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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|---|---|---|---|---|---|---|---|---|---|---|
inv_inv : commute a b → commute a⁻¹ b⁻¹ | semiconj_by.inv_inv_symm | lemma | commute.inv_inv | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"inv_inv",
"semiconj_by.inv_inv_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_inv_iff : commute a⁻¹ b⁻¹ ↔ commute a b | semiconj_by.inv_inv_symm_iff | lemma | commute.inv_inv_iff | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"semiconj_by.inv_inv_symm_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv (hab : commute a b) : (a * b)⁻¹ = a⁻¹ * b⁻¹ | by rw [hab.eq, mul_inv_rev] | lemma | commute.mul_inv | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"mul_inv",
"mul_inv_rev"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv (hab : commute a b) : (a * b)⁻¹ = a⁻¹ * b⁻¹ | by rw [hab.eq, mul_inv_rev] | lemma | commute.inv | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"mul_inv_rev"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_mul_div_comm (hbd : commute b d) (hbc : commute b⁻¹ c) :
a / b * (c / d) = a * c / (b * d) | by simp_rw [div_eq_mul_inv, mul_inv_rev, hbd.inv_inv.symm.eq, hbc.mul_mul_mul_comm] | lemma | commute.div_mul_div_comm | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"div_eq_mul_inv",
"div_mul_div_comm",
"mul_inv_rev"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_mul_comm (hcd : commute c d) (hbc : commute b c⁻¹) :
a * b / (c * d) = a / c * (b / d) | (hcd.div_mul_div_comm hbc.symm).symm | lemma | commute.mul_div_mul_comm | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"mul_div_mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_div_div_comm (hbc : commute b c) (hbd : commute b⁻¹ d)
(hcd : commute c⁻¹ d) : a / b / (c / d) = a / c / (b / d) | by simp_rw [div_eq_mul_inv, mul_inv_rev, inv_inv, hbd.symm.eq, hcd.symm.eq,
hbc.inv_inv.mul_mul_mul_comm] | lemma | commute.div_div_div_comm | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"div_div_div_comm",
"div_eq_mul_inv",
"inv_inv",
"mul_inv_rev"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_right : commute a b → commute a b⁻¹ | semiconj_by.inv_right | theorem | commute.inv_right | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"semiconj_by.inv_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_right_iff : commute a b⁻¹ ↔ commute a b | semiconj_by.inv_right_iff | theorem | commute.inv_right_iff | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"semiconj_by.inv_right_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_left : commute a b → commute a⁻¹ b | semiconj_by.inv_symm_left | theorem | commute.inv_left | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"semiconj_by.inv_symm_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_left_iff : commute a⁻¹ b ↔ commute a b | semiconj_by.inv_symm_left_iff | theorem | commute.inv_left_iff | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"semiconj_by.inv_symm_left_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_cancel (h : commute a b) : a⁻¹ * b * a = b | by rw [h.inv_left.eq, inv_mul_cancel_right] | theorem | commute.inv_mul_cancel | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"inv_mul_cancel",
"inv_mul_cancel_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_cancel_assoc (h : commute a b) : a⁻¹ * (b * a) = b | by rw [← mul_assoc, h.inv_mul_cancel] | theorem | commute.inv_mul_cancel_assoc | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_cancel (h : commute a b) : a * b * a⁻¹ = b | by rw [h.eq, mul_inv_cancel_right] | theorem | commute.mul_inv_cancel | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"mul_inv_cancel",
"mul_inv_cancel_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_cancel_assoc (h : commute a b) : a * (b * a⁻¹) = b | by rw [← mul_assoc, h.mul_inv_cancel] | theorem | commute.mul_inv_cancel_assoc | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute",
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_cancel_comm : a * b * a⁻¹ = b | (commute.all a b).mul_inv_cancel | lemma | mul_inv_cancel_comm | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute.all",
