Modulo Addition has Identity

Theorem
Addition modulo $m$ has an identity:

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

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

$$ $$ $$ $$

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