Modulo Addition is Closed

Theorem
Addition modulo $m$ is closed:

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

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