Isomorphism Preserves Inverses/Proof 2

Theorem

Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures.

Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be an isomorphism.

Let $\struct {S, \circ}$ have an identity $e_S$.

Then $x^{-1}$ is an inverse of $x$ for $\circ$ if and only if $\map \phi {x^{-1} }$ is an inverse of $\map \phi x$ for $*$.

That is, if and only if:

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

Proof

We have that an isomorphism is a homomorphism which is also a bijection.

By definition, an epimorphism is a homomorphism which is also a surjection.

That is, an isomorphism is an epimorphism which is also an injection.

Thus Epimorphism Preserves Inverses can be applied.

$\blacksquare$