Zero Vector is Unique

Theorem
Let $\struct {\mathbf V, +, \circ}_{\mathbb F}$ be a vector space over $\mathbb F$, as defined by the vector space axioms.

Then the zero vector in $\mathbf V$ is unique:


 * $\exists! \mathbf 0 \in \mathbf V: \forall \mathbf x \in \mathbf V: \mathbf x + \mathbf 0 = \mathbf x$

Proof of Existence
Follows from the vector space axioms.

Proof of Uniqueness
Let $\mathbf 0$, $\mathbf 0'$ be zero vectors.

Utilizing the vector space axioms:

Also see

 * Additive Inverse in Vector Space is Unique