Modulo Multiplication has Identity

Theorem
Multiplication modulo $m$ has an identity:


 * $\forall \left[\!\left[{x}\right]\!\right]_m \in \Z_m: \left[\!\left[{x}\right]\!\right]_m \times_m \left[\!\left[{1}\right]\!\right]_m = \left[\!\left[{x}\right]\!\right]_m = \left[\!\left[{1}\right]\!\right]_m \times_m \left[\!\left[{x}\right]\!\right]_m$

Proof
Follows directly from the definition of multiplication modulo $m$:

Thus $\left[\!\left[{1}\right]\!\right]_m$ is the identity for multiplication modulo $m$.