Modulo Addition is Closed/Integers

Theorem
Let $m \in \Z$ be an integer.

Then addition modulo $m$ on the set of integers 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 the definition of 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 $m$.

Also see

 * Modulo Addition is Closed/Real Numbers