Non-singularity of Group Variety

Question

I'm preparing a presentation about Abelian Varieties(definition and proof of the commutativity of the group law) and I'm following Milne's notes. At some point, he states that any group variety is non-singular and he says that one can prove it by showing that the union of all the translates of the non-singular locus is the entire variety. More precisely, if $V$ is my variety, $U$ is the set of non-singular points and $$U_a = \{xa : x \in U\}, \: a \in V$$ are my translates, I want to show that $$V = \bigcup_{a \in V} U_a$$ One inclusion is trivial, but I can't see the other one. In some notes of mine, I wrote this line $$U = U_e \Longrightarrow x \in V : x = xe = xa^{-1}a \in U_a$$ where with $e$ I'm denoting the identity and I'm taking $a \in U$. At the time this line seemed pretty clear to me, but now it seems wrong. Please can someone help me with this inclusion and if what I wrote is correct explain it to me? I couldn't find a specific qustion so I decided to ask a new one, but if there already exist please redirect me.