Equivalence of Definitions of P-adic Norms/Lemma 1

Theorem
Let $p \in \N$ be a prime number.

Let $\nu_p: \Z \to \N \cup \set {+\infty}$ be the $p$-adic valuation on the integers.

Then:
 * $\forall x \in Z: \map {\nu_p} x = k : x = p^k y : p \nmid y$