# List of lemmas

This following is a list of lemmas (or, "lemmata", i.e. minor theorems, or sometimes intermediate technical results factored out of proofs). See also list of axioms, list of theorems and list of conjectures.

## Algebra

* Abhyankar's lemma
* Aubin–Lions lemma
* Bergman's diamond lemma
* Fitting lemma
* Injective test lemma
* Hua's lemma (exponential sums)
* Krull's separation lemma
* Schanuel's lemma (projective modules)
* Schwartz–Zippel lemma
* Shapiro's lemma
* Stewart–Walker lemma (tensors)
* Whitehead's lemma (Lie algebras)
* Zariski's lemma

### Algebraic geometry

* Abhyankar's lemma
* Fundamental lemma (Langlands program)

### Category theory

* Five lemma
* Horseshoe lemma
* Nine lemma
* Short five lemma
* Snake lemma
* Splitting lemma
* Yoneda lemma

### Linear algebra

* Matrix determinant lemma
* Matrix inversion lemma

### Group theory

* Burnside's lemma also known as the Cauchy–Frobenius lemma
* Frattini's lemma (finite groups)
* Goursat's lemma
* Mautner's lemma (representation theory)
* Ping-pong lemma (geometric group theory)
* Schreier's subgroup lemma
* Schur's lemma (representation theory)
* Zassenhaus lemma

### Polynomials

* Gauss's lemma (polynomials)
* Schwartz–Zippel lemma

### Ring theory and commutative algebra

* Artin–Rees lemma
* Hensel's lemma (commutative rings)
* Nakayama lemma
* Noether's normalization lemma
* Prime avoidance lemma

### Universal algebra

* Jónsson's lemma

## Analysis

* Fekete's lemma
* Fundamental lemma of calculus of variations
* Hopf lemma
* Sard's lemma (singularity theory)
* Stechkin's lemma (functional and numerical analysis)
* Vitali covering lemma (real analysis)
* Watson's lemma

### Complex analysis

* Estimation lemma (contour integrals)
* Hartogs's lemma (several complex variables)
* Jordan's lemma
* Lemma on the Logarithmic derivative
* Schwarz lemma

### Fourier analysis

* Riemann–Lebesgue lemma

### Differential equations

* Borel's lemma (partial differential equations)
* Grönwall's lemma
* Lax–Milgram lemma
* Pugh's closing lemma
* Weyl's lemma (Laplace equation) (partial differential equations)

### Differential forms

* Poincaré lemma of closed and exact differential forms

### Functional analysis

* Cotlar–Stein lemma
* Ehrling's lemma
* Riesz's lemma

### Mathematical series

* Abel's lemma
* Kronecker's lemma

### Numerical analysis

* Bramble–Hilbert lemma
* Céa's lemma

## Applied mathematics

* Danielson–Lanczos lemma (Fourier transforms)
* Farkas's lemma (linear programming)
* Feld–Tai lemma (electromagnetism)
* Little's lemma (queuing theory)
* Finsler's lemma

### Control theory

* Finsler's lemma
* Hautus lemma
* Kalman–Yakubovich–Popov lemma

### Computational complexity theory

* Isolation lemma
* Switching lemma

#### Cryptography

* Forking lemma
* Leftover hash lemma
* Piling-up lemma (linear cryptanalysis)
* Yao's XOR lemma

#### Formal languages

* Interchange lemma
* Newman's lemma (term rewriting)
* Ogden's lemma
* Pumping lemma sometimes called the Bar-Hillel lemma

### Microeconomics

* Hotelling's lemma
* Shephard's lemma

## Combinatorics

* Cousin's lemma (integrals)
* Dickson's lemma
* Littlewood–Offord lemma
* Pólya–Burnside lemma
* Sperner's lemma
* Ky Fan lemma (combinatorial geometry)

### Graph theory

* Berge's lemma
* Counting lemma
* Crossing lemma
* Expander mixing lemma
* Handshaking lemma
* Kelly's lemma
* Kőnig's lemma
* Szemerédi regularity lemma

### Order theory

* Higman's lemma
* Ultrafilter lemma

## Dynamical systems

* Barbalat's lemma
* Kac's lemma (