Modulo Multiplication is Closed

Theorem
Multiplication modulo $m$ is closed on the set of integers modulo $m$:


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

Proof
From the definition of multiplication modulo $m$, we have:
 * $\left[\!\left[{x}\right]\!\right]_m \times_m \left[\!\left[{y}\right]\!\right]_m = \left[\!\left[{x y}\right]\!\right]_m$.

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 $m$.