Zero Vector Space Product iff Factor is Zero/Proof 2

Theorem

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:

$\lambda \circ \mathbf v = \bszero \iff \paren {\lambda = 0_F \lor x = \bszero}$

Proof

The sufficient condition is proved in Vector Scaled by Zero is Zero Vector, and in Zero Vector Scaled is Zero Vector.

The necessary condition is proved in Vector Product is Zero only if Factor is Zero.

$\blacksquare$