Modulo Addition has Identity

Theorem
Addition modulo $m$ has an identity:

$$\forall \left[\left[{x}\right]\right]_m \in \mathbb{Z}_m: \left[\left[{x}\right]\right]_m +_m \left[\left[{0}\right]\right]_m = \left[\left[{x}\right]\right]_m = \left[\left[{0}\right]\right]_m +_m \left[\left[{x}\right]\right]_m$$.

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

Thus $$\left[\left[{0}\right]\right]_m$$ is the identity for addition modulo $m$.