemma | eq_div_of_mul_eq' | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_eq_of_eq_mul'' (h : a = c * b) : a / b = c | by simp [h] | lemma | div_eq_of_eq_mul'' | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_mul_of_div_eq (h : a / c = b) : a = b * c | by simp [← h] | lemma | eq_mul_of_div_eq | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_of_eq_div (h : a = c / b) : a * b = c | by simp [h] | lemma | mul_eq_of_eq_div | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_right_inj : a / b = a / c ↔ b = c | div_right_injective.eq_iff | lemma | div_right_inj | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_left_inj : b / a = c / a ↔ b = c | by { rw [div_eq_mul_inv, div_eq_mul_inv], exact mul_left_inj _ } | lemma | div_left_inj | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_eq_mul_inv",
"mul_left_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_mul_div_cancel' (a b c : G) : (a / b) * (b / c) = a / c | by rw [← mul_div_assoc, div_mul_cancel'] | lemma | div_mul_div_cancel' | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_mul_cancel'",
"mul_div_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_div_div_cancel_right' (a b c : G) : (a / c) / (b / c) = a / b | by rw [← inv_div c b, div_inv_eq_mul, div_mul_div_cancel'] | lemma | div_div_div_cancel_right' | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_inv_eq_mul",
"div_mul_div_cancel'",
"inv_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_eq_one : a / b = 1 ↔ a = b | ⟨eq_of_div_eq_one, λ h, by rw [h, div_self']⟩ | theorem | div_eq_one | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_self'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_ne_one : a / b ≠ 1 ↔ a ≠ b | not_congr div_eq_one | theorem | div_ne_one | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_eq_self : a / b = a ↔ b = 1 | by rw [div_eq_mul_inv, mul_right_eq_self, inv_eq_one] | theorem | div_eq_self | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_eq_mul_inv",
"inv_eq_one",
"mul_right_eq_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_div_iff_mul_eq' : a = b / c ↔ a * c = b | by rw [div_eq_mul_inv, eq_mul_inv_iff_mul_eq] | theorem | eq_div_iff_mul_eq' | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_eq_mul_inv",
"eq_mul_inv_iff_mul_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_eq_iff_eq_mul : a / b = c ↔ a = c * b | by rw [div_eq_mul_inv, mul_inv_eq_iff_eq_mul] | theorem | div_eq_iff_eq_mul | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_eq_mul_inv",
"mul_inv_eq_iff_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_iff_eq_of_div_eq_div (H : a / b = c / d) : a = b ↔ c = d | by rw [← div_eq_one, H, div_eq_one] | theorem | eq_iff_eq_of_div_eq_div | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_inverse_div_mul_left (c : G) : function.left_inverse (λ x, x / c) (λ x, x * c) | assume x, mul_div_cancel'' x c | theorem | left_inverse_div_mul_left | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"mul_div_cancel''"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_inverse_mul_left_div (c : G) : function.left_inverse (λ x, x * c) (λ x, x / c) | assume x, div_mul_cancel' x c | theorem | left_inverse_mul_left_div | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_mul_cancel'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_inverse_mul_right_inv_mul (c : G) :
function.left_inverse (λ x, c * x) (λ x, c⁻¹ * x) | assume x, mul_inv_cancel_left c x | theorem | left_inverse_mul_right_inv_mul | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"mul_inv_cancel_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_inverse_inv_mul_mul_right (c : G) :
function.left_inverse (λ x, c⁻¹ * x) (λ x, c * x) | assume x, inv_mul_cancel_left c x | theorem | left_inverse_inv_mul_mul_right | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"inv_mul_cancel_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_npow_eq_one_of_zpow_eq_one {n : ℤ} (hn : n ≠ 0) {x : G} (h : x ^ n = 1) :
∃ n : ℕ, 0 < n ∧ x ^ n = 1 | begin
cases n with n n,
{ rw zpow_of_nat at h,
refine ⟨n, nat.pos_of_ne_zero (λ n0, hn _), h⟩, rw n0, refl },
{ rw [zpow_neg_succ_of_nat, inv_eq_one] at h,
refine ⟨n + 1, n.succ_pos, h⟩ }
end | lemma | exists_npow_eq_one_of_zpow_eq_one | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"inv_eq_one",
"zpow_neg_succ_of_nat",
"zpow_of_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_eq_of_eq_mul' {a b c : G} (h : a = b * c) : a / b = c | by rw [h, div_eq_mul_inv, mul_comm, inv_mul_cancel_left] | lemma | div_eq_of_eq_mul' | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_eq_mul_inv",
"inv_mul_cancel_left",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_mul_left_eq_div (a b c : G) : (c * a) / (c * b) = a / b | by simp | lemma | mul_div_mul_left_eq_div | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_div_of_mul_eq'' (h : c * a = b) : a = b / c | by simp [h.symm] | lemma | eq_div_of_mul_eq'' | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_mul_of_div_eq' (h : a / b = c) : a = b * c | by simp [h.symm] | lemma | eq_mul_of_div_eq' | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_of_eq_div' (h : b = c / a) : a * b = c | begin simp [h], rw [mul_comm c, mul_inv_cancel_left] end | lemma | mul_eq_of_eq_div' | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"mul_comm",
