I don't know how can I to prove the following (this isn't homework, i'm just flipping through a book):
if a ∈ F, v ∈ V, and av=0
then a=0 or v=0
If it hasn't been proven already, try proving the preliminary facts that $0_Fv=0_V$ and $a0_V=0_V$ for $a\in F$ and $v\in V$. This follows by using that fact that $0=0+0$ (for both the zero element of $F$ and of $V$), and using distributivity.
If $a=0$, you are done. So suppose $a\neq 0$, and hence has a multiplicative inverse. Try multiplying both sides of your equation by the inverse of $a$, and recall the axiom that $1v=v$ where $1$ is the multiplicative identity of $F$, to get your conclusion.