P-adic Valuation is Valuation

Theorem
The $p$-adic valuation $\nu_p: \Q \to \Z \cup \left\{{+\infty}\right\}$ 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 $(V3)$
From P-adic Valuation on Integers follows that:

Hence:
 * $\nu_p \left({\dfrac a b + \dfrac c d}\right) \ge \min \left\{ {\nu_p \left({\dfrac a b}\right), \nu_p \left({\dfrac c d}\right)}\right\}$

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