Modulo Addition has Identity

Theorem
Let $m \in \Z$ be an integer.

Then addition modulo $m$ has an identity:


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

That is:
 * $\forall a \in \Z: a + 0 \equiv a \equiv 0 + a \pmod m$

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