# Finite Ordinal Times Ordinal

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## Theorem

Let $m$ and $n$ be finite ordinals.

Let $m \ne 0$, where $0$ is the zero ordinal.

Let $x$ be a limit ordinal.

Then:

- $m \times \left({ x + n }\right) = x + \left({ m \times n }\right)$

## Proof

By Ordinal Multiplication is Left Distributive, it follows that:

- $m \times \left({ x + n }\right) = \left({ m \times x }\right) + \left({ m \times n }\right)$

It remains to prove that $x = \left({ m \times x }\right)$.

Since $x$ is a limit ordinal, it follows that:

\(\, \displaystyle \exists y \in \operatorname{On}: \, \) | \(\displaystyle x\) | \(=\) | \(\displaystyle \left({\omega \times y}\right)\) | Factorization of Limit Ordinals | |||||||||

\(\displaystyle m \times x\) | \(=\) | \(\displaystyle m \times \left({\omega \times y}\right)\) | Substitutivity of Class Equality | ||||||||||

\(\displaystyle m \times x\) | \(=\) | \(\displaystyle \left({m \times \omega}\right) \times y\) | Ordinal Multiplication is Associative | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \omega \times y\) | by lemma |

$\blacksquare$

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 8.29$