Integer Addition is Well-Defined/Proof 2

Theorem

Let $\struct {\N \times \N, \oplus}$ be the (external) direct product of $\struct {\N, +}$ with itself, where $\oplus$ is the operation on $\N \times \N$ induced by $+$ on $\N$.

Let $\boxtimes$ be the cross-relation defined on $\N \times \N$ by:

$\tuple {x_1, y_1} \boxtimes \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1$

The operation $\oplus$ on these equivalence classes is well-defined, in the sense that:

$\ds \eqclass {a_1, b_1} {}$

$=$

$\ds \eqclass {a_2, b_2} {}$

$\ds \eqclass {c_1, d_1} {}$

$=$

$\ds \eqclass {c_2, d_2} {}$

$\ds \leadsto \ \ $

$\ds \eqclass {a_1, b_1} {} \oplus \eqclass {c_1, d_1} {}$

$=$

$\ds \eqclass {a_2, b_2} {} \oplus \eqclass {c_2, d_2} {}$

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:

$\ds u' + v'$

$=$

$\ds \eqclass {b + u, b} {} + \eqclass {c + v, c} {}$

$\ds $

$=$

$\ds \eqclass {b + c + u + v, b + c} {}$

$\ds $

$=$

$\ds \eqclass {b + u + v, b} {}$

$\ds $

$=$

$\ds \paren {u + v}'$

$\blacksquare$