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[Merged by Bors] - chore(algebra/invertible): units coerced to their monoid are invertible #8195

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[Merged by Bors] - chore(algebra/invertible): units coerced to their monoid are invertible #8195

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chore(algebra/invertible): units coerced to their monoid are invertible
eric-wieser Jul 5, 2021
5c39439
Update invertible.lean
eric-wieser Jul 5, 2021
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9 changes: 8 additions & 1 deletion src/algebra/invertible.lean
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Expand Up @@ -104,6 +104,13 @@ def unit_of_invertible [monoid α] (a : α) [invertible a] : units α :=
lemma is_unit_of_invertible [monoid α] (a : α) [invertible a] : is_unit a :=
⟨unit_of_invertible a, rfl⟩

/-- Units are invertible in their associated monoid. -/
def units.invertible [monoid α] (u : units α) : invertible (u : α) :=
{ inv_of := ↑(u⁻¹), inv_of_mul_self := u.inv_mul, mul_inv_of_self := u.mul_inv }

@[simp] lemma inv_of_units [monoid α] (u : units α) [invertible (u : α)] : ⅟(u : α) = ↑(u⁻¹) :=
inv_of_eq_right_inv u.mul_inv

/-- Each element of a group is invertible. -/
def invertible_of_group [group α] (a : α) : invertible a :=
⟨a⁻¹, inv_mul_self a, mul_inv_self a⟩
Expand All @@ -113,7 +120,7 @@ inv_of_eq_right_inv (mul_inv_self a)

/-- `1` is the inverse of itself -/
def invertible_one [monoid α] : invertible (1 : α) :=
⟨ 1, mul_one _, one_mul _ ⟩
⟨1, mul_one _, one_mul _⟩

@[simp] lemma inv_of_one [monoid α] [invertible (1 : α)] : ⅟(1 : α) = 1 :=
inv_of_eq_right_inv (mul_one _)
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