Integer Multiplication is Commutative

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Theorem

The operation of multiplication on the set of integers $\Z$ is commutative:

$\forall x, y \in \Z: x \times y = y \times x$

Let $x = \left[\!\left[{a, b}\right]\!\right]$ and $y = \left[\!\left[{c, d}\right]\!\right]$ for some $x, y \in \Z$.

Then:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle x \times y$	$=$	$\displaystyle \left[\!\left[{a, b}\right]\!\right] \times \left[\!\left[{c, d}\right]\!\right]$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \left[\!\left[{ac + bd, ad + bc}\right]\!\right]$	$\displaystyle $	$\displaystyle $	$\displaystyle $	by definition of integer multiplication
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \left[\!\left[{ca + db, da + cb}\right]\!\right]$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Natural Number Multiplication is Commutative
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \left[\!\left[{ca + db, cb + da}\right]\!\right]$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Natural Number Addition is Commutative
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \left[\!\left[{c, d}\right]\!\right] \times \left[\!\left[{a, b}\right]\!\right]$	$\displaystyle $	$\displaystyle $	$\displaystyle $	by definition of integer multiplication
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle y \times x$	$\displaystyle $	$\displaystyle $	$\displaystyle $

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle x \times y\)	\(=\)	\(\displaystyle \left[\!\left[{a, b}\right]\!\right] \times \left[\!\left[{c, d}\right]\!\right]\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left[\!\left[{ac + bd, ad + bc}\right]\!\right]\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition of integer multiplication
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left[\!\left[{ca + db, da + cb}\right]\!\right]\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Natural Number Multiplication is Commutative
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left[\!\left[{ca + db, cb + da}\right]\!\right]\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Natural Number Addition is Commutative
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left[\!\left[{c, d}\right]\!\right] \times \left[\!\left[{a, b}\right]\!\right]\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition of integer multiplication
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle y \times x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)