"mul_inv_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_cancel_comm_assoc : a * (b * a⁻¹) = b | (commute.all a b).mul_inv_cancel_assoc | lemma | mul_inv_cancel_comm_assoc | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute.all"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_cancel_comm : a⁻¹ * b * a = b | (commute.all a b).inv_mul_cancel | lemma | inv_mul_cancel_comm | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute.all",
"inv_mul_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_cancel_comm_assoc : a⁻¹ * (b * a) = b | (commute.all a b).inv_mul_cancel_assoc | lemma | inv_mul_cancel_comm_assoc | algebra.group | src/algebra/group/commute.lean | [
"algebra.group.semiconj"
] | [
"commute.all"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_conj (a b : α) | ∃ c : αˣ, semiconj_by ↑c a b | def | is_conj | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"semiconj_by"
] | We say that `a` is conjugate to `b` if for some unit `c` we have `c * a * c⁻¹ = b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_conj.refl (a : α) : is_conj a a | ⟨1, semiconj_by.one_left a⟩ | lemma | is_conj.refl | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"is_conj",
"semiconj_by.one_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_conj.symm {a b : α} : is_conj a b → is_conj b a | | ⟨c, hc⟩ := ⟨c⁻¹, hc.units_inv_symm_left⟩ | lemma | is_conj.symm | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"is_conj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_conj_comm {g h : α} : is_conj g h ↔ is_conj h g | ⟨is_conj.symm, is_conj.symm⟩ | lemma | is_conj_comm | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"is_conj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_conj.trans {a b c : α} : is_conj a b → is_conj b c → is_conj a c | | ⟨c₁, hc₁⟩ ⟨c₂, hc₂⟩ := ⟨c₂ * c₁, hc₂.mul_left hc₁⟩ | lemma | is_conj.trans | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"is_conj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_conj_iff_eq {α : Type*} [comm_monoid α] {a b : α} : is_conj a b ↔ a = b | ⟨λ ⟨c, hc⟩, begin
rw [semiconj_by, mul_comm, ← units.mul_inv_eq_iff_eq_mul, mul_assoc, c.mul_inv, mul_one] at hc,
exact hc,
end, λ h, by rw h⟩ | lemma | is_conj_iff_eq | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"comm_monoid",
"is_conj",
"mul_assoc",
"mul_comm",
"mul_one",
"semiconj_by",
"units.mul_inv_eq_iff_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.map_is_conj (f : α →* β) {a b : α} : is_conj a b → is_conj (f a) (f b) | | ⟨c, hc⟩ := ⟨units.map f c, by rw [units.coe_map, semiconj_by, ← f.map_mul, hc.eq, f.map_mul]⟩ | lemma | monoid_hom.map_is_conj | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"is_conj",
"semiconj_by",
"units.coe_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_conj_one_right {a : α} : is_conj 1 a ↔ a = 1 | ⟨λ ⟨c, hc⟩, mul_right_cancel (hc.symm.trans ((mul_one _).trans (one_mul _).symm)), λ h, by rw [h]⟩ | lemma | is_conj_one_right | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"is_conj",
"mul_one",
"mul_right_cancel",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_conj_one_left {a : α} : is_conj a 1 ↔ a = 1 | calc is_conj a 1 ↔ is_conj 1 a : ⟨is_conj.symm, is_conj.symm⟩
... ↔ a = 1 : is_conj_one_right | lemma | is_conj_one_left | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"is_conj",
"is_conj_one_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_conj_iff {a b : α} :
is_conj a b ↔ ∃ c : α, c * a * c⁻¹ = b | ⟨λ ⟨c, hc⟩, ⟨c, mul_inv_eq_iff_eq_mul.2 hc⟩, λ ⟨c, hc⟩,
⟨⟨c, c⁻¹, mul_inv_self c, inv_mul_self c⟩, mul_inv_eq_iff_eq_mul.1 hc⟩⟩ | lemma | is_conj_iff | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"inv_mul_self",
"is_conj",
"mul_inv_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_inv {a b : α} : (b * a * b⁻¹)⁻¹ = b * a⁻¹ * b⁻¹ | ((mul_aut.conj b).map_inv a).symm | lemma | conj_inv | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"map_inv",
"mul_aut.conj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_mul {a b c : α} : (b * a * b⁻¹) * (b * c * b⁻¹) = b * (a * c) * b⁻¹ | ((mul_aut.conj b).map_mul a c).symm | lemma | conj_mul | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"map_mul",
"mul_aut.conj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_pow {i : ℕ} {a b : α} : (a * b * a⁻¹) ^ i = a * (b ^ i) * a⁻¹ | begin
