Modulo Addition is Commutative

Theorem
Modulo addition is commutative:


 * $\forall x, y, z \in \Z: x + y \pmod m = y + x \pmod m$

Proof
From the definition of modulo addition, this is also written:


 * $\forall m \in \Z: \forall \left[\!\left[{x}\right]\!\right]_m, \left[\!\left[{y}\right]\!\right]_m \in \Z_m: \left[\!\left[{x}\right]\!\right]_m +_m \left[\!\left[{y}\right]\!\right]_m = \left[\!\left[{y}\right]\!\right]_m +_m \left[\!\left[{x}\right]\!\right]_m$

Hence: