Lemma (mathematics)

Theorem for proving more complex theorems

Not to be confused with Lemma (morphology).

In mathematics and other fields,[a] a lemma (pl.: lemmas or lemmata) is a generally minor, proven proposition which is used to prove a larger statement. For that reason, it is also known as a "helping theorem" or an "auxiliary theorem".[3][4] 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.[5]

## Etymology

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From the Ancient Greek λῆμμα, (perfect passive εἴλημμαι) something received or taken. Thus something taken for granted in an argument.[6]

## Comparison with theorem

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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 more substantial theorem – a step in the direction of proof.[5]

## Well-known lemmas

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Some powerful results in mathematics are known as lemmas, first named for their originally minor purpose. These include, among others:

* Bézout's lemma
* Burnside's lemma
* Dehn's lemma
* Euclid's lemma
* Farkas' lemma
* Fatou's lemma
* Gauss's lemma (any of several named after Carl Friedrich Gauss)
* Greendlinger's lemma
* Itô's lemma
* Jordan's lemma
* Lovász local lemma
* Nakayama's lemma
* Poincaré's lemma
* Riesz's lemma
* Schur's lemma
* Schwarz's lemma
* Sperner's lemma
* Urysohn's lemma
* Vitali covering lemma
* Yoneda's lemma
* Zorn's lemma

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.

## See also

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Look up lemma in Wiktionary, the free dictionary.

* Axiom
* Corollary
* Co-premise
* Fundamental lemma
* Inference objection
* List of lemmas
* Objection
* Porism
* Theorem
* Theorem terminology

## Notes

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1. ^ Such as informal logic, argument mapping, and philosophy.[1][2]

## References

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1. ^ [1] "Lemma." Merriam-Webster.com Dictionary, Merriam-Webster.
2. ^ Loewen, Nathan R. B. Beyond the Problem of Evil. Lexington Books. March 12, 2018. ISBN 9781498555739 p. 47
3. ^ Higham, Nicholas J. (1998). Handbook of Writing for the Mathematical Sciences. Society for Industrial and Applied Mathematics. pp. 16. ISBN 0-89871-420-6.
4. ^ "Definition of lemma | Dictionary.com". www.dictionary.com. Retrieved 2019-11-28.
5. ^ a b 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.
6. ^ "Oxford English Dictionary". www.oed.com. Oxford University Press. Retrieved 26 April 2023.

## External links

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* Doron Zeilberger, Opinion 82: A Good Lemma is Worth a Thousand Theorems

This article incorporates material from Lemma on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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