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_eq_of_mul_eq_one_left {a : α} (h : a * u = 1) : ↑u⁻¹ = a | calc ↑u⁻¹ = 1 * ↑u⁻¹ : by rw one_mul
... = a : by rw [←h, mul_inv_cancel_right] | lemma | units.inv_eq_of_mul_eq_one_left | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"inv_eq_of_mul_eq_one_left",
"mul_inv_cancel_right",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_eq_of_mul_eq_one_right {a : α} (h : ↑u * a = 1) : ↑u⁻¹ = a | calc ↑u⁻¹ = ↑u⁻¹ * 1 : by rw mul_one
... = a : by rw [←h, inv_mul_cancel_left] | lemma | units.inv_eq_of_mul_eq_one_right | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"inv_eq_of_mul_eq_one_right",
"inv_mul_cancel_left",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_inv_of_mul_eq_one_left {a : α} (h : ↑u * a = 1) : a = ↑u⁻¹ | (units.inv_eq_of_mul_eq_one_right h).symm | lemma | units.eq_inv_of_mul_eq_one_left | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"eq_inv_of_mul_eq_one_left",
"units.inv_eq_of_mul_eq_one_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_inv_of_mul_eq_one_right {a : α} (h : a * u = 1) : a = ↑u⁻¹ | (units.inv_eq_of_mul_eq_one_left h).symm | lemma | units.eq_inv_of_mul_eq_one_right | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"eq_inv_of_mul_eq_one_right",
"units.inv_eq_of_mul_eq_one_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_eq_one {a : α} : a * ↑u⁻¹ = 1 ↔ a = u | ⟨inv_inv u ▸ units.eq_inv_of_mul_eq_one_right, λ h, mul_inv_of_eq h.symm⟩ | lemma | units.mul_inv_eq_one | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"mul_inv_eq_one",
"units.eq_inv_of_mul_eq_one_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_eq_one {a : α} : ↑u⁻¹ * a = 1 ↔ ↑u = a | ⟨inv_inv u ▸ units.inv_eq_of_mul_eq_one_right, inv_mul_of_eq⟩ | lemma | units.inv_mul_eq_one | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"inv_mul_eq_one",
"units.inv_eq_of_mul_eq_one_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_one_iff_eq_inv {a : α} : a * u = 1 ↔ a = ↑u⁻¹ | by rw [←mul_inv_eq_one, inv_inv] | lemma | units.mul_eq_one_iff_eq_inv | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"inv_inv",
"mul_eq_one_iff_eq_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_one_iff_inv_eq {a : α} : ↑u * a = 1 ↔ ↑u⁻¹ = a | by rw [←inv_mul_eq_one, inv_inv] | lemma | units.mul_eq_one_iff_inv_eq | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"inv_inv",
"mul_eq_one_iff_inv_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_unique {u₁ u₂ : αˣ} (h : (↑u₁ : α) = ↑u₂) : (↑u₁⁻¹ : α) = ↑u₂⁻¹ | units.inv_eq_of_mul_eq_one_right $ by rw [h, u₂.mul_inv] | lemma | units.inv_unique | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"inv_unique",
"units.inv_eq_of_mul_eq_one_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inv {M : Type*} [division_monoid M] (u : units M) : ↑u⁻¹ = (u⁻¹ : M) | eq.symm $ inv_eq_of_mul_eq_one_right u.mul_inv | lemma | units.coe_inv | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"division_monoid",
"inv_eq_of_mul_eq_one_right",
"units"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units.mk_of_mul_eq_one [comm_monoid α] (a b : α) (hab : a * b = 1) :
αˣ | ⟨a, b, hab, (mul_comm b a).trans hab⟩ | def | units.mk_of_mul_eq_one | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"comm_monoid",
"mul_comm"
] | For `a, b` in a `comm_monoid` such that `a * b = 1`, makes a unit out of `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
units.coe_mk_of_mul_eq_one [comm_monoid α] {a b : α} (h : a * b = 1) :
(units.mk_of_mul_eq_one a b h : α) = a | rfl | lemma | units.coe_mk_of_mul_eq_one | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"comm_monoid",
"units.mk_of_mul_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
divp (a : α) (u) : α | a * (u⁻¹ : αˣ) | def | divp | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [] | Partial division. It is defined when the