induction i with i hi,
{ simp },
{ simp [pow_succ, hi] }
end | lemma | conj_pow | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_zpow {i : ℤ} {a b : α} : (a * b * a⁻¹) ^ i = a * (b ^ i) * a⁻¹ | begin
induction i,
{ simp },
{ simp [zpow_neg_succ_of_nat, conj_pow] }
end | lemma | conj_zpow | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"conj_pow",
"zpow_neg_succ_of_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_injective {x : α} : function.injective (λ (g : α), x * g * x⁻¹) | (mul_aut.conj x).injective | lemma | conj_injective | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"mul_aut.conj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_conj_iff₀ [group_with_zero α] {a b : α} :
is_conj a b ↔ ∃ c : α, c ≠ 0 ∧ c * a * c⁻¹ = b | ⟨λ ⟨c, hc⟩, ⟨c, begin
rw [← units.coe_inv, units.mul_inv_eq_iff_eq_mul],
exact ⟨c.ne_zero, hc⟩,
end⟩, λ ⟨c, c0, hc⟩,
⟨units.mk0 c c0, begin
rw [semiconj_by, ← units.mul_inv_eq_iff_eq_mul, units.coe_inv, units.coe_mk0],
exact hc
end⟩⟩ | lemma | is_conj_iff₀ | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"group_with_zero",
"is_conj",
"semiconj_by",
"units.coe_inv",
"units.coe_mk0",
"units.mul_inv_eq_iff_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
setoid (α : Type*) [monoid α] : setoid α | { r := is_conj, iseqv := ⟨is_conj.refl, λa b, is_conj.symm, λa b c, is_conj.trans⟩ } | def | is_conj.setoid | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"is_conj",
"is_conj.symm",
"monoid"
] | The setoid of the relation `is_conj` iff there is a unit `u` such that `u * x = y * u` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
conj_classes (α : Type*) [monoid α] : Type* | quotient (is_conj.setoid α) | def | conj_classes | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"is_conj.setoid",
"monoid"
] | The quotient type of conjugacy classes of a group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk {α : Type*} [monoid α] (a : α) : conj_classes α | ⟦a⟧ | def | conj_classes.mk | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"conj_classes",
"monoid"
] | The canonical quotient map from a monoid `α` into the `conj_classes` of `α` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_eq_mk_iff_is_conj {a b : α} :
conj_classes.mk a = conj_classes.mk b ↔ is_conj a b | iff.intro quotient.exact quot.sound | theorem | conj_classes.mk_eq_mk_iff_is_conj | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"conj_classes.mk",
"is_conj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_mk_eq_mk (a : α) : ⟦ a ⟧ = conj_classes.mk a | rfl | theorem | conj_classes.quotient_mk_eq_mk | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"conj_classes.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_mk_eq_mk (a : α) : quot.mk setoid.r a = conj_classes.mk a | rfl | theorem | conj_classes.quot_mk_eq_mk | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"conj_classes.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forall_is_conj {p : conj_classes α → Prop} :
(∀a, p a) ↔ (∀a, p (conj_classes.mk a)) | iff.intro
(assume h a, h _)
(assume h a, quotient.induction_on a h) | theorem | conj_classes.forall_is_conj | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"conj_classes",
"conj_classes.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_surjective : function.surjective (@conj_classes.mk α _) | forall_is_conj.2 (λ a, ⟨a, rfl⟩) | theorem | conj_classes.mk_surjective | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"conj_classes.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_eq_mk_one : (1 : conj_classes α) = conj_classes.mk 1 | rfl | theorem | conj_classes.one_eq_mk_one | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"conj_classes",
"conj_classes.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_rep (a : conj_classes α) : ∃ a0 : α, conj_classes.mk a0 = a | quot.exists_rep a | lemma | conj_classes.exists_rep | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"conj_classes",
"conj_classes.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (f : α →* β) : conj_classes α → conj_classes β | quotient.lift (conj_classes.mk ∘ f) (λ a b ab, mk_eq_mk_iff_is_conj.2 (f.map_is_conj ab)) | def | conj_classes.map | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"conj_classes",
