Modulo Multiplication is Associative/Proof 1

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Theorem

Multiplication modulo $m$ is associative:

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


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

$\blacksquare$


Sources

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