Editing Lemma (mathematics)

{{Short description|Theorem for proving more complex theorems}}
{{Distinguish|Lemma (morphology)}}

In [[mathematics]] and other fields,{{efn|Such as [[informal logic]], [[argument mapping]], and [[philosophy]].<ref>[https://www.merriam-webster.com/dictionary/lemma.] "Lemma." Merriam-Webster.com Dictionary, Merriam-Webster.</ref><ref>Loewen, Nathan R. B. ''Beyond the Problem of Evil.'' Lexington Books. March 12, 2018. {{ISBN|9781498555739}} p. 47</ref>}} a '''lemma''' ({{plural form}}: '''lemmas''' or '''lemmata''') is a generally minor, proven [[Theorem#Terminology|proposition]] which is used to prove a larger statement. For that reason, it is also known as a "helping [[theorem]]" or an "auxiliary theorem".<ref>{{cite book |last= Higham |first= Nicholas J. |title= Handbook of Writing for the Mathematical Sciences |publisher= [[Society for Industrial and Applied Mathematics]] |year= 1998 |isbn= 0-89871-420-6 |pages= [https://archive.org/details/handbookofwritin0000high/page/16 16] |url= https://archive.org/details/handbookofwritin0000high/page/16 }}</ref><ref name=":0">{{Cite web|url=https://www.dictionary.com/browse/lemma|title=Definition of lemma {{!}} Dictionary.com|website=www.dictionary.com|language=en|access-date=2019-11-28}}</ref> In many cases, a lemma derives its importance from the theorem it aims to [[mathematical proof|prove]]; however, a lemma can also turn out to be more important than originally thought.<ref name=":1">{{Cite web|url=https://divisbyzero.com/2008/09/22/what-is-the-difference-between-a-theorem-a-lemma-and-a-corollary/|title=What is the difference between a theorem, a lemma, and a corollary?|last=Richeson|first=Dave|date=2008-09-23|website=David Richeson: Division by Zero|language=en|access-date=2019-11-28}}</ref>

==Etymology==
From the [[Ancient Greek]] λῆμμα, (perfect passive εἴλημμαι) something received or taken. Thus something taken for granted in an argument.<ref name="OED">{{cite web |title=Oxford English Dictionary |url=https://www.oed.com |website=www.oed.com |publisher=Oxford University Press |access-date=26 April 2023 |language=en}}</ref>

==Comparison with theorem==
There is no formal distinction between a lemma and a [[theorem]], only one of intention (see [[Theorem#Terminology|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.<ref name=":1"/>

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

{{div col|colwidth=18em}}
* [[Bézout's identity|Bézout's lemma]]
* [[Burnside's lemma]]
* [[Dehn's lemma]]
* [[Euclid's lemma]]
* [[Farkas' lemma]]
* [[Fatou's lemma]]
* [[Gauss's lemma (disambiguation)|Gauss's lemma]] (any of several named after [[Carl Friedrich Gauss]])
* [[Small cancellation theory#Greendlinger.27s_lemma|Greendlinger's lemma]]
* [[Itô's lemma]]
* [[Jordan's lemma]]
* [[Lovász local lemma]]
* [[Nakayama's lemma]]
* [[Closed and exact differential forms#Poincar.C3.A9_lemma|Poincaré's lemma]]
* [[Riesz's lemma]]
* [[Schur's lemma]]
* [[Schwarz's lemma]]
* [[Sperner's lemma]]
* [[Urysohn's lemma]]
* [[Vitali covering lemma]]
* [[Yoneda lemma|Yoneda's lemma]]
* [[Zorn's lemma]]
{{div col end}}

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==
{{wiktionary|lemma}}

{{div col|colwidth=30em}}
*[[Axiom]]
*[[Corollary]]
*[[Co-premise]]
*[[Fundamental lemma]]
*[[Inference objection]]
*[[List of lemmas]]
*[[Objection (argument)|Objection]]
*[[Porism]]
*[[Theorem]]
*[[Theorem#Terminology|Theorem terminology]]
{{div col end}}

==Notes==
{{notelist|25em}}

==References==
{{reflist|25em}}

==External links==
*[[Doron Zeilberger]], [http://www.math.rutgers.edu/~zeilberg/Opinion82.html Opinion 82: A Good Lemma is Worth a Thousand Theorems]

{{PlanetMath attribution|id=4492|title=Lemma}}


{{Mathematical logic}}

[[Category:Mathematical terminology]]
[[Category:Lemmas|*]]