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 $m$.