Vector Inverse is Negative Vector

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.

Then:


 * $\forall \mathbf v \in \mathbf V: -\mathbf v = -1_F \circ \mathbf v$

Proof
so $-1_F \circ \mathbf v$ is an additive inverse of $\mathbf v$.

From Additive Inverse in Vector Space is Unique:
 * $-1_F \circ \mathbf v = -\mathbf v$