Fermat's Little Theorem/Proof 2

Theorem
If $p$ is a prime number and $p \nmid n$, then $n^{p-1} \equiv 1 \pmod p$.

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

and Euler's Theorem can be applied.

Thus:
 * $n^{\phi \left({p}\right)} \equiv 1 \pmod p$

But from Euler Phi Function of Prime Power:
 * $\phi \left({p}\right) = p \left({1 - \dfrac 1 p}\right) = p - 1$

and the result follows.