## Theorem

Let $V$ be a vector space over a field $F$.

Let $\mathbf a, \mathbf b \in V$.

Let $\mathbf a + \mathbf b = \mathbf a$.

Then:

$\mathbf b = \bszero$

where $\bszero$ is the zero vector of $V$.

## Proof

 $\displaystyle \mathbf a + \mathbf b$ $=$ $\displaystyle \mathbf a$ $\displaystyle \leadsto \ \$ $\displaystyle \paren {-\mathbf a} + \paren {\mathbf a + \mathbf b}$ $=$ $\displaystyle \paren {-\mathbf a} + \mathbf a$ $\displaystyle \leadsto \ \$ $\displaystyle \paren {\paren {-\mathbf a} + \mathbf a} + \mathbf b$ $=$ $\displaystyle \paren {-\mathbf a} + \mathbf a$ Vector Space Axiom $\text V 2$: Associativity $\displaystyle \leadsto \ \$ $\displaystyle \bszero + \mathbf b$ $=$ $\displaystyle \bszero$ Vector Space Axiom $\text V 4$: Inverses $\displaystyle \leadsto \ \$ $\displaystyle \mathbf b$ $=$ $\displaystyle \bszero$ Vector Space Axiom $\text V 3$: Identity

$\blacksquare$