Positive Integer is Well-Defined
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Theorem
"Positive" as applied to an integer is well-defined.
Proof
Let us define $\eqclass {\tuple {a, b} } \boxminus$ as in the formal definition of integers.
That is, $\eqclass {\tuple {a, b} } \boxminus$ is an equivalence class of ordered pairs of natural numbers under the congruence relation $\boxminus$.
$\boxminus$ is the congruence relation defined on $\N \times \N$ by $\tuple {x_1, y_1} \boxminus \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1$.
In order to streamline the notation, we will use $\eqclass {a, b} {}$ to mean $\eqclass {\tuple {a, b} } \boxminus$, as suggested.
Thus, what we are trying to prove is:
- $\eqclass {a, b} {} = \eqclass {c, d} {} \land b < a \implies d < c$
By definition:
- $\eqclass {a, b} {} = \eqclass {c, d} {} \iff a + d = b + c$
So:
\(\ds b\) | \(<\) | \(\ds a\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \exists p \in \N: \, \) | \(\ds a\) | \(=\) | \(\ds b + p\) | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds b + p + d\) | \(=\) | \(\ds b + c\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds p + d\) | \(=\) | \(\ds c\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds d\) | \(<\) | \(\ds c\) |
$\blacksquare$