Modulo Addition has Identity

Theorem
Let $m \in \R$ be a real number.

Then addition modulo $m$ has an identity:


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

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

Proof
Follows directly from the definition of modulo addition:

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