So I have seen many proofs of the Strong Triangle Identity for the p-adic numbers. Namely, that: $$ \| x + y \|_p \leq \text{max}\left( \|x\|_p, \|y\|_p\right) $$
All such proofs use the assumption that $x \neq y$, stating that if this were the case, then the equality above would hold. However, I can't seem to convince myself of this. I get:
$$x = y = p^k \frac{m}{n} \hspace{4pt}\text{with} \hspace{4pt} m, n, k \in \mathbb{Z}, \hspace{4pt} \text{and} \hspace{4pt} m,n \hspace{4pt} \text{coprime to p} $$ $$ x + y = 2x = p^k \frac{2m}{n}$$
My trouble is, $2x$ is only properly defined when the numertaor, ($2m$), is coprime with p. But why should this be the case? I do not see any reason why p must be an odd prime. Any clarificaiton would be appreciated.