Vector Product is Zero only if Factor is Zero

Theorem
Let $\left({\mathbf V, +, \circ}\right)_{\mathbb F}$ be a vector space over $\mathbb F$, as defined by the vector space axioms.

Then:


 * $\forall \lambda \in \mathbb F: \forall \mathbf v \in \mathbf V: \lambda \circ \mathbf v = \mathbf 0 \implies \left({\lambda = 0 \lor \mathbf v = \mathbf 0}\right)$

where $\mathbf 0 \in \mathbf V$ is the zero vector and $0 \in \mathbb F$ is the zero scalar.

Proof
Suppose not, that is, assume:


 * $\exists \lambda \in \mathbb F: \exists \mathbf v \in \mathbf V: \lambda \circ \mathbf v = \mathbf 0 \land \lambda \ne 0 \land \mathbf v \ne \mathbf 0$.

...which clearly is the negation of the exposition of the theorem.

Utilizing the vector space axioms:

which contradicts the assumption that $\mathbf v \ne \mathbf 0$.

Also see

 * Zero Vector Scaled is Zero Vector
 * Vector Scaled by Zero is Zero Vector
 * Zero Vector Space Product iff Factor is Zero