Elementary Identities in a Vector Space

Let $K$ be a field and let $V$ be a vector space over $K$, with the conditions 1-8 of that definition. Then the following hold.

1. (Uniqueness of the zero vector) There is exactly one element $0_{V}\in V$ with $v+0_{V}=v$ for every $v\in V$.

2. (Uniqueness of additive inverses) For every $v\in V$ there is exactly one $w\in V$ with $v+w=0_{V}$. It is denoted $-v$, and for $u,v\in V$ one writes $u-v=u+(-v)$.

3. (Zero scalar) $0v=0_{V}$ for every $v\in V$, where $0$ is the additive identity of $K$.

4. (Zero vector) $\lambda 0_{V}=0_{V}$ for every $\lambda\in K$.

5. (Negation) $(-1)v=-v$ for every $v\in V$, where $-1$ is the additive inverse of $1$ in $K$.

6. (No zero divisors) If $\lambda\in K$, $v\in V$ and $\lambda v=0_{V}$, then $\lambda=0$ or $v=0_{V}$.