"conj_classes.mk"
] | A `monoid_hom` maps conjugacy classes of one group to conjugacy classes of another. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_surjective {f : α →* β} (hf : function.surjective f) :
function.surjective (conj_classes.map f) | begin
intros b,
obtain ⟨b, rfl⟩ := conj_classes.mk_surjective b,
obtain ⟨a, rfl⟩ := hf b,
exact ⟨conj_classes.mk a, rfl⟩,
end | lemma | conj_classes.map_surjective | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"conj_classes.map",
"conj_classes.mk_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_injective : function.injective (@conj_classes.mk α _) | λ _ _, (mk_eq_mk_iff_is_conj.trans is_conj_iff_eq).1 | lemma | conj_classes.mk_injective | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"conj_classes.mk",
"is_conj_iff_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_bijective : function.bijective (@conj_classes.mk α _) | ⟨mk_injective, mk_surjective⟩ | lemma | conj_classes.mk_bijective | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"conj_classes.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_equiv : α ≃ conj_classes α | ⟨conj_classes.mk, quotient.lift id (λ (a : α) b, is_conj_iff_eq.1), quotient.lift_mk _ _,
begin
rw [function.right_inverse, function.left_inverse, forall_is_conj],
intro x,
rw [← quotient_mk_eq_mk, ← quotient_mk_eq_mk, quotient.lift_mk, id.def],
end⟩ | def | conj_classes.mk_equiv | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"conj_classes",
"quotient.lift_mk"
] | The bijection between a `comm_group` and its `conj_classes`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
conjugates_of (a : α) : set α | {b | is_conj a b} | def | conjugates_of | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"is_conj"
] | Given an element `a`, `conjugates a` is the set of conjugates. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_conjugates_of_self {a : α} : a ∈ conjugates_of a | is_conj.refl _ | lemma | mem_conjugates_of_self | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"conjugates_of",
"is_conj.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_conj.conjugates_of_eq {a b : α} (ab : is_conj a b) :
conjugates_of a = conjugates_of b | set.ext (λ g, ⟨λ ag, (ab.symm).trans ag, λ bg, ab.trans bg⟩) | lemma | is_conj.conjugates_of_eq | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"conjugates_of",
"is_conj",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_conj_iff_conjugates_of_eq {a b : α} :
is_conj a b ↔ conjugates_of a = conjugates_of b | ⟨is_conj.conjugates_of_eq, λ h, begin
have ha := mem_conjugates_of_self,
rwa ← h at ha,
end⟩ | lemma | is_conj_iff_conjugates_of_eq | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"conjugates_of",
"is_conj",
"mem_conjugates_of_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
carrier : conj_classes α → set α | quotient.lift conjugates_of (λ (a : α) b ab, is_conj.conjugates_of_eq ab) | def | conj_classes.carrier | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"conj_classes",
"conjugates_of",
"is_conj.conjugates_of_eq"
] | Given a conjugacy class `a`, `carrier a` is the set it represents. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_carrier_mk {a : α} : a ∈ carrier (conj_classes.mk a) | is_conj.refl _ | lemma | conj_classes.mem_carrier_mk | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"conj_classes.mk",
"is_conj.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_carrier_iff_mk_eq {a : α} {b : conj_classes α} :
a ∈ carrier b ↔ conj_classes.mk a = b | begin
revert b,
rw forall_is_conj,
intro b,
rw [carrier, eq_comm, mk_eq_mk_iff_is_conj, ← quotient_mk_eq_mk, quotient.lift_mk],
refl,
end | lemma | conj_classes.mem_carrier_iff_mk_eq | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"conj_classes",
"conj_classes.mk",
"quotient.lift_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
carrier_eq_preimage_mk {a : conj_classes α} :
a.carrier = conj_classes.mk ⁻¹' {a} | set.ext (λ x, mem_carrier_iff_mk_eq) | lemma | conj_classes.carrier_eq_preimage_mk | algebra.group | src/algebra/group/conj.lean | [
"algebra.group.semiconj",
"algebra.group_with_zero.basic",
"algebra.hom.aut",
"algebra.hom.group"
] | [
"conj_classes",
"conj_classes.mk",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_vadd (G : Type*) (P : Type*) | (vadd : G → P → P) | class | has_vadd | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | Type class for the `+ᵥ` notation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_vsub (G : out_param Type*) (P : Type*) | (vsub : P → P → G) | class | has_vsub | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | Type class for the `-ᵥ` notation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_smul (M : Type*) (α : Type*) | (smul : M → α → α) | class | has_smul | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | Typeclass for types with a scalar multiplication operation, denoted `•` (`\bu`) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_mul : G → G → G | λ g : G, λ x : G, g * x | def | left_mul | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | `left_mul g` denotes left multiplication by `g` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_mul : G → G → G | λ g : G, λ x : G, x * g | def | right_mul | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | `right_mul g` denotes right multiplication by `g` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_left_cancel_mul (G : Type u) [has_mul G] : Prop | (mul_left_cancel : ∀ a b c : G, a * b = a * c → b = c) | class | is_left_cancel_mul | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"mul_left_cancel"
] | A mixin for left cancellative multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_right_cancel_mul (G : Type u) [has_mul G] : Prop | (mul_right_cancel : ∀ a b c : G, a * b = c * b → a = c) | class | is_right_cancel_mul | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"mul_right_cancel"
] | A mixin for right cancellative multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_cancel_mul (G : Type u) [has_mul G]
extends is_left_cancel_mul G, is_right_cancel_mul G : Prop | class | is_cancel_mul | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"is_left_cancel_mul",
"is_right_cancel_mul"
] | A mixin for cancellative multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_left_cancel_add (G : Type u) [has_add G] : Prop | (add_left_cancel : ∀ a b c : G, a + b = a + c → b = c) | class | is_left_cancel_add | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | A mixin for left cancellative addition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_right_cancel_add (G : Type u) [has_add G] : Prop | (add_right_cancel : ∀ a b c : G, a + b = c + b → a = c) | class | is_right_cancel_add | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | A mixin for right cancellative addition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_cancel_add (G : Type u) [has_add G]
extends is_left_cancel_add G, is_right_cancel_add G : Prop | class | is_cancel_add | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"is_left_cancel_add",
"is_right_cancel_add"
] | A mixin for cancellative addition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_cancel : a * b = a * c → b = c | is_left_cancel_mul.mul_left_cancel a b c | lemma | mul_left_cancel | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_cancel_iff : a * b = a * c ↔ b = c | ⟨mul_left_cancel, congr_arg _⟩ | lemma | mul_left_cancel_iff | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_injective (a : G) : function.injective ((*) a) | λ b c, mul_left_cancel | theorem | mul_right_injective | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"mul_left_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_inj (a : G) {b c : G} : a * b = a * c ↔ b = c | (mul_right_injective a).eq_iff | theorem | mul_right_inj | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"mul_right_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_ne_mul_right (a : G) {b c : G} : a * b ≠ a * c ↔ b ≠ c | (mul_right_injective a).ne_iff | theorem | mul_ne_mul_right | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"mul_right_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_cancel : a * b = c * b → a = c | is_right_cancel_mul.mul_right_cancel a b c | lemma | mul_right_cancel | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_cancel_iff : b * a = c * a ↔ b = c | ⟨mul_right_cancel, congr_arg _⟩ | lemma | mul_right_cancel_iff | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_injective (a : G) : function.injective (λ x, x * a) | λ b c, mul_right_cancel | theorem | mul_left_injective | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"mul_right_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_inj (a : G) {b c : G} : b * a = c * a ↔ b = c | (mul_left_injective a).eq_iff | theorem | mul_left_inj | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"mul_left_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_ne_mul_left (a : G) {b c : G} : b * a ≠ c * a ↔ b ≠ c | (mul_left_injective a).ne_iff | theorem | mul_ne_mul_left | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"mul_left_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
semigroup (G : Type u) extends has_mul G | (mul_assoc : ∀ a b c : G, a * b * c = a * (b * c)) | class | semigroup | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"mul_assoc"
] | A semigroup is a type with an associative `(*)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_semigroup (G : Type u) extends has_add G | (add_assoc : ∀ a b c : G, a + b + c = a + (b + c)) | class | add_semigroup | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | An additive semigroup is a type with an associative `(+)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_assoc : ∀ a b c : G, a * b * c = a * (b * c) | semigroup.mul_assoc | lemma | mul_assoc | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
semigroup.to_is_associative : is_associative G (*) | ⟨mul_assoc⟩ | instance | semigroup.to_is_associative | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_semigroup (G : Type u) extends semigroup G | (mul_comm : ∀ a b : G, a * b = b * a) | class | comm_semigroup | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"mul_comm",
"semigroup"
] | A commutative semigroup is a type with an associative commutative `(*)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_comm_semigroup (G : Type u) extends add_semigroup G | (add_comm : ∀ a b : G, a + b = b + a) | class | add_comm_semigroup | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"add_semigroup"
] | A commutative additive semigroup is a type with an associative commutative `(+)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_comm : ∀ a b : G, a * b = b * a | comm_semigroup.mul_comm | lemma | mul_comm | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_semigroup.to_is_commutative : is_commutative G (*) | ⟨mul_comm⟩ | instance | comm_semigroup.to_is_commutative | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_semigroup.is_right_cancel_mul.to_is_left_cancel_mul (G : Type u) [comm_semigroup G]
[is_right_cancel_mul G] : is_left_cancel_mul G | ⟨λ a b c h, mul_right_cancel $ (mul_comm _ _).trans (h.trans $ mul_comm _ _)⟩ | lemma | comm_semigroup.is_right_cancel_mul.to_is_left_cancel_mul | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"comm_semigroup",
"is_left_cancel_mul",
"is_right_cancel_mul",
"mul_comm",
"mul_right_cancel"
] | Any `comm_semigroup G` that satisfies `is_right_cancel_mul G` also satisfies
`is_left_cancel_mul G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comm_semigroup.is_left_cancel_mul.to_is_right_cancel_mul (G : Type u) [comm_semigroup G]
[is_left_cancel_mul G] : is_right_cancel_mul G | ⟨λ a b c h, mul_left_cancel $ (mul_comm _ _).trans (h.trans $ mul_comm _ _)⟩ | lemma | comm_semigroup.is_left_cancel_mul.to_is_right_cancel_mul | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"comm_semigroup",
"is_left_cancel_mul",
"is_right_cancel_mul",
"mul_comm",
"mul_left_cancel"
] | Any `comm_semigroup G` that satisfies `is_left_cancel_mul G` also satisfies
`is_right_cancel_mul G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comm_semigroup.is_left_cancel_mul.to_is_cancel_mul (G : Type u) [comm_semigroup G]
[is_left_cancel_mul G] : is_cancel_mul G | { .. ‹is_left_cancel_mul G›, .. comm_semigroup.is_left_cancel_mul.to_is_right_cancel_mul G } | lemma | comm_semigroup.is_left_cancel_mul.to_is_cancel_mul | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"comm_semigroup",
"comm_semigroup.is_left_cancel_mul.to_is_right_cancel_mul",
"is_cancel_mul",
"is_left_cancel_mul"
] | Any `comm_semigroup G` that satisfies `is_left_cancel_mul G` also satisfies