"mul_inv_cancel_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_div_self' (a b : G) : a / (a / b) = b | by simpa using mul_inv_cancel_left a b | lemma | div_div_self' | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"mul_inv_cancel_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_eq_div_mul_div (a b c : G) : a / b = c / b * (a / c) | by simp [mul_left_comm c] | lemma | div_eq_div_mul_div | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"mul_left_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_div_cancel (a b : G) : a / (a / b) = b | div_div_self' a b | lemma | div_div_cancel | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_div_self'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_div_cancel_left (a b : G) : a / b / a = b⁻¹ | by simp | lemma | div_div_cancel_left | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_div_iff_mul_eq'' : a = b / c ↔ c * a = b | by rw [eq_div_iff_mul_eq', mul_comm] | lemma | eq_div_iff_mul_eq'' | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"eq_div_iff_mul_eq'",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_eq_iff_eq_mul' : a / b = c ↔ a = b * c | by rw [div_eq_iff_eq_mul, mul_comm] | lemma | div_eq_iff_eq_mul' | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_eq_iff_eq_mul",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_cancel''' (a b : G) : a * b / a = b | by rw [div_eq_inv_mul, inv_mul_cancel_left] | lemma | mul_div_cancel''' | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_eq_inv_mul",
"inv_mul_cancel_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_cancel'_right (a b : G) : a * (b / a) = b | by rw [← mul_div_assoc, mul_div_cancel'''] | lemma | mul_div_cancel'_right | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"mul_div_assoc",
"mul_div_cancel'''"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_mul_cancel'' (a b : G) : a / (a * b) = b⁻¹ | by rw [← inv_div, mul_div_cancel'''] | lemma | div_mul_cancel'' | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"inv_div",
"mul_div_cancel'''"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mul_inv_cancel'_right (a b : G) : a * (b * a⁻¹) = b | by rw [← div_eq_mul_inv, mul_div_cancel'_right a b] | lemma | mul_mul_inv_cancel'_right | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_eq_mul_inv",
"mul_div_cancel'_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mul_div_cancel (a b c : G) : (a * c) * (b / c) = a * b | by rw [mul_assoc, mul_div_cancel'_right] | lemma | mul_mul_div_cancel | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"mul_assoc",
"mul_div_cancel'_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_mul_mul_cancel (a b c : G) : (a / c) * (b * c) = a * b | by rw [mul_left_comm, div_mul_cancel', mul_comm] | lemma | div_mul_mul_cancel | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_mul_cancel'",
"mul_comm",
"mul_left_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_mul_div_cancel'' (a b c : G) : (a / b) * (c / a) = c / b | by rw mul_comm; apply div_mul_div_cancel' | lemma | div_mul_div_cancel'' | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_mul_div_cancel'",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_div_cancel (a b c : G) : (a * b) / (a / c) = b * c | by rw [← div_mul, mul_div_cancel'''] | lemma | mul_div_div_cancel | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_mul",
"mul_div_cancel'''"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_div_div_cancel_left (a b c : G) : (c / a) / (c / b) = b / a | by rw [← inv_div b c, div_inv_eq_mul, mul_comm, div_mul_div_cancel'] | lemma | div_div_div_cancel_left | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_inv_eq_mul",
"div_mul_div_cancel'",
"inv_div",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_eq_div_iff_mul_eq_mul : a / b = c / d ↔ a * d = c * b | begin
rw [div_eq_iff_eq_mul, div_mul_eq_mul_div, eq_comm, div_eq_iff_eq_mul'],
simp only [mul_comm, eq_comm]
end | lemma | div_eq_div_iff_mul_eq_mul | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_eq_iff_eq_mul",
"div_eq_iff_eq_mul'",
"div_mul_eq_mul_div",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_eq_div_iff_div_eq_div : a / b = c / d ↔ a / c = b / d | by rw [div_eq_iff_eq_mul, div_mul_eq_mul_div, div_eq_iff_eq_mul', mul_div_assoc] | lemma | div_eq_div_iff_div_eq_div | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"div_eq_iff_eq_mul",
"div_eq_iff_eq_mul'",
"div_mul_eq_mul_div",
"mul_div_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_sub (a b : M) : bit0 (a - b) = bit0 a - bit0 b | sub_add_sub_comm _ _ _ _ | lemma | bit0_sub | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit1_sub [has_one M] (a b : M) : bit1 (a - b) = bit1 a - bit0 b | (congr_arg (+ (1 : M)) $ bit0_sub a b : _).trans $ sub_add_eq_add_sub _ _ _ | lemma | bit1_sub | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"bit0_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiplicative_of_symmetric_of_is_total
