English
For a group G, the inverse of a submonoid S ⊆ G is the submonoid S⁻¹ := { g⁻¹ : g ∈ S }. This operation is well-defined and yields a submonoid.
Русский
Для группы G обращение к подмономоиду S ⊆ G определяется как S⁻¹ := { g⁻¹ : g ∈ S }. Это образует подмономоиду.
LaTeX
$$$S^{-1} := \\{ g^{-1} : g \\in S \\}$ и это подмономоид.$$
Lean4
/-- The submonoid with every element inverted. -/
@[to_additive /-- The additive submonoid with every element negated. -/
]
protected def inv : Inv (Submonoid G) where
inv
S :=
{ carrier := (S : Set G)⁻¹
mul_mem' := fun ha hb => by rw [mem_inv, mul_inv_rev]; exact mul_mem hb ha
one_mem' := mem_inv.2 <| by rw [inv_one]; exact S.one_mem' }
