Ordinal Multiplication is Associative

Theorem
Let $x$, $y$, and $z$ be ordinals.

Let $\times$ denote ordinal multiplication.

Then:
 * $x \times \left({ y \times z }\right) = \left({ x \times y }\right) \times z$

Proof
The proof shall proceed by Transfinite Induction on $z$.

Basis for the Induction
Let $0$ denote the zero ordinal.

This proves the basis for the induction.

Induction Step
This proves the induction step.

Limit Case
The inductive hypothesis for the limit case states that:


 * $x \times ( y \times w ) = ( x \times y ) \times w$ for all $w \in z$ where $z$ is a limit ordinal.

The proof shall proceed by cases:

Case 1
If $y = 0$, then:

Case 2
If $y \ne 0$, then $y \times z$ is a limit ordinal by Limit Ordinals Preserved Under Ordinal Multiplication.

It follows that:

If $u < ( y \times z )$, then $u < ( y \times w )$ for some $w \in z$ by Ordinal Less than Ordinal Times Limit.

Generalizing, the result follows for all $u \in ( y \times z )$, so by Supremum Inequality for Ordinals:


 * $x \times ( y \times z ) \le ( x \times y ) \times z$

Conversely, take any $w < z$.

It follows that:


 * $( x \times y ) \times z \le x \times ( y \times z )$ by Supremum Inequality for Ordinals.

By Equality of Sets, $x \times ( y \times z ) = ( x \times y ) \times z$.

This proves the limit case.