(hsymm : symmetric p) (hf_swap : ∀ {a b}, p a b → f a b * f b a = 1)
(hmul : ∀ {a b c}, r a b → r b c → p a b → p b c → p a c → f a c = f a b * f b c)
{a b c : α} (pab : p a b) (pbc : p b c) (pac : p a c) : f a c = f a b * f b c | begin
suffices : ∀ {b c}, r b c → p a b → p b c → p a c → f a c = f a b * f b c,
{ obtain rbc | rcb := total_of r b c,
{ exact this rbc pab pbc pac },
{ rw [this rcb pac (hsymm pbc) pab, mul_assoc, hf_swap (hsymm pbc), mul_one] } },
intros b c rbc pab pbc pac,
obtain rab | rba := total_of r a b,
{ exa... | lemma | multiplicative_of_symmetric_of_is_total | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"mul_assoc",
"mul_one",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiplicative_of_is_total (p : α → Prop)
(hswap : ∀ {a b}, p a → p b → f a b * f b a = 1)
(hmul : ∀ {a b c}, r a b → r b c → p a → p b → p c → f a c = f a b * f b c)
{a b c : α} (pa : p a) (pb : p b) (pc : p c) : f a c = f a b * f b c | begin
apply multiplicative_of_symmetric_of_is_total (λ a b, p a ∧ p b) r f (λ _ _, and.swap),
{ simp_rw and_imp, exact @hswap },
{ exact λ a b c rab rbc pab pbc pac, hmul rab rbc pab.1 pab.2 pac.2 },
exacts [⟨pa, pb⟩, ⟨pb, pc⟩, ⟨pa, pc⟩],
end | lemma | multiplicative_of_is_total | algebra.group | src/algebra/group/basic.lean | [
"algebra.group.defs"
] | [
"and_imp",
"multiplicative_of_symmetric_of_is_total"
] | If a binary function from a type equipped with a total relation `r` to a monoid is
anti-symmetric (i.e. satisfies `f a b * f b a = 1`), in order to show it is multiplicative
(i.e. satisfies `f a c = f a b * f b c`), we may assume `r a b` and `r b c` are satisfied.
We allow restricting to a subset specified by a p... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
commutator_element {G : Type*} [group G] : has_bracket G G | ⟨λ g₁ g₂, g₁ * g₂ * g₁⁻¹ * g₂⁻¹⟩ | instance | commutator_element | algebra.group | src/algebra/group/commutator.lean | [
"algebra.group.defs",
"data.bracket"
] | [
"group",
"has_bracket"
] | The commutator of two elements `g₁` and `g₂`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
commutator_element_def {G : Type*} [group G] (g₁ g₂ : G) :
⁅g₁, g₂⁆ = g₁ * g₂ * g₁⁻¹ * g₂⁻¹ | rfl | lemma | commutator_element_def | algebra.group | src/algebra/group/commutator.lean | [
"algebra.group.defs",
"data.bracket"
] | [
"group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute {S : Type*} [has_mul S] (a b : S) : Prop | semiconj_by a b b | def | commute | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"semiconj_by"
] | Two elements commute if `a * b = b * a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq {a b : S} (h : commute a b) : a * b = b * a | h | lemma | commute.eq | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute"
] | Equality behind `commute a b`; useful for rewriting. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl (a : S) : commute a a | eq.refl (a * a) | lemma | commute.refl | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute"
] | Any element commutes with itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm {a b : S} (h : commute a b) : commute b a | eq.symm h | lemma | commute.symm | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute"
] | If `a` commutes with `b`, then `b` commutes with `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
semiconj_by {a b : S} (h : commute a b) : semiconj_by a b b | h | theorem | commute.semiconj_by | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_iff {a b : S} : commute a b ↔ commute b a | ⟨commute.symm, commute.symm⟩ | theorem | commute.symm_iff | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
on_is_refl {f : G → S} : is_refl G (λ a b, commute (f a) (f b)) | ⟨λ _, commute.refl _⟩ | instance | commute.on_is_refl | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"commute.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right (hab : commute a b) (hac : commute a c) : commute a (b * c) | hab.mul_right hac | lemma | commute.mul_right | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute"
] | If `a` commutes with both `b` and `c`, then it commutes with their product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_left (hac : commute a c) (hbc : commute b c) : commute (a * b) c | hac.mul_left hbc | lemma | commute.mul_left | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute"
] | If both `a` and `b` commute with `c`, then their product commutes with `c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_comm (h : commute b c) (a : S) :
a * b * c = a * c * b | by simp only [mul_assoc, h.eq] | lemma | commute.right_comm | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_comm (h : commute a b) (c) :
a * (b * c) = b * (a * c) | by simp only [← mul_assoc, h.eq] | lemma | commute.left_comm | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mul_mul_comm (hbc : commute b c) (a d : S) :
(a * b) * (c * d) = (a * c) * (b * d) | by simp only [hbc.left_comm, mul_assoc] | lemma | commute.mul_mul_mul_comm | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"mul_assoc",
"mul_mul_mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
