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:


 * $\forall \eqclass x m, \eqclass y m \in \Z_m: \eqclass x m +_m \eqclass y m \in \Z_m$.

Proof
From the definition of addition modulo $m$, we have:
 * $\eqclass x m +_m \eqclass y m = \eqclass {x + y} m$

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

Therefore for all $0 \le r < m$:
 * $\eqclass {x + y} m = \eqclass r m$.

Therefore from the definition of integers modulo $m$:
 * $\eqclass {x + y} m \in \Z_m$

Also see

 * Modulo Addition is Closed/Real Numbers