Equivalence classes defined as $\overline{a} = \{ b \in \mathbb{Z} : b \equiv a \mod n \}$ .
I need to prove the distributive law in $\mathbb{Z}_{n}$. So, if $\overline{a}$, $\overline{b}$, and $\overline{c}$ are arbitrary elements in $\mathbb{Z}_{n}$, then $\overline{a}*(\overline{b}+\overline{c}) = \overline{a}*\overline{b}+\overline{a}*\overline{c}$.
I've seen examples proving the distributive law with real numbers and understood, but I'm not sure if the same proof works with equivalence classes.
The definition of the operations on $\mathbb{Z}_n$ are $$ \bar{a}+\bar{b}=\overline{a+b} \qquad \bar{a}\,\bar{b}=\overline{ab} $$ Then $$ \bar{a}(\bar{b}+\bar{c})= \bar{a}\overline{(b+c)}= \overline{a(b+c)}=\overline{ab+ac} $$ Can you finish?