P-adic Valuation is Valuation

Theorem
The $p$-adic valuation $\nu_p: \Q \to \Z \cup \set {+\infty}$ is a valuation on $\Q$.

Proof
To prove that $\nu_p$ is a valuation it is necessary to demonstrate:

Let $q := \dfrac a b, r := \dfrac c d \in \Q$.

Axiom $(\text V 3)$
From Restricted P-adic Valuation is Valuation follows that:

Hence:
 * $\map {\nu_p} {\dfrac a b + \dfrac c d} \ge \min \set {\map {\nu_p} {\dfrac a b}, \map {\nu_p} {\dfrac c d} }$

Thus $\nu_p: \Q \to \Z \cup \set {+\infty}$ is a valuation on $\Q$ by definition.