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
Let $\dfrac a b = \dfrac c d \in \Q$.

Thus:
 * $a d = b c \in \Z$

Then:

So:

Thus, by definition, $\nu_p: \Q \to \Z \cup \left\{{+\infty}\right\}$ is well defined.