In mathematics, a lemma is a proven statement used as a stepping‑stone toward the proof of another statement, typically a theorem. Lemmas are intermediate results that simplify the logical structure of a larger argument by isolating auxiliary facts that are required repeatedly or that clarify the main line of reasoning.
Definition and purpose
A lemma is a proposition that has been demonstrated to be true on the basis of previously established results such as axioms, definitions, or earlier theorems. While a lemma may be of interest in its own right, its primary function is to facilitate the proof of a more significant result. The distinction between a lemma and a theorem is conventional rather than formal; the naming often reflects the author's assessment of the relative importance of the statements.
Typical usage
Mathematical texts frequently introduce lemmas to:
- Break complex proofs into manageable parts.
- Provide reusable components that apply across multiple theorems.
- Highlight technical subtleties that would otherwise obscure the main argument.
Examples of widely cited lemmas include:
- Zorn’s Lemma – an equivalent formulation of the Axiom of Choice, used to prove the existence of maximal elements in partially ordered sets.
- Urysohn’s Lemma – a result in topology that guarantees the existence of continuous functions separating disjoint closed sets in normal spaces.
- Fatou’s Lemma – a fundamental inequality in measure theory concerning the limit inferior of integrals of non‑negative functions.
- Schur’s Lemma – a statement in representation theory about intertwiners of irreducible representations.
Relation to theorems and corollaries
In the hierarchical structure of mathematical literature, a typical progression is:
- Lemma – auxiliary result.
- Theorem – principal result, often proved using one or more lemmas.
- Corollary – immediate consequence of a theorem.
However, the classification is not rigid; a result originally labeled a lemma may later be recognized as a theorem when its importance becomes apparent, and vice versa.
Historical note
The term derives from the Latin lemma, meaning “something taken” or “premise,” reflecting the role of a lemma as a premise adopted for subsequent reasoning. Its mathematical usage dates back to classical works, where authors such as Euclid employed auxiliary propositions without explicitly distinguishing them from theorems.
Formal considerations
From a logical perspective, a lemma is a formula ϕ for which a proof exists within a given formal system. The proof of a theorem τ may contain a sub‑proof that establishes ϕ; that sub‑proof is then identified as a lemma. No additional axiomatic status is conferred upon a lemma beyond that of any other provable statement.
References
Standard mathematical textbooks and research monographs routinely distinguish lemmas from theorems in accordance with the conventions described above.