`is_cancel_mul G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comm_semigroup.is_right_cancel_mul.to_is_cancel_mul (G : Type u) [comm_semigroup G]
[is_right_cancel_mul G] : is_cancel_mul G | { .. ‹is_right_cancel_mul G›, .. comm_semigroup.is_right_cancel_mul.to_is_left_cancel_mul G } | lemma | comm_semigroup.is_right_cancel_mul.to_is_cancel_mul | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"comm_semigroup",
"comm_semigroup.is_right_cancel_mul.to_is_left_cancel_mul",
"is_cancel_mul",
"is_right_cancel_mul"
] | Any `comm_semigroup G` that satisfies `is_right_cancel_mul G` also satisfies
`is_cancel_mul G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_cancel_semigroup (G : Type u) extends semigroup G | (mul_left_cancel : ∀ a b c : G, a * b = a * c → b = c) | class | left_cancel_semigroup | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"mul_left_cancel",
"semigroup"
] | A `left_cancel_semigroup` is a semigroup such that `a * b = a * c` implies `b = c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_left_cancel_semigroup (G : Type u) extends add_semigroup G | (add_left_cancel : ∀ a b c : G, a + b = a + c → b = c) | class | add_left_cancel_semigroup | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"add_semigroup"
] | An `add_left_cancel_semigroup` is an additive semigroup such that
`a + b = a + c` implies `b = c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_cancel_semigroup.to_is_left_cancel_mul (G : Type u) [left_cancel_semigroup G] :
is_left_cancel_mul G | { mul_left_cancel := left_cancel_semigroup.mul_left_cancel } | instance | left_cancel_semigroup.to_is_left_cancel_mul | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"is_left_cancel_mul",
"left_cancel_semigroup",
"mul_left_cancel"
] | Any `left_cancel_semigroup` satisfies `is_left_cancel_mul`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_cancel_semigroup (G : Type u) extends semigroup G | (mul_right_cancel : ∀ a b c : G, a * b = c * b → a = c) | class | right_cancel_semigroup | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"mul_right_cancel",
"semigroup"
] | A `right_cancel_semigroup` is a semigroup such that `a * b = c * b` implies `a = c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_right_cancel_semigroup (G : Type u) extends add_semigroup G | (add_right_cancel : ∀ a b c : G, a + b = c + b → a = c) | class | add_right_cancel_semigroup | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"add_semigroup"
] | An `add_right_cancel_semigroup` is an additive semigroup such that
`a + b = c + b` implies `a = c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_cancel_semigroup.to_is_right_cancel_mul (G : Type u) [right_cancel_semigroup G] :
is_right_cancel_mul G | { mul_right_cancel := right_cancel_semigroup.mul_right_cancel } | instance | right_cancel_semigroup.to_is_right_cancel_mul | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"is_right_cancel_mul",
"mul_right_cancel",
"right_cancel_semigroup"
] | Any `right_cancel_semigroup` satisfies `is_right_cancel_mul`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_one_class (M : Type u) extends has_one M, has_mul M | (one_mul : ∀ (a : M), 1 * a = a)
(mul_one : ∀ (a : M), a * 1 = a) | class | mul_one_class | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"mul_one",
"one_mul"
] | Typeclass for expressing that a type `M` with multiplication and a one satisfies
`1 * a = a` and `a * 1 = a` for all `a : M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_zero_class (M : Type u) extends has_zero M, has_add M | (zero_add : ∀ (a : M), 0 + a = a)
(add_zero : ∀ (a : M), a + 0 = a) | class | add_zero_class | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | Typeclass for expressing that a type `M` with addition and a zero satisfies
`0 + a = a` and `a + 0 = a` for all `a : M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_one_class.ext {M : Type u} : ∀ ⦃m₁ m₂ : mul_one_class M⦄, m₁.mul = m₂.mul → m₁ = m₂ | begin
rintros ⟨one₁, mul₁, one_mul₁, mul_one₁⟩ ⟨one₂, mul₂, one_mul₂, mul_one₂⟩ (rfl : mul₁ = mul₂),
congr,
exact (one_mul₂ one₁).symm.trans (mul_one₁ one₂),
end | lemma | mul_one_class.ext | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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