second argument is invertible, and unlike the division operator
in `division_ring` it is not totalized at zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
divp_self (u : αˣ) : (u : α) /ₚ u = 1 | units.mul_inv _ | theorem | divp_self | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"units.mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
divp_one (a : α) : a /ₚ 1 = a | mul_one _ | theorem | divp_one | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
divp_assoc (a b : α) (u : αˣ) : a * b /ₚ u = a * (b /ₚ u) | mul_assoc _ _ _ | theorem | divp_assoc | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
divp_assoc' (x y : α) (u : αˣ) : x * (y /ₚ u) = (x * y) /ₚ u | (divp_assoc _ _ _).symm | lemma | divp_assoc' | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"divp_assoc"
] | `field_simp` needs the reverse direction of `divp_assoc` to move all `/ₚ` to the right. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
divp_inv (u : αˣ) : a /ₚ u⁻¹ = a * u | rfl | theorem | divp_inv | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
divp_mul_cancel (a : α) (u : αˣ) : a /ₚ u * u = a | (mul_assoc _ _ _).trans $ by rw [units.inv_mul, mul_one] | theorem | divp_mul_cancel | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"mul_assoc",
"mul_one",
"units.inv_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_divp_cancel (a : α) (u : αˣ) : (a * u) /ₚ u = a | (mul_assoc _ _ _).trans $ by rw [units.mul_inv, mul_one] | theorem | mul_divp_cancel | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"mul_assoc",
"mul_one",
"units.mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
divp_left_inj (u : αˣ) {a b : α} : a /ₚ u = b /ₚ u ↔ a = b | units.mul_left_inj _ | theorem | divp_left_inj | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"units.mul_left_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
divp_divp_eq_divp_mul (x : α) (u₁ u₂ : αˣ) :
(x /ₚ u₁) /ₚ u₂ = x /ₚ (u₂ * u₁) | by simp only [divp, mul_inv_rev, units.coe_mul, mul_assoc] | theorem | divp_divp_eq_divp_mul | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"divp",
"mul_assoc",
"mul_inv_rev",
"units.coe_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
divp_eq_iff_mul_eq {x : α} {u : αˣ} {y : α} : x /ₚ u = y ↔ y * u = x | u.mul_left_inj.symm.trans $ by rw [divp_mul_cancel]; exact ⟨eq.symm, eq.symm⟩ | theorem | divp_eq_iff_mul_eq | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"divp_mul_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_divp_iff_mul_eq {x : α} {u : αˣ} {y : α} : x = y /ₚ u ↔ x * u = y | by rw [eq_comm, divp_eq_iff_mul_eq] | theorem | eq_divp_iff_mul_eq | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"divp_eq_iff_mul_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
divp_eq_one_iff_eq {a : α} {u : αˣ} : a /ₚ u = 1 ↔ a = u | (units.mul_left_inj u).symm.trans $ by rw [divp_mul_cancel, one_mul] | theorem | divp_eq_one_iff_eq | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"divp_mul_cancel",
"one_mul",
"units.mul_left_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_divp (u : αˣ) : 1 /ₚ u = ↑u⁻¹ | one_mul _ | theorem | one_divp | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_eq_one_divp (u : αˣ) : ↑u⁻¹ = 1 /ₚ u | by rw one_divp | lemma | inv_eq_one_divp | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"one_divp"
] | Used for `field_simp` to deal with inverses of units. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_eq_one_divp' (u : αˣ) :
((1 / u : αˣ) : α) = 1 /ₚ u | by rw [one_div, one_divp] | lemma | inv_eq_one_divp' | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"one_div",
"one_divp"
] | Used for `field_simp` to deal with inverses of units. This form of the lemma
