Sufficient Condition for Vector Equals Inverse iff 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:


 * $\forall \mathbf v, -\mathbf v \in \mathbf V: \mathbf v = - \mathbf v \iff \mathbf v = \mathbf 0$

Sufficient Condition
Utilizing the vector space axioms: