Modulo Addition is Closed

Theorem
Addition modulo $m$ is closed:

$$\forall \left[\!\left[{x}\right]\!\right]_m, \left[\!\left[{y}\right]\!\right]_m \in \Z_m: \left[\!\left[{x}\right]\!\right]_m +_m \left[\!\left[{y}\right]\!\right]_m \in \Z_m$$.

Proof
From Addition modulo $m$, we have $$\left[\!\left[{x}\right]\!\right]_m +_m \left[\!\left[{y}\right]\!\right]_m = \left[\!\left[{x + y}\right]\!\right]_m$$.

By the Division Theorem, $$x + y = q m + r$$ where $$0 \le r < m$$.

Therefore $$\left[\!\left[{x + y}\right]\!\right]_m = \left[\!\left[{r}\right]\!\right]_m, 0 \le r < m$$.

Therefore $$\left[\!\left[{x + y}\right]\!\right]_m \in \Z_m$$, from the definition of integers modulo $m$.