is essential since `field_simp` likes to use `inv_eq_one_div` to rewrite
`↑u⁻¹ = ↑(1 / u)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_div_eq_divp (u₁ u₂ : αˣ) : ↑(u₁ / u₂) = ↑u₁ /ₚ u₂ | by rw [divp, division_def, units.coe_mul] | lemma | coe_div_eq_divp | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"divp",
"units.coe_mul"
] | `field_simp` moves division inside `αˣ` to the right, and this lemma
lifts the calculation to `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
divp_mul_eq_mul_divp (x y : α) (u : αˣ) : x /ₚ u * y = x * y /ₚ u | by simp_rw [divp, mul_assoc, mul_comm] | theorem | divp_mul_eq_mul_divp | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"divp",
"mul_assoc",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
divp_eq_divp_iff {x y : α} {ux uy : αˣ} :
x /ₚ ux = y /ₚ uy ↔ x * uy = y * ux | by rw [divp_eq_iff_mul_eq, divp_mul_eq_mul_divp, divp_eq_iff_mul_eq] | lemma | divp_eq_divp_iff | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"divp_eq_iff_mul_eq",
"divp_mul_eq_mul_divp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
divp_mul_divp (x y : α) (ux uy : αˣ) :
(x /ₚ ux) * (y /ₚ uy) = (x * y) /ₚ (ux * uy) | by rw [divp_mul_eq_mul_divp, divp_assoc', divp_divp_eq_divp_mul] | lemma | divp_mul_divp | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"divp_assoc'",
"divp_divp_eq_divp_mul",
"divp_mul_eq_mul_divp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_one_of_mul_right (h : a * b = 1) : a = 1 | congr_arg units.inv $ subsingleton.elim (units.mk _ _ (by rwa mul_comm) h) 1 | lemma | eq_one_of_mul_right | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_one_of_mul_left (h : a * b = 1) : b = 1 | congr_arg units.inv $ subsingleton.elim (units.mk _ _ h $ by rwa mul_comm) 1 | lemma | eq_one_of_mul_left | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_one : a * b = 1 ↔ a = 1 ∧ b = 1 | ⟨λ h, ⟨eq_one_of_mul_right h, eq_one_of_mul_left h⟩, by { rintro ⟨rfl, rfl⟩, exact mul_one _ }⟩ | lemma | mul_eq_one | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"eq_one_of_mul_left",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit [monoid M] (a : M) : Prop | ∃ u : Mˣ, (u : M) = a | def | is_unit | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"monoid"
] | An element `a : M` of a monoid is a unit if it has a two-sided inverse.
The actual definition says that `a` is equal to some `u : Mˣ`, where
`Mˣ` is a bundled version of `is_unit`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unit_of_subsingleton [monoid M] [subsingleton M] (a : M) : is_unit a | ⟨⟨a, a, subsingleton.elim _ _, subsingleton.elim _ _⟩, rfl⟩ | lemma | is_unit_of_subsingleton | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"is_unit",
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units.is_unit [monoid M] (u : Mˣ) : is_unit (u : M) | ⟨u, rfl⟩ | lemma | units.is_unit | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"is_unit",
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_one [monoid M] : is_unit (1:M) | ⟨1, rfl⟩ | theorem | is_unit_one | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"is_unit",
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_of_mul_eq_one [comm_monoid M]
(a b : M) (h : a * b = 1) : is_unit a | ⟨units.mk_of_mul_eq_one a b h, rfl⟩ | theorem | is_unit_of_mul_eq_one | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"comm_monoid",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit.exists_right_inv [monoid M]
{a : M} (h : is_unit a) : ∃ b, a * b = 1 | by { rcases h with ⟨⟨a, b, hab, _⟩, rfl⟩, exact ⟨b, hab⟩ } | theorem | is_unit.exists_right_inv | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"is_unit",
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit.exists_left_inv [monoid M]
