P-adic Valuation of Rational Number is Well Defined

Theorem
The $p$-adic valuation:
 * $\nu_p: \Q \to \Z \cup \set {+\infty}$

is well defined.

Proof
Let $\dfrac a b = \dfrac c d \in \Q _{\ne 0}$.

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

Then:

So:

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