all {S : Type*} [comm_semigroup S] (a b : S) : commute a b | mul_comm a b | theorem | commute.all | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"comm_semigroup",
"commute",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_right (a : M) : commute a 1 | semiconj_by.one_right a | theorem | commute.one_right | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"semiconj_by.one_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_left (a : M) : commute 1 a | semiconj_by.one_left a | theorem | commute.one_left | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"semiconj_by.one_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_right (h : commute a b) (n : ℕ) : commute a (b ^ n) | h.pow_right n | theorem | commute.pow_right | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_left (h : commute a b) (n : ℕ) : commute (a ^ n) b | (h.symm.pow_right n).symm | theorem | commute.pow_left | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_pow (h : commute a b) (m n : ℕ) : commute (a ^ m) (b ^ n) | (h.pow_left m).pow_right n | theorem | commute.pow_pow | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_pow (a : M) (n : ℕ) : commute a (a ^ n) | (commute.refl a).pow_right n | theorem | commute.self_pow | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"commute.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_self (a : M) (n : ℕ) : commute (a ^ n) a | (commute.refl a).pow_left n | theorem | commute.pow_self | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"commute.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_pow_self (a : M) (m n : ℕ) : commute (a ^ m) (a ^ n) | (commute.refl a).pow_pow m n | theorem | commute.pow_pow_self | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"commute.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.pow_succ' (a : M) (n : ℕ) : a ^ (n + 1) = a ^ n * a | (pow_succ a n).trans (self_pow _ _) | theorem | pow_succ' | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units_inv_right : commute a u → commute a ↑u⁻¹ | semiconj_by.units_inv_right | theorem | commute.units_inv_right | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"semiconj_by.units_inv_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units_inv_right_iff :
commute a ↑u⁻¹ ↔ commute a u | semiconj_by.units_inv_right_iff | theorem | commute.units_inv_right_iff | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"semiconj_by.units_inv_right_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units_inv_left : commute ↑u a → commute ↑u⁻¹ a | semiconj_by.units_inv_symm_left | theorem | commute.units_inv_left | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"semiconj_by.units_inv_symm_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units_inv_left_iff: commute ↑u⁻¹ a ↔ commute ↑u a | semiconj_by.units_inv_symm_left_iff | theorem | commute.units_inv_left_iff | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"semiconj_by.units_inv_symm_left_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units_coe : commute u₁ u₂ → commute (u₁ : M) u₂ | semiconj_by.units_coe | theorem | commute.units_coe | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"semiconj_by.units_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units_of_coe : commute (u₁ : M) u₂ → commute u₁ u₂ | semiconj_by.units_of_coe | theorem | commute.units_of_coe | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"semiconj_by.units_of_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units_coe_iff : commute (u₁ : M) u₂ ↔ commute u₁ u₂ | semiconj_by.units_coe_iff | theorem | commute.units_coe_iff | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"semiconj_by.units_coe_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.units.left_of_mul (u : Mˣ) (a b : M) (hu : a * b = u) (hc : commute a b) : Mˣ | { val := a,
inv := b * ↑u⁻¹,
val_inv := by rw [← mul_assoc, hu, u.mul_inv],
inv_val := have commute a u, from hu ▸ (commute.refl _).mul_right hc,
by rw [← this.units_inv_right.right_comm, ← hc.eq, hu, u.mul_inv] } | def | units.left_of_mul | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"commute.refl",
"mul_assoc"
] | If the product of two commuting elements is a unit, then the left multiplier is a unit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.units.right_of_mul (u : Mˣ) (a b : M) (hu : a * b = u) (hc : commute a b) : Mˣ | u.left_of_mul b a (hc.eq ▸ hu) hc.symm | def | units.right_of_mul | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute"
] | If the product of two commuting elements is a unit, then the right multiplier is a unit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unit_mul_iff (h : commute a b) :
is_unit (a * b) ↔ is_unit a ∧ is_unit b | ⟨λ ⟨u, hu⟩, ⟨(u.left_of_mul a b hu.symm h).is_unit, (u.right_of_mul a b hu.symm h).is_unit⟩,
λ H, H.1.mul H.2⟩ | lemma | commute.is_unit_mul_iff | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_unit_mul_self_iff :
is_unit (a * a) ↔ is_unit a | (commute.refl a).is_unit_mul_iff.trans (and_self _) | lemma | is_unit_mul_self_iff | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute.refl",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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