Zero_Vector_Space_Product_iff_Factor_is_Zero

Let $F$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $\struct {\mathbf V, +, \circ}_F$ be a vector space over $F$, as defined by the vector space axioms. Let $\mathbf v \in \mathbf V, \lambda \in F$. Then $\lambda \circ \mathbf v = \bszero \iff \paren {\lambda = 0_F \lor x = \bszero}$