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:


 * $\forall m \in \Z: \forall \eqclass x m, \eqclass y m \in \Z_m: \eqclass x m +_m \eqclass y m = \eqclass y m +_m \eqclass x m$

Hence: