Need help proving an equality

Question

I'm looking for help proving this equality:

$\forall m:m \times 0 = 0 = 0 \times m$

Any help would be greatly appreciated. Thank you very much.

Answer 1

This is assuming that you know distributivity and that $\forall n \in \mathbb{Z}, n + 0 = 0 + n = 0$.

\begin{align} 0 &= 0\\ 0 &= 0 + 0\\ a \cdot 0 &= a \cdot (0 + 0)\\ (a \cdot 0) &= (a \cdot 0) + (a \cdot 0)\\ \color{blue}{-(a \cdot 0) + (a \cdot 0)} &= (a \cdot 0) + \color{blue}{(a \cdot 0)+ -(a \cdot 0)}\\ 0 &= (a \cdot 0) + 0\\ 0 &= a \cdot 0 \end{align}

Commutativity then implies that $0 = a \cdot 0 = 0 \cdot a$.