Prove commutative law of multiplication using peano axioms.

Question

That is, prove $∀x∀y(x \cdot y=y \cdot x)$.

I have tried induction but it seems not work well. It may require the rule of additive cancellation to be proved. could someone please prove it please? Thanks in advance!

Answer 1

HINT

From previous posts I know you are working with the following two axioms for multiplication:

$\forall x \: x \cdot 0 = 0$

$\forall x \forall y \: x \cdot s(y) = (x\cdot y) + x$

So here multiplication is recursively defined over the right operand. To prove commutation for multiplication you will have to prove its left-recursive counterparts:

$\forall x \: 0 \cdot x = 0$

$\forall x \forall y \: s(x) \cdot y = (x \cdot y) + y$

Both can be proven by induction. Once you have those, it is a fairly easy proof (again using induction) to get:

$\forall x \forall y \: x\cdot y = y \cdot x$