# Modulo Multiplication is Associative/Proof 1

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## Theorem

$\forall \eqclass x m, \eqclass y m, \eqclass z m \in \Z_m: \paren {\eqclass x m \times_m \eqclass y m} \times_m \eqclass z m = \eqclass x m \times_m \paren {\eqclass y m \times_m \eqclass z m}$

That is:

$\forall x, y, z \in \Z_m: \paren {x \cdot_m y} \cdot_m z = x \cdot_m \paren {y \cdot_m z}$

## Proof

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

$\blacksquare$