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 $m$.