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.