Proof of arithmetical identities $-(-m)=m$ and $-0=0$

Question

I have two prepositions to prove with the basic axioms for addition and multiplication:

1. For all $m \in\mathbb Z$, $-(-m) = m$

\begin{align*} (-m) + -(-m) &= 0\\ (-m) + m &= 0 \\ (-m) + -(-m) &= (-m) + m \\ m + (-m) + -(-m) &= m + (-m) + m\\ 0 + -(-m) &= 0 + m \\ -(-m) &= m \end{align*}

2. $-0 = 0$

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

The second step is based on the first preposition.

I would greatly appreciate the community's feedback. Thank you!

Answer 1

Your proofs are sufficiently correct.