What's the difference between theorem, lemma and corollary?

Can anybody explain me what is the basic difference between theorem, lemma and corollary?

We have been using it for a long time but I never paid any attention. I am just curious to know.

• Lemmas are smaller results to be used in a bigger (more important) result. The big result is usually a theorem. Corollaries are special cases of theorems. – Cameron Williams Aug 9 '13 at 5:23
• @CameronWilliams... Oh.... thanks a lot. Your reply in few words cleared my doubt :) – monalisa Aug 9 '13 at 5:24
• I'm glad I could help :) – Cameron Williams Aug 9 '13 at 5:25
• @Cameron is, of course, correct but let me add that lemmas needn't always be small results. Typically, they are but some results such as Urysohn's lemma could also be referred to as theorems. See, e.g., math.stackexchange.com/questions/111428/lemma-vs-theorem for additional discussion. – Amitesh Datta Aug 9 '13 at 5:33
• See also en.wikipedia.org/wiki/List_of_lemmas for a list of famous lemmas. – lhf Nov 25 '14 at 11:31

Lemma is generally used to describe a "helper" fact that is used in the proof of a more significant result.

Significant results are frequently called theorems.

Short, easy results of theorems are called corollaries.

But the words aren't exactly that set in stone.

• so it seems like, the one we don't really need to write down the proof is corollary?? – ArtificiallyIntelligence Sep 22 '18 at 5:53

A lot of authors like to use lemma to mean "small theorem." Often a group of lemmas are used to prove a larger result, a "theorem."

A corollary is something that follows trivially from any one of a theorem, lemma, or other corollary.

However, when it boils down to it, all of these things are equivalent as they denote the truth of a statement.

• I want to stress that calling something a lemma, theorem, or corollary is purely a choice made for organizational purposes. That's probably the most important thing to take away from my answer: a lemma is just as true as a theorem! – parsiad Aug 9 '13 at 5:27

Terence Tao (Analysis I, p. 25, n. 4):

From a logical point of view, there is no difference between a lemma, proposition, theorem, or corollary - they are all claims waiting to be proved. However, we use these terms to suggest different levels of importance and difficulty.

A lemma is an easily proved claim which is helpful for proving other propositions and theorems, but is usually not particularly interesting in its own right.

A proposition is a statement which is interesting in its own right, while

a theorem is a more important statement than a proposition which says something definitive on the subject, and often takes more effort to prove than a proposition or lemma.

A corollary is a quick consequence of a proposition or theorem that was proven recently.

Here is some information from this link:

Theorem — a mathematical statement that is proved using rigorous mathematical reasoning. In a mathematical paper, the term theorem is often reserved for the most important results.

Lemma — a minor result whose sole purpose is to help in proving a theorem. It is a stepping stone on the path to proving a theorem. Very occasionally lemmas can take on a life of their own (Zorn’s lemma, Urysohn’s lemma, Burnside’s lemma, Sperner’s lemma).

Corollary — a result in which the (usually short) proof relies heavily on a given theorem (we often say that “this is a corollary of Theorem A”).

• So a result which relies heavily on a lemma or proposition (but not a theorem) is not a corollary? – ashpool Aug 5 '15 at 5:32
• Copying text verbatim is not a helpful answer. Especially when the source is not mentioned/linked (divisbyzero.com/2008/09/22/…) – Neowizard Aug 2 '16 at 16:57
• It's still quite helpful – tjt263 Aug 20 at 11:48

A lemma are those minor results which are used into proving a definite results of a theorem.

When writing a paper, i usually call lemma a technical result that will be used many times in the remaining part of the paper. The idea is to avoid repeating a similar argument in different proofs. For that reason a lemma is not always worth remembering in itself (and if it is, one can include it in a broader theorem).

lemma: A basic result which are used to prove theorems
theorem :Relatively more important and big result which has to be proved corollary: special case result which intuitively comes from theorem. conjecture:A result which is assumed to be true but still not prove exists. Proposition: A result which is either true or false. Axioms or postulates: A set of statement without proof which is assumed to be true and used building blocks to prove several mathematical theorems and results.