# Ordinal Multiplication is Left Distributive

## Theorem

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

Let $\times$ denote ordinal multiplication.

Let $+$ denote ordinal addition.

Then:

$x \times \left({ y + z }\right) = \left({ x \times y }\right) + \left({ x \times z }\right)$

## Proof

The proof shall proceed by Transfinite Induction, as follows:

### Basis for the Induction

Let $0$ denote the ordinal zero.

 $\displaystyle x \times \left({ y + 0 }\right)$ $=$ $\displaystyle x \times y$ Definition of Ordinal Addition $\displaystyle$ $=$ $\displaystyle \left({ x \times y }\right) + 0$ Definition of Ordinal Addition $\displaystyle$ $=$ $\displaystyle \left({ x \times y }\right) + \left({ x \times 0 }\right)$ Definition of Ordinal Multiplication

This proves the basis for the induction.

### Induction Step

 $\displaystyle x \times \left({ y + z }\right)$ $=$ $\displaystyle \left({ x \times y }\right) + \left({ x \times z }\right)$ Inductive Hypothesis $\displaystyle \implies \ \$ $\displaystyle x \times \left({ y + z^+ }\right)$ $=$ $\displaystyle x \times \left({ y + z }\right)^+$ Definition of Ordinal Addition $\displaystyle$ $=$ $\displaystyle \left({ x \times \left({ y + z }\right) }\right) + x$ Definition of Ordinal Multiplication $\displaystyle$ $=$ $\displaystyle \left({ \left({ x \times y }\right) + \left({ x \times z }\right) }\right) + x$ Inductive Hypothesis $\displaystyle$ $=$ $\displaystyle \left({ x \times y }\right) + \left({ \left({ x \times z }\right) + x }\right)$ Ordinal Addition is Associative $\displaystyle$ $=$ $\displaystyle \left({ x \times y }\right) + \left({ x \times z^+ }\right)$ Definition of Ordinal Multiplication

This proves the induction step.

### Limit Case

The inductive hypothesis for the limit case states that:

$x \times \left({ y + w }\right) = \left({ x \times y }\right) + \left({ x \times w }\right)$ for all $w \in z$ and $z$ is a limit ordinal.

The proof shall proceed by cases:

### Case 1

Suppose $x = 0$.

 $\displaystyle x \times \left({ y + z }\right)$ $=$ $\displaystyle 0$ Ordinal Multiplication by Zero $\displaystyle$ $=$ $\displaystyle 0 + 0$ Definition of Ordinal Addition $\displaystyle$ $=$ $\displaystyle \left({ x \times y }\right) + \left({ x \times z }\right)$ Ordinal Multiplication by Zero

### Case 2

Suppose that $x \ne 0$.

Since $w$ is a limit ordinal, $y + w$ and $x \times w$ are limit ordinals by Limit Ordinals Preserved Under Ordinal Addition and Limit Ordinals Preserved Under Ordinal Multiplication.

 $\displaystyle x \times \left({ y + z }\right)$ $=$ $\displaystyle \bigcup_{w \mathop \in \left({ y + z }\right)} \left({ x \times w }\right)$ Definition of Ordinal Multiplication $\displaystyle \left({ x \times y }\right) + \left({ x \times z }\right)$ $=$ $\displaystyle \bigcup_{v \mathop \in \left({ x \times z }\right)} \left({ x \times y }\right) + v$ Definition of Ordinal Addition

Take any $w \in \left({ y + z }\right)$.

It follows that $w \in y \lor \left({ y \subseteq w \land w \in \left({ y + z }\right) }\right)$ by Relation between Two Ordinals and Transitive Set is Proper Subset of Ordinal iff Element of Ordinal.

Thus, $\left({ w \in y \lor w = \left({ y + u }\right) }\right)$ for some $u \in z$ by Ordinal Subtraction when Possible is Unique.

If $w < y$, then:

 $\displaystyle \left({ x \times w }\right)$ $\in$ $\displaystyle \left({ x \times y }\right)$ Membership is Left Compatible with Ordinal Multiplication $\displaystyle$ $\subseteq$ $\displaystyle \left({ x \times y }\right) + v$ Ordinal is Less than Sum

If $w = \left({ y + u }\right)$, then:

 $\displaystyle x \times w$ $=$ $\displaystyle x \times \left({ y + u }\right)$ definition of $w$ $\displaystyle$ $=$ $\displaystyle \left({ x \times y }\right) + \left({ x \times u }\right)$ Inductive Hypothesis $\displaystyle \left({ x \times u }\right)$ $\in$ $\displaystyle \left({ x \times z }\right)$ Membership is Left Compatible with Ordinal Multiplication $\displaystyle$ $=$ $\displaystyle \left({ x \times y }\right) + v$ setting $v$ to $x \times u$
$x \times \left({ y + z }\right) \subseteq \left({ x \times y }\right) + \left({ x \times z }\right)$

Conversely, if $v \in \left({ x \times z }\right)$, then:

 $\displaystyle \exists w \in z: \ \$ $\displaystyle v$ $\in$ $\displaystyle \left({ x \times w }\right)$ Ordinal is Less than Ordinal times Limit $\displaystyle \implies \ \$ $\displaystyle \left({ x \times y }\right) + v$ $=$ $\displaystyle \left({ x \times y }\right) + \left({ x \times w }\right)$ Substitutivity of Class Equality $\displaystyle$ $=$ $\displaystyle x \times \left({ y + w }\right)$ Inductive Hypothesis
$\left({ x \times y }\right) + \left({ x \times z }\right) \subseteq x \times \left({ y + z }\right)$

By definition of set equality:

$x \times \left({ y + z }\right) = \left({ x \times y }\right) + \left({ x \times z }\right)$

This proves the limit case.

$\blacksquare$