I'm reading Number Theory I, by Kato, et al. The authors claim that for $x$, $y\in \mathbb{Q}$ and $m$ a square free positive integer, if $2x$, $x^2-my^2 \in \mathbb{Z}$, then for any odd prime $p$ $\mathrm{ord}_p(m)+2\mathrm{ord}_p(y) \geq 0$.
I get that $\mathrm{ord}_p(my^2)=\mathrm{ord}_p(m)+2\mathrm{ord}_p(y)$ and $\mathrm{ord}_p(x)\geq 0$. I'm just not sure that I can say that $\mathrm{ord}_p(x^2-my^2)=\min\{\mathrm{ord}_p(x^2), \mathrm{ord}_p(my^2)\}$.
I'm having a bit of trouble seeing this. Does anyone know what to do here?