Provide reasons (mostly vector space properties) as justification for each of the six steps of the following proof.

Theorem For any vector $\vect{u}\in\complex{m}$, $0\vect{u}=\zerovector.$

Proof: Let $\vect{u}\in\complex{m}$.

\begin{align*} \zerovector &= 0\vect{u} +(-0\vect{u})\\ &= (0+0)\vect{u} + (-0\vect{u})\\ &= (0\vect{u}+0\vect{u}) + (-0\vect{u})\\ &= 0\vect{u} + (0\vect{u} + (-0\vect{u}))\\ &= 0\vect{u} + \zerovector\\ &= 0\vect{u} \end{align*}