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

Theorem For any scalar $c$, $c\,\zerovector = \zerovector$.

Proof: Let $c$ be an arbitrary scalar.

\begin{align*} \zerovector &= c\zerovector + (-c\zerovector)\\ &= c(\zerovector + \zerovector) + (-c\zerovector)\\ &= (c\zerovector + c\zerovector) + (-c\zerovector)\\ &= c\zerovector + (c\zerovector + (-c\zerovector))\\ &= c\zerovector + \zerovector\\ &= c\zerovector \end{align*}