Homomorphism of Power of Group Element

Theorem
Let $\struct {G, \circ}$ and $\struct {H, \ast}$ be groups.

Let $\phi: S \to T$ be a group homomorphism.

Then:
 * $\forall n \in \Z: \forall g \in G: \map \phi {g^n} = \paren {\map \phi g}^n$

Proof
The result for $n \in \N_{>0}$ follows directly from General Morphism Property for Semigroups.

For $n = 0$, we use Homomorphism with Cancellable Codomain Preserves Identity.

For $n < 0$, we use Homomorphism with Identity Preserves Inverses, along with Index Laws for Monoids: Negative Index.