Integer Multiplication is Well-Defined

Theorem
Integer multiplication is well-defined.

Proof
Let us define $\left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxminus$ as in the formal definition of integers.

That is, $\left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxminus$ 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 $\left({x_1, y_1}\right) \boxminus \left({x_2, y_2}\right) \iff x_1 + y_2 = x_2 + y_1$.

In order to streamline the notation, we will use $\left[\!\left[{a, b}\right]\!\right]$ to mean $\left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxminus$, as suggested.

We need to show that $\left[\!\left[{a, b}\right]\!\right] = \left[\!\left[{p, q}\right]\!\right] \land \left[\!\left[{c, d}\right]\!\right] = \left[\!\left[{r, s}\right]\!\right] \Longrightarrow \left[\!\left[{a, b}\right]\!\right] \times \left[\!\left[{c, d}\right]\!\right] = \left[\!\left[{p, q}\right]\!\right] \times \left[\!\left[{r, s}\right]\!\right]$.

We have $\left[\!\left[{a, b}\right]\!\right] = \left[\!\left[{p, q}\right]\!\right] \land \left[\!\left[{c, d}\right]\!\right] = \left[\!\left[{r, s}\right]\!\right] \iff a + q = b + p \land c + s = d + r$ by the definition of $\boxminus$.

From the definition of integer multiplication, we have:


 * $\forall a, b, c, d \in \N: \left[\!\left[{a, b}\right]\!\right] \times \left[\!\left[{c, d}\right]\!\right] = \left[\!\left[{ac + bd, ad + bc}\right]\!\right]$.

So, suppose that $\left[\!\left[{a, b}\right]\!\right] = \left[\!\left[{p, q}\right]\!\right]$ and $\left[\!\left[{c, d}\right]\!\right] = \left[\!\left[{r, s}\right]\!\right]$.

Both $+$ and $\times$ are commutative and associative on $\N$. Thus:

So we have $ac + bd + ps + qr = ad + bc + pr + qs$ and so, by the definition of $\boxminus$, we have:


 * $\left[\!\left[{ac + bd, ad + bc}\right]\!\right] = \left[\!\left[{pr + qs, ps + qr}\right]\!\right]$

So, by the definition of integer multiplication, this leads to:


 * $\left[\!\left[{a, b}\right]\!\right] \times \left[\!\left[{c, d}\right]\!\right] = \left[\!\left[{p, q}\right]\!\right] \times \left[\!\left[{r, s}\right]\!\right]$

Thus integer multiplication has been shown to be well-defined.