In the proof of a technical lemma in the Arithmetic of Elliptic curves, Silverman uses the following: Assume $v$ is any valuation and $p\in \mathbb Z$ such that $0<v(p)<\infty$. Then $$v(n!)=\sum_{i=1}^\infty \left[\frac{n}{p^i}\right]v(p).$$ The chapter is dedicated to formal groups on complete local fields, but I suppose this formula is meant to work in a general setting (?).
It might be obvious but I am a little confused on how to derive this. I am aware of the Legendre formula $v_p(n!)=\sum \lfloor \frac{n}{p^i}\rfloor$ but this looks different. Unless for any valuation $v$ we have that $v_p(n)=\frac{v(n)}{v(p)}$ which I don't really see why should that be true.