Integer Multiplication is Well-Defined/Proof 2

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Theorem

Integer multiplication is well-defined.


Proof

Consider the formal definition of the integers: $x = \eqclass {a, b} {}$ is an equivalence class of ordered pairs of natural numbers.


Consider the mapping $\phi: \N_{>0} \to \Z_{>0}$ defined as:

$\forall u \in \N_{>0}: \map \phi u = u'$

where $u' \in \Z$ be the (strictly) positive integer $\eqclass {b + u, b} {}$.

Let $v' = \eqclass {c + v, c} {}$.

Then:

\(\ds u' v'\) \(=\) \(\ds \eqclass {b + u, b} {} \times \eqclass {c + v, c} {}\)
\(\ds \) \(=\) \(\ds \eqclass {\paren {b + u} \paren {c + v} + b c, \paren {b + u} c + b \paren {c + v} } {}\)
\(\ds \) \(=\) \(\ds \eqclass {b c + b v + c u + u v + b c, b c + u c + b c + b v} {}\)
\(\ds \) \(=\) \(\ds \eqclass {b c + u v, b c} {}\)
\(\ds \) \(=\) \(\ds \eqclass {b + u v, b} {}\)
\(\ds \) \(=\) \(\ds \paren {u v}'\)

$\blacksquare$


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