Let $O$ be a discrete valuation ring with valuation $v$. We normalize by $v(\pi) =1$, with $\pi$ a prime element in $O$.
By definition, for all $x,y$ in $O$, we have $v(x+y) \geq \min (v(x),v(y))$. For some reason, the texts (Neukirch and Serre) suggest that one has equality when $v(x) \neq v(y)$. Where does this come from?
Does one also have an upper bound for $v(x+y)$?
I actually have elements $x_1,\ldots,x_n$ and I want to show that $v(x_1+\ldots+x_n) = \min(v(x_i))$. Does it suffice to show that $v(x_i) \neq v(x_j)$ for all $i\neq j$?