Aside from field and vector space definitions, the only two other propositions I should limit myself to using are $0v =0$ for all $v\in V$ and $a0 =0$ for all $a \in \mathbb{F}$.
I feel like I'm missing something obvious. One implication was straightforward: $$av=a0 \implies a^{-1}av = a^{-1}a0 \implies v =0.$$ As for the other (showing $a=0$ is a possibility) I haven't been able to make much progress. At most I showed that $$(\underbrace{a+a+\cdots +a}_\text{$n$-many})v=0v$$ for positive integers $n \geq 1$, but since $V$ lacks multaplicative inverses it feels like I hit a dead end and can't isolate $a$ somehow.
Could someone at most provide a nudge in the right direction?