P-adic Valuation of Difference of Powers with Coprime Exponent

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

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

Let $p$ be a prime number.

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

and:
 * $p \nmid x y n$.

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