Powers of Group Elements

Definition
For notational simplicity, let $$(a*a*\cdots*a)_n$$ represent $$n$$ (a positive integer) copies of the group element $$a$$ being combined by the operation $$*$$. Note that this is not standard notation and will only be used for the sake of this proof.

Let $$(G,*)$$ be a group and $$a \in G$$ and $$n \in \mathbb{Z}$$. We define

$$ a^n = \begin{cases} (a*a*\cdots*a)_n, & \mbox{if }n > 0 \\ e, & \mbox{if }n = 0 \\ (a^{-1}*a^{-1}*\cdots*a^{-1})_{-n}, & \mbox{if }n < 0 \end{cases} $$

Theorem
For any element $$g$$ in a group $$G$$ and $$m,n \in \mathbb{Z}$$,

$$g^m * g^n = g^{m+n}$$ and $$(g^m)^n = g^{mn}$$.

Proof
Case 1: Let $$m>0$$.


 * Sub-case 1: Suppose $$n>0$$.  Then,

Also,


 * Sub-Case 2: Suppose $$n=0$$.  Then,

Also,

Sub-case 3: Suppose $$n<0$$.

If $$m>-n$$, then

If $$m = -n$$, then

If $$m<-n$$, then

Also,

Case 2: Let $$m=0$$ and $$n \in \mathbb{Z}$$. Then,

Also,

Case 3: Let $$m<0$$.


 * Sub-case 1: Suppose $$n>0$$.

If $$n>-m$$, then

If $$n=-m$$, then

If $$n<-m$$, then

Also,


 * Sub-Case 2: Suppose $$n=0$$.

Also,


 * Sub-Case 3: Suppose $$n<0$$.

Also,

QED