Vector Inverse is Negative Vector

Theorem

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

Then:

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

Proof

Utilizing the vector space axioms:

 $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle \mathbf v + \left({-1_{\mathbb F} \circ \mathbf v }\right)$$ $$=$$ $$\displaystyle$$ $$\displaystyle \left({1_{\mathbb F} \circ \mathbf v}\right) + \left({-1_{\mathbb F} \circ \mathbf v }\right)$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$ $$\displaystyle \left({1_{\mathbb F} - 1_{\mathbb F} }\right) \circ \mathbf v$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$ $$\displaystyle 0 \circ \mathbf v$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$ $$\displaystyle \mathbf 0$$ $$\displaystyle$$ $$\displaystyle$$ Vector Scaled by Zero is Zero Vector

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

From Vector Inverse Unique, $-1_{\mathbb F} \circ \mathbf v = -\mathbf v$.

$\blacksquare$