Vector Inverse is Negative Vector

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

Then:


 * $\forall \mathbf v, -\mathbf v \in \mathbf V: -\mathbf v = -1_{\mathbb F} \circ \mathbf v$

Proof
Utilizing the vector space axioms:

so $-1_{\mathbb F} \circ \mathbf v$ is an additive inverse of $\mathbf v$.

From Additive Inverse in Vector Space is Unique:
 * $-1_{\mathbb F} \circ \mathbf v = -\mathbf v$