P-adic Valuation of Rational Number is Well Defined

Theorem
The $p$-adic valuation $\nu_p: \Q \to \Z \cup \left\{{+\infty}\right\}$ is well defined.

Proof
Assume now that $\dfrac{a}{b}=\dfrac{c}{d} \in \Q$; that is, let $a d = b c \in \Z$.

Hence:

Now, in conclusion:

Hence, $\nu_p^\Q: \Q \to \Z \cup \left\{{+\infty}\right\}$ is well defined.