{a : M} (h : is_unit a) : ∃ b, b * a = 1 | by { rcases h with ⟨⟨a, b, _, hba⟩, rfl⟩, exact ⟨b, hba⟩ } | theorem | is_unit.exists_left_inv | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"is_unit",
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_iff_exists_inv [comm_monoid M]
{a : M} : is_unit a ↔ ∃ b, a * b = 1 | ⟨λ h, h.exists_right_inv,
λ ⟨b, hab⟩, is_unit_of_mul_eq_one _ b hab⟩ | theorem | is_unit_iff_exists_inv | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"comm_monoid",
"is_unit",
"is_unit_of_mul_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_iff_exists_inv' [comm_monoid M]
{a : M} : is_unit a ↔ ∃ b, b * a = 1 | by simp [is_unit_iff_exists_inv, mul_comm] | theorem | is_unit_iff_exists_inv' | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"comm_monoid",
"is_unit",
"is_unit_iff_exists_inv",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit.mul [monoid M] {x y : M} : is_unit x → is_unit y → is_unit (x * y) | by { rintros ⟨x, rfl⟩ ⟨y, rfl⟩, exact ⟨x * y, units.coe_mul _ _⟩ } | lemma | is_unit.mul | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"is_unit",
"monoid",
"units.coe_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units.is_unit_mul_units [monoid M] (a : M) (u : Mˣ) :
is_unit (a * u) ↔ is_unit a | iff.intro
(assume ⟨v, hv⟩,
have is_unit (a * ↑u * ↑u⁻¹), by existsi v * u⁻¹; rw [←hv, units.coe_mul],
by rwa [mul_assoc, units.mul_inv, mul_one] at this)
(λ v, v.mul u.is_unit) | theorem | units.is_unit_mul_units | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"is_unit",
"monoid",
"mul_assoc",
"mul_one",
"units.coe_mul",
"units.mul_inv"
] | Multiplication by a `u : Mˣ` on the right doesn't affect `is_unit`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
units.is_unit_units_mul {M : Type*} [monoid M] (u : Mˣ) (a : M) :
is_unit (↑u * a) ↔ is_unit a | iff.intro
(assume ⟨v, hv⟩,
have is_unit (↑u⁻¹ * (↑u * a)), by existsi u⁻¹ * v; rw [←hv, units.coe_mul],
by rwa [←mul_assoc, units.inv_mul, one_mul] at this)
u.is_unit.mul | theorem | units.is_unit_units_mul | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"is_unit",
"monoid",
"one_mul",
"units.coe_mul",
"units.inv_mul"
] | Multiplication by a `u : Mˣ` on the left doesn't affect `is_unit`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unit_of_mul_is_unit_left [comm_monoid M] {x y : M}
(hu : is_unit (x * y)) : is_unit x | let ⟨z, hz⟩ := is_unit_iff_exists_inv.1 hu in
is_unit_iff_exists_inv.2 ⟨y * z, by rwa ← mul_assoc⟩ | theorem | is_unit_of_mul_is_unit_left | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"comm_monoid",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_of_mul_is_unit_right [comm_monoid M] {x y : M}
(hu : is_unit (x * y)) : is_unit y | @is_unit_of_mul_is_unit_left _ _ y x $ by rwa mul_comm | theorem | is_unit_of_mul_is_unit_right | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"comm_monoid",
"is_unit",
"is_unit_of_mul_is_unit_left",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_iff [comm_monoid M] {x y : M} : is_unit (x * y) ↔ is_unit x ∧ is_unit y | ⟨λ h, ⟨is_unit_of_mul_is_unit_left h, is_unit_of_mul_is_unit_right h⟩,
λ h, is_unit.mul h.1 h.2⟩ | lemma | is_unit.mul_iff | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"comm_monoid",
"is_unit",
"is_unit.mul",
"is_unit_of_mul_is_unit_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit (h : is_unit a) : Mˣ | (classical.some h).copy a (classical.some_spec h).symm _ rfl | def | is_unit.unit | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"is_unit"
] | The element of the group of units, corresponding to an element of a monoid which is a unit. When
`α` is a `division_monoid`, use `is_unit.unit'` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unit_of_coe_units {a : Mˣ} (h : is_unit (a : M)) : h.unit = a | units.ext $ rfl | lemma | is_unit.unit_of_coe_units | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"is_unit",
