In mathematics, a lemma (plural lemmas or lemmata) is a generally minor, proven proposition which is used as a stepping stone to a larger result. For that reason, it is also known as a "helping theorem" or an "auxiliary theorem". In many cases, a lemma derives its importance from the theorem it aims to prove, however, a lemma can also turn out to be more important than originally thought. The word "lemma" derives from the Ancient Greek λῆμμα ("anything which is received", such as a gift, profit, or a bribe).
Comparison with theorem
There is no formal distinction between a lemma and a theorem, only one of intention (see Theorem terminology). However, a lemma can be considered a minor result whose sole purpose is to help prove a theorem – a step in the direction of proof – or a short theorem appearing at an intermediate stage in a proof.
A good stepping stone can lead to many others. Some powerful results in mathematics are known as lemmas. These include, among others:
While these results originally seemed too simple or too technical to warrant independent interest, they have eventually turned out to be central to the theories in which they occur.
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- "The Definitive Glossary of Higher Mathematical Jargon — Lemma". Math Vault. 2019-08-01. Retrieved 2019-11-28.
- Higham, Nicholas J. (1998). Handbook of Writing for the Mathematical Sciences. Society for Industrial and Applied Mathematics. p. 16. ISBN 0-89871-420-6.
- "Definition of lemma | Dictionary.com". www.dictionary.com. Retrieved 2019-11-28.
- Richeson, Dave (2008-09-23). "What is the difference between a theorem, a lemma, and a corollary?". David Richeson: Division by Zero. Retrieved 2019-11-28.
- Wolfram, Stephen (2002). A New Kind of Science. Wolfram Media, Inc. p. 1176. ISBN 1-57955-008-8.