Theorem

${latex.inline(-1)v = -v\quad \forall v \in V}

Proof

For every ${latex.inlinev \in V} we have ${latex.inlinev + (-1)v = 1v + (-1)v = (1 + -1)v = 0v =0}. This means that -1v is the additive inverse of v which is equal to -v.