"units.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_spec (h : is_unit a) : ↑h.unit = a | rfl | lemma | is_unit.unit_spec | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inv_mul (h : is_unit a) : ↑(h.unit)⁻¹ * a = 1 | units.mul_inv _ | lemma | is_unit.coe_inv_mul | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"is_unit",
"units.mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_coe_inv (h : is_unit a) : a * ↑(h.unit)⁻¹ = 1 | by convert h.unit.mul_inv | lemma | is_unit.mul_coe_inv | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_inj (h : is_unit a) : b * a = c * a ↔ b = c | let ⟨u, hu⟩ := h in hu ▸ u.mul_left_inj | lemma | is_unit.mul_left_inj | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"is_unit",
"mul_left_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_inj (h : is_unit a) : a * b = a * c ↔ b = c | let ⟨u, hu⟩ := h in hu ▸ u.mul_right_inj | lemma | is_unit.mul_right_inj | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"is_unit",
"mul_right_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_cancel (h : is_unit a) : a * b = a * c → b = c | h.mul_right_inj.1 | lemma | is_unit.mul_left_cancel | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"is_unit",
"mul_left_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_cancel (h : is_unit b) : a * b = c * b → a = c | h.mul_left_inj.1 | lemma | is_unit.mul_right_cancel | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"is_unit",
"mul_right_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_injective (h : is_unit a) : injective ((*) a) | λ _ _, h.mul_left_cancel | lemma | is_unit.mul_right_injective | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"is_unit",
"mul_right_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_injective (h : is_unit b) : injective (* b) | λ _ _, h.mul_right_cancel | lemma | is_unit.mul_left_injective | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"is_unit",
"mul_left_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_cancel : is_unit a → a⁻¹ * a = 1 | by { rintro ⟨u, rfl⟩, rw [← units.coe_inv, units.inv_mul] } | lemma | is_unit.inv_mul_cancel | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"inv_mul_cancel",
"is_unit",
"units.coe_inv",
"units.inv_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_cancel : is_unit a → a * a⁻¹ = 1 | by { rintro ⟨u, rfl⟩, rw [← units.coe_inv, units.mul_inv] } | lemma | is_unit.mul_inv_cancel | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"is_unit",
"mul_inv_cancel",
"units.coe_inv",
"units.mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
group_of_is_unit [hM : monoid M] (h : ∀ (a : M), is_unit a) : group M | { inv := λ a, ↑((h a).unit)⁻¹,
mul_left_inv := λ a, by
{ change ↑((h a).unit)⁻¹ * a = 1,
rw [units.inv_mul_eq_iff_eq_mul, (h a).unit_spec, mul_one] },
.. hM } | def | group_of_is_unit | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"group",
"is_unit",
"monoid",
"mul_left_inv",
"mul_one",
"units.inv_mul_eq_iff_eq_mul"
] | Constructs a `group` structure on a `monoid` consisting only of units. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comm_group_of_is_unit [hM : comm_monoid M] (h : ∀ (a : M), is_unit a) :
comm_group M | { inv := λ a, ↑((h a).unit)⁻¹,
mul_left_inv := λ a, by
{ change ↑((h a).unit)⁻¹ * a = 1,
rw [units.inv_mul_eq_iff_eq_mul, (h a).unit_spec, mul_one] },
.. hM } | def | comm_group_of_is_unit | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"comm_group",
"comm_monoid",
"is_unit",
"mul_left_inv",
"mul_one",
"units.inv_mul_eq_iff_eq_mul"
] | Constructs a `comm_group` structure on a `comm_monoid` consisting only of units. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mul_hom [has_mul α] : α →ₙ* (with_one α) | { to_fun := coe, map_mul' := λ x y, rfl } | def | with_one.coe_mul_hom | algebra.group.with_one | src/algebra/group/with_one/basic.lean | [
