# Ordinal Multiplication is Left Distributive

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## Theorem

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

Let $\times$ denote ordinal multiplication.

Let $+$ denote ordinal addition.

Then:

$x \times \paren {y + z} = \paren {x \times y} + \paren {x \times z}$

## Proof

The proof shall proceed by Transfinite Induction, as follows:

### Basis for the Induction

Let $0$ denote the ordinal zero.

 $\ds x \times \paren {y + 0}$ $=$ $\ds x \times y$ Definition of Ordinal Addition $\ds$ $=$ $\ds \paren {x \times y} + 0$ Definition of Ordinal Addition $\ds$ $=$ $\ds \paren {x \times y} + \paren {x \times 0}$ Definition of Ordinal Multiplication

This proves the basis for the induction.

### Induction Step

 $\ds x \times \paren {y + z}$ $=$ $\ds \paren {x \times y} + \paren {x \times z}$ Inductive Hypothesis $\ds \leadsto \ \$ $\ds x \times \paren {y + z^+}$ $=$ $\ds x \times \paren {y + z}^+$ Definition of Ordinal Addition $\ds$ $=$ $\ds \paren {x \times \paren {y + z} } + x$ Definition of Ordinal Multiplication $\ds$ $=$ $\ds \paren {\paren {x \times y} + \paren {x \times z} } + x$ Inductive Hypothesis $\ds$ $=$ $\ds \paren {x \times y} + \paren {\paren {x \times z} + x}$ Ordinal Addition is Associative $\ds$ $=$ $\ds \paren {x \times y} + \paren {x \times z^+}$ Definition of Ordinal Multiplication

This proves the induction step.

### Limit Case

The inductive hypothesis for the limit case states that:

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

The proof shall proceed by cases:

### Case 1

Suppose $x = 0$.

 $\ds x \times \paren {y + z}$ $=$ $\ds 0$ Ordinal Multiplication by Zero $\ds$ $=$ $\ds 0 + 0$ Definition of Ordinal Addition $\ds$ $=$ $\ds \paren {x \times y} + \paren {x \times z}$ 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.

 $\ds x \times \paren {y + z}$ $=$ $\ds \bigcup_{w \mathop \in \paren {y + z} } \paren {x \times w}$ Definition of Ordinal Multiplication $\ds \paren {x \times y} + \paren {x \times z}$ $=$ $\ds \bigcup_{v \mathop \in \paren {x \times z} } \paren {x \times y} + v$ Definition of Ordinal Addition

Take any $w \in y + z$.

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

Thus, $w \in y \lor w = y + u$ for some $u \in z$ by Ordinal Subtraction when Possible is Unique.

If $w < y$, then:

 $\ds x \times w$ $\in$ $\ds x \times y$ Membership is Left Compatible with Ordinal Multiplication $\ds$ $\subseteq$ $\ds \paren {x \times y} + v$ Ordinal is Less than Sum

If $w = y + u$, then:

 $\ds x \times w$ $=$ $\ds x \times \paren {y + u}$ Definition of $w$ $\ds$ $=$ $\ds \paren {x \times y} + \paren {x \times u}$ Inductive Hypothesis $\ds x \times u$ $\in$ $\ds x \times z$ Membership is Left Compatible with Ordinal Multiplication $\ds$ $=$ $\ds \paren {x \times y} + v$ setting $v$ to $x \times u$
$x \times \paren {y + z} \subseteq \paren {x \times y} + \paren {x \times z}$

Conversely, if $v \in x \times z$, then:

 $\ds \exists w \in z: \,$ $\ds v$ $\in$ $\ds x \times w$ Ordinal is Less than Ordinal times Limit $\ds \leadsto \ \$ $\ds \paren {x \times y} + v$ $=$ $\ds \paren {x \times y} + \paren {x \times w}$ Substitutivity of Class Equality $\ds$ $=$ $\ds x \times \paren {y + w}$ Inductive Hypothesis
$\paren {x \times y} + \paren {x \times z} \subseteq x \times \paren {y + z}$

By definition of set equality:

$x \times \paren {y + z} = \paren {x \times y} + \paren {x \times z}$

This proves the limit case.

$\blacksquare$