Inverse of Inverse

Theorem

General Algebraic Structure

Let $\struct {S, \circ}$ be an algebraic structure with an identity element $e$.

Let $x \in S$ be invertible, and let $y$ be an inverse of $x$.

Then $x$ is also an inverse of $y$.

Monoid

Let $\struct {S, \circ}$ be a monoid.

Let $x \in S$ be invertible, and let its inverse be $x^{-1}$.

Then $x^{-1}$ is also invertible, and:

$\paren {x^{-1} }^{-1} = x$

Group

Let $\struct {G, \circ}$ be a group.

Let $g \in G$, with inverse $g^{-1}$.

Then:

$\paren {g^{-1} }^{-1} = g$

Ring

Let $\struct {R, +, \circ}$ be a ring.

Let $a \in R$ and let $-a$ be the ring negative of $a$.

Then:

$-\paren {-a} = a$

Field

Let $\struct {F, +, \times}$ be a field whose zero is $0_F$.

Let $a \in F$ and let $-a$ be the field negative of $a$.

Then:

$-\paren {-a} = a$