Lifting The Exponent Lemma

Theorem
Let $x, y \in \Z$ be integers.

Let $n \in \Z_{\ge 0}$ be a positive integer.

Let $p$ be an odd prime number.

Let:
 * $p \mathrel \backslash x - y$

and:
 * $p \nmid x y$

where $\backslash$ and $\nmid$ denote divisibility and non-divisibility respectively.

Then
 * $\nu_p \left({x^n - y^n}\right) = \nu_p \left({x - y}\right) + \nu_p \left({n}\right)$

where $\nu_p$ denotes $p$-adic valuation.