Proof of -(-v)=v in a vector space The exercise is: Prove that $-(-v)=v$ for every $v \in V$

Proof
Suppose  $v \in V$ and $V$ is a vector space.
Then $-(-v) \in V$ as result of the scalar multiplication property and
$-(-v)=-(-1\cdot v)=-1 \cdot(-1\cdot v) =(-1 \cdot-1)\cdot v = 1 \cdot v = v $
The desired result $-(-v)=v$ holds.

The solution manual gives the proof:

Proof


I just wanted to make sure that the way I did the proof isn't missing anything?Thanks

Your solution seems good, acknowledging the fact that V is a vector space, and therefore satisfies the axioms pertaining to the proof.
