Modulo Multiplication has Identity

Theorem
Multiplication modulo $m$ has an identity:


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

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

$$ $$ $$ $$

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