Modulo Multiplication is Commutative

Theorem

$\forall \eqclass x m, \eqclass y m \in \Z_m: \eqclass x m \times_m \eqclass y m = \eqclass y m \times_m \eqclass x m$

Proof

 $\displaystyle \eqclass x m \times_m \eqclass y m$ $=$ $\displaystyle \eqclass {x y} m$ Definition of Modulo Multiplication $\displaystyle$ $=$ $\displaystyle \eqclass {y x} m$ Integer Multiplication is Commutative $\displaystyle$ $=$ $\displaystyle \eqclass y m \times_m \eqclass x m$ Definition of Modulo Multiplication

$\blacksquare$