"algebra.group.with_one.defs",
"algebra.hom.equiv.basic"
] | [
"with_one"
] | `coe` as a bundled morphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift : (α →ₙ* β) ≃ (with_one α →* β) | { to_fun := λ f,
{ to_fun := λ x, option.cases_on x 1 f,
map_one' := rfl,
map_mul' := λ x y,
with_one.cases_on x (by { rw one_mul, exact (one_mul _).symm }) $ λ x,
with_one.cases_on y (by { rw mul_one, exact (mul_one _).symm }) $ λ y,
f.map_mul x y },
inv_fun := λ F, F.to_mul_hom.comp coe_... | def | with_one.lift | algebra.group.with_one | src/algebra/group/with_one/basic.lean | [
"algebra.group.with_one.defs",
"algebra.hom.equiv.basic"
] | [
"inv_fun",
"lift",
"monoid_hom.ext",
"mul_hom.ext",
"mul_one",
"one_mul",
"with_one",
"with_one.cases_on"
] | Lift a semigroup homomorphism `f` to a bundled monoid homorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_coe (x : α) : lift f x = f x | rfl | lemma | with_one.lift_coe | algebra.group.with_one | src/algebra/group/with_one/basic.lean | [
"algebra.group.with_one.defs",
"algebra.hom.equiv.basic"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_one : lift f 1 = 1 | rfl | lemma | with_one.lift_one | algebra.group.with_one | src/algebra/group/with_one/basic.lean | [
"algebra.group.with_one.defs",
"algebra.hom.equiv.basic"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_unique (f : with_one α →* β) : f = lift (f.to_mul_hom.comp coe_mul_hom) | (lift.apply_symm_apply f).symm | theorem | with_one.lift_unique | algebra.group.with_one | src/algebra/group/with_one/basic.lean | [
"algebra.group.with_one.defs",
"algebra.hom.equiv.basic"
] | [
"lift",
"lift_unique",
"with_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (f : α →ₙ* β) : with_one α →* with_one β | lift (coe_mul_hom.comp f) | def | with_one.map | algebra.group.with_one | src/algebra/group/with_one/basic.lean | [
"algebra.group.with_one.defs",
"algebra.hom.equiv.basic"
] | [
"lift",
"with_one"
] | Given a multiplicative map from `α → β` returns a monoid homomorphism
from `with_one α` to `with_one β` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_coe (f : α →ₙ* β) (a : α) : map f (a : with_one α) = f a | lift_coe _ _ | lemma | with_one.map_coe | algebra.group.with_one | src/algebra/group/with_one/basic.lean | [
"algebra.group.with_one.defs",
"algebra.hom.equiv.basic"
] | [
"with_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id : map (mul_hom.id α) = monoid_hom.id (with_one α) | by { ext, induction x using with_one.cases_on; refl } | lemma | with_one.map_id | algebra.group.with_one | src/algebra/group/with_one/basic.lean | [
"algebra.group.with_one.defs",
"algebra.hom.equiv.basic"
] | [
"map_id",
"monoid_hom.id",
"mul_hom.id",
"with_one",
"with_one.cases_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_map (f : α →ₙ* β) (g : β →ₙ* γ) (x) :
map g (map f x) = map (g.comp f) x | by { induction x using with_one.cases_on; refl } | lemma | with_one.map_map | algebra.group.with_one | src/algebra/group/with_one/basic.lean | [
"algebra.group.with_one.defs",
"algebra.hom.equiv.basic"
] | [
"with_one.cases_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp (f : α →ₙ* β) (g : β →ₙ* γ) :
map (g.comp f) = (map g).comp (map f) | monoid_hom.ext $ λ x, (map_map f g x).symm | lemma | with_one.map_comp | algebra.group.with_one | src/algebra/group/with_one/basic.lean | [
"algebra.group.with_one.defs",
"algebra.hom.equiv.basic"
] | [
"map_comp",
"monoid_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.mul_equiv.with_one_congr (e : α ≃* β) : with_one α ≃* with_one β | { to_fun := map e.to_mul_hom,
inv_fun := map e.symm.to_mul_hom,
left_inv := λ x, (map_map _ _ _).trans $ by induction x using with_one.cases_on; { simp },
right_inv := λ x, (map_map _ _ _).trans $ by induction x using with_one.cases_on; { simp },
.. map e.to_mul_hom } | def | mul_equiv.with_one_congr | algebra.group.with_one | src/algebra/group/with_one/basic.lean | [
"algebra.group.with_one.defs",
"algebra.hom.equiv.basic"
] | [
"inv_fun",
"with_one",
"with_one.cases_on"
] | A version of `equiv.option_congr` for `with_one`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.mul_equiv.with_one_congr_refl : (mul_equiv.refl α).with_one_congr = mul_equiv.refl _ | mul_equiv.to_monoid_hom_injective map_id | lemma | mul_equiv.with_one_congr_refl | algebra.group.with_one | src/algebra/group/with_one/basic.lean | [
"algebra.group.with_one.defs",
"algebra.hom.equiv.basic"
] | [
"map_id",
"mul_equiv.refl",
"mul_equiv.to_monoid_hom_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.mul_equiv.with_one_congr_symm (e : α ≃* β) :
e.with_one_congr.symm = e.symm.with_one_congr | rfl | lemma | mul_equiv.with_one_congr_symm | algebra.group.with_one | src/algebra/group/with_one/basic.lean | [
"algebra.group.with_one.defs",
"algebra.hom.equiv.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.mul_equiv.with_one_congr_trans (e₁ : α ≃* β) (e₂ : β ≃* γ) :
e₁.with_one_congr.trans e₂.with_one_congr = (e₁.trans e₂).with_one_congr | mul_equiv.to_monoid_hom_injective (map_comp _ _).symm | lemma | mul_equiv.with_one_congr_trans | algebra.group.with_one | src/algebra/group/with_one/basic.lean | [
"algebra.group.with_one.defs",
"algebra.hom.equiv.basic"
] | [
"map_comp",
"mul_equiv.to_monoid_hom_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_one (α) | option α | def | with_one | algebra.group.with_one | src/algebra/group/with_one/defs.lean | [
"order.with_bot",
"algebra.ring.defs"
] | [] | Add an extra element `1` to a type | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rec_one_coe {C : with_one α → Sort*} (h₁ : C 1) (h₂ : Π (a : α), C a) :
Π (n : with_one α), C n | option.rec h₁ h₂ | def | with_one.rec_one_coe | algebra.group.with_one | src/algebra/group/with_one/defs.lean | [
"order.with_bot",
"algebra.ring.defs"
] | [
"with_one"
] | Recursor for `with_one` using the preferred forms `1` and `↑a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unone {x : with_one α} (hx : x ≠ 1) : α | with_bot.unbot x hx | def | with_one.unone | algebra.group.with_one | src/algebra/group/with_one/defs.lean | [
"order.with_bot",
"algebra.ring.defs"
] | [
"with_bot.unbot",
"with_one"
] | Deconstruct a `x : with_one α` to the underlying value in `α`, given a proof that `x ≠ 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unone_coe {x : α} (hx : (x : with_one α) ≠ 1) : unone hx = x | rfl | lemma | with_one.unone_coe | algebra.group.with_one | src/algebra/group/with_one/defs.lean | [
"order.with_bot",
"algebra.ring.defs"
] | [
"with_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_unone {x : with_one α} (hx : x ≠ 1) : ↑(unone hx) = x | with_bot.coe_unbot x hx | lemma | with_one.coe_unone | algebra.group.with_one | src/algebra/group/with_one/defs.lean | [
"order.with_bot",
"algebra.ring.defs"
] | [
"with_bot.coe_unbot",
"with_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
some_eq_coe {a : α} : (some a : with_one α) = ↑a | rfl | lemma | with_one.some_eq_coe | algebra.group.with_one | src/algebra/group/with_one/defs.lean | [
"order.with_bot",
"algebra.ring.defs"
] | [
"with_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ne_one {a : α} : (a : with_one α) ≠ (1 : with_one α) | option.some_ne_none a | lemma | with_one.coe_ne_one | algebra.group.with_one | src/algebra/group/with_one/defs.lean | [
"order.with_bot",
"algebra.ring.defs"
] | [
"option.some_ne_none",
"with_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_ne_coe {a : α} : (1 : with_one α) ≠ a | coe_ne_one.symm | lemma | with_one.one_ne_coe | algebra.group.with_one | src/algebra/group/with_one/defs.lean | [
"order.with_bot",
"algebra.ring.defs"
] | [
"with_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_one_iff_exists {x : with_one α} : x ≠ 1 ↔ ∃ (a : α), ↑a = x | option.ne_none_iff_exists | lemma | with_one.ne_one_iff_exists | algebra.group.with_one | src/algebra/group/with_one/defs.lean | [
"order.with_bot",
"algebra.ring.defs"
] | [
"option.ne_none_iff_exists",
"with_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
can_lift : can_lift (with_one α) α coe (λ a, a ≠ 1) | { prf := λ a, ne_one_iff_exists.1 } | instance | with_one.can_lift | algebra.group.with_one | src/algebra/group/with_one/defs.lean | [
"order.with_bot",
"algebra.ring.defs"
] | [
"can_lift",
"with_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inj {a b : α} : (a : with_one α) = b ↔ a = b | option.some_inj | lemma | with_one.coe_inj | algebra.group.with_one | src/algebra/group/with_one/defs.lean | [
"order.with_bot",
"algebra.ring.defs"
] | [
"option.some_inj",
"with_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cases_on {P : with_one α → Prop} :
∀ (x : with_one α), P 1 → (∀ a : α, P a) → P x | option.cases_on | lemma | with_one.cases_on | algebra.group.with_one | src/algebra/group/with_one/defs.lean | [
"order.with_bot",
"algebra.ring.defs"
] | [
"with_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul [has_mul α] (a b : α) : ((a * b : α) : with_one α) = a * b | rfl | lemma | with_one.coe_mul | algebra.group.with_one | src/algebra/group/with_one/defs.lean | [
"order.with_bot",
"algebra.ring.defs"
] | [
"with_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inv [has_inv α] (a : α) : ((a⁻¹ : α) : with_one α) = a⁻¹ | rfl | lemma | with_one.coe_inv | algebra.group.with_one | src/algebra/group/with_one/defs.lean | [
"order.with_bot",
"algebra.ring.defs"
] | [
"with_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one [has_one α] : ((1 : α) : with_zero α) = 1 | rfl | lemma | with_zero.coe_one | algebra.group.with_one | src/algebra/group/with_one/defs.lean | [
"order.with_bot",
"algebra.ring.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul {α : Type u} [has_mul α]
{a b : α} : ((a * b : α) : with_zero α) = a * b | rfl | lemma | with_zero.coe_mul | algebra.group.with_one | src/algebra/group/with_one/defs.lean | [
"order.with_bot",
"algebra.ring.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_pow [has_one α] [has_pow α ℕ] {a : α} (n : ℕ) :
↑(a ^ n : α) = (↑a ^ n : with_zero α) | rfl | lemma | with_zero.coe_pow | algebra.group.with_one | src/algebra/group/with_one/defs.lean | [
"order.with_bot",
"algebra.ring.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inv [has_inv α] (a : α) : ((a⁻¹ : α) : with_zero α) = a⁻¹ | rfl | lemma | with_zero.coe_inv | algebra.group.with_one | src/algebra/group/with_one/defs.lean | [
"order.with_bot",
"algebra.ring.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_zero [has_inv α] : (0 : with_zero α)⁻¹ = 0 | rfl | lemma | with_zero.inv_zero | algebra.group.with_one | src/algebra/group/with_one/defs.lean | [
"order.with_bot",
"algebra.ring.defs"
] | [
"inv_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_div [has_div α] (a b : α) : ↑(a / b : α) = (a / b : with_zero α) | rfl | lemma | with_zero.coe_div | algebra.group.with_one | src/algebra/group/with_one/defs.lean | [
"order.with_bot",
"algebra.ring.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zpow [div_inv_monoid α] {a : α} (n : ℤ) :
↑(a ^ n : α) = (↑a ^ n : with_zero α) | rfl | lemma | with_zero.coe_zpow | algebra.group.with_one | src/algebra/group/with_one/defs.lean | [
"order.with_bot",
"algebra.ring.defs"
] | [
"div_inv_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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