Vector Space has Unique Additive Identity

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Theorem

A vector space has unique additive identity.

Proof 1

Let $V$ be a vector space.

Let $0$ and $0'$ both be additive identities for $V$.

Because $0$ is an additive identity:

$0' = 0' + 0$

From commutativity of vector space addition:

$0' + 0 = 0 + 0'$

Because $0'$ is an additive identity:

$0 + 0 = 0$

Hence:

$0' = 0$

Thus $V$ has a unique additive identity.

$\blacksquare$