Proof of Uniqueness of the Zero Vector and of Negatives

The zero vector. Some $z\in V$ satisfies $v+z=v$ for every $v\in V$, by Vector Space over a Field (iii). If $z'\in V$ satisfies $v+z'=v$ for every $v\in V$ as well, then, applying the property of $z$ at $v=z'$, Vector Space over a Field (ii), and the property of $z'$ at $v=z$,

$$z'=z'+z=z+z'=z .$$

Negatives. Let $v\in V$. Since $v+z=v$ for every $v$, Vector Space over a Field (iv) gives some $w\in V$ with $v+w=z$. If also $w'\in V$ satisfies $v+w'=z$, then by the property of $z$, Vector Space over a Field (i) and Vector Space over a Field (ii),

$$w=w+z=w+(v+w')=(w+v)+w'=(v+w)+w'=z+w'=w'+z=w' .$$