Vector Space has Unique Additive Identity
A vector space has unique additive identity.
Suppose $0$ and $0'$ are both additive identities for some vector space $V$. Then $0'=0'+0$ because $0$ is an additive identity. $0'+0=0+0'$ follows from the commutativity of vector spaces. Now, $0+0=0$ because $0'$ is an additive identity, and hence $0'=0$. Thus $V$ has a unique additive identity.