It is from Serre's local field. Let $a$ and $b$ be two fractional ideals of Dedekind domain $A$, then $v_p((a:b))=v_p(a)-v_p(b)=v_p(ab^{-1})$.
I can understand the last equality without difficulty. However, to show the first equality, I considered showing $(a:b)b=a$, but could not find a way to start... I'm aware $(A:a)=a^{-1}$ for any fractional ideal $a$ but I don't think the same argument would apply here.
Any hint would be appreciated.