What is a Lemma in Mathematics?

In mathematics, a lemma is a proven statement or proposition that serves as a stepping-stone to help prove a larger, more significant result, such as a theorem. Lemmas are typically not the main focus of mathematical work; instead, they provide essential supporting results that simplify or make possible the proof of a more complex statement.

The purpose of a lemma is to break down complicated proofs into manageable parts. By establishing certain foundational truths as lemmas, mathematicians can avoid redundant work and streamline their arguments, making complex proofs clearer and easier to follow.

Key Points About Lemmas

1. Intermediate Results: Lemmas are often intermediate results that bridge the gap between known information and the final result.
2. Can Stand Alone: While a lemma usually supports a theorem, it may be useful in multiple proofs or applications.
3. Famous Example: Some lemmas become so widely known and used that they attain near-theorem status. An example is Zorn's Lemma, which is central to proofs in set theory and functional analysis.

For instance, in proving the Pythagorean Theorem, certain geometric lemmas about similar triangles are often used to simplify the proof.