Ordinal Multiplication by One

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Theorem

Let $x$ be an ordinal.

Let $1$ denote the successor of $\varnothing$.

\(\displaystyle \left({x \cdot 1}\right)\) \(=\) \(\displaystyle x\)
\(\displaystyle \left({1 \cdot x}\right)\) \(=\) \(\displaystyle x\)


Proof

\(\displaystyle \left({x \cdot 1}\right)\) \(=\) \(\displaystyle \left({x \cdot \varnothing^+}\right)\) Definition of One ($1$)
\(\displaystyle \) \(=\) \(\displaystyle \left({\left({x \cdot \varnothing}\right) + x}\right)\) Definition of Ordinal Multiplication
\(\displaystyle \) \(=\) \(\displaystyle \left({\varnothing + x}\right)\) Definition of Ordinal Multiplication
\(\displaystyle \) \(=\) \(\displaystyle x\) Ordinal Addition by Zero

$\Box$


The proof of the other equality shall proceed by Transfinite Induction.


Basis for the Induction

\(\displaystyle \left({1 \cdot \varnothing}\right)\) \(=\) \(\displaystyle \varnothing\) Definition of Ordinal Multiplication

This proves the basis for the induction.


Induction Step

\(\displaystyle \left({1 \cdot x}\right)\) \(=\) \(\displaystyle x\) Inductive Hypothesis
\(\displaystyle \implies \ \ \) \(\displaystyle \left({\left({1 \cdot x}\right) + 1}\right)\) \(=\) \(\displaystyle x^+\) Ordinal Addition by One
\(\displaystyle \left({1 \cdot x^+}\right)\) \(=\) \(\displaystyle \left({\left({1 \cdot x}\right) + 1}\right)\) Definition of Ordinal Multiplication
\(\displaystyle \implies \ \ \) \(\displaystyle \left({1 \cdot x^+}\right)\) \(=\) \(\displaystyle x^+\) Equality is Transitive

This proves the induction step.


Limit Case

\(\displaystyle \forall y \in x: \ \ \) \(\displaystyle \left({1 \cdot y}\right)\) \(=\) \(\displaystyle y\) Hypothesis
\(\displaystyle \implies \ \ \) \(\displaystyle \bigcup_{y \mathop \in x} \left({1 \cdot y}\right)\) \(=\) \(\displaystyle \bigcup_{y \mathop \in x} y\) Indexed Union Equality
\(\displaystyle \implies \ \ \) \(\displaystyle \left({1 \cdot x}\right)\) \(=\) \(\displaystyle \bigcup_{y \mathop \in x} y\) Definition of Ordinal Multiplication
\(\displaystyle \implies \ \ \) \(\displaystyle \left({1 \cdot x}\right)\) \(=\) \(\displaystyle x\) Union of Limit Ordinal

This proves the limit case.

$\blacksquare$


Sources