Fermat's Little Theorem/Proof 2

Proof
By Prime not Divisor implies Coprime:
 * $p \nmid n \implies p \perp n$

and Euler's Theorem can be applied.

Thus:
 * $n^{\map \phi p} \equiv 1 \pmod p$

But from Euler Phi Function of Prime Power:
 * $\map \phi p = p \paren {1 - \dfrac 1 p} = p - 1$

and the result follows.