Subtraction on Integers is Extension of Natural Numbers

From ProofWiki
Jump to navigation Jump to search

Theorem

Integer subtraction is an extension of the definition of subtraction on the natural numbers.


Proof

Let $m, n \in \N: m \le n$.

From natural number subtraction, $\exists p \in \N: m + p = n$ such that $n - m = p$.

As $m, n, p \in \N$, it follows that $m, n, p \in \Z$ as well.

However, as $\Z$ is the inverse completion of $\N$, it follows that $-m \in \Z$ as well, so it makes sense to express the following:


\(\displaystyle \paren {n + \paren {-m} } + m\) \(=\) \(\displaystyle n + \paren {\paren {-m} + m}\)
\(\displaystyle \) \(=\) \(\displaystyle n\)
\(\displaystyle \) \(=\) \(\displaystyle p + m\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {n - m} + m\)


Thus, as all elements of $\Z$ are cancellable, it follows that $n + \paren {-m} = n - m$.

So:

$\forall m, n \in \Z, m \le n: n + \paren {-m} = n - m = n -_\N m$

and the result follows.

$\blacksquare$