# Modulo Multiplication is Associative/Proof 1

## Theorem

$\forall \left[\!\left[{x}\right]\!\right]_m, \left[\!\left[{y}\right]\!\right]_m, \left[\!\left[{z}\right]\!\right]_m \in \Z_m: \left({\left[\!\left[{x}\right]\!\right]_m \times_m \left[\!\left[{y}\right]\!\right]_m}\right) \times_m \left[\!\left[{z}\right]\!\right]_m = \left[\!\left[{x}\right]\!\right]_m \times_m \left({\left[\!\left[{y}\right]\!\right]_m \times_m \left[\!\left[{z}\right]\!\right]_m}\right)$

That is:

$\forall x, y, z \in \Z_m: \left({x \cdot_m y}\right) \cdot_m z = x \cdot_m \left({y \cdot_m z}\right)$

## Proof

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

$\blacksquare$