monoid_properties:> isMonoid premonoid
}.
-notation "1" with precedence 89
-for @{ 'munit }.
-
-interpretation "Monoid unit" 'munit =
- (cic:/matita/algebra/monoids/e.con _).
+interpretation "Monoid unit" 'neutral = (e _).
definition is_left_inverse ≝
λM:Monoid.
λopp: M → M.
- ∀x:M. (opp x)·x = 1.
+ ∀x:M. (opp x)·x = ⅇ.
definition is_right_inverse ≝
λM:Monoid.
λopp: M → M.
- ∀x:M. x·(opp x) = 1.
+ ∀x:M. x·(opp x) = ⅇ.
theorem is_left_inverse_to_is_right_inverse_to_eq:
∀M:Monoid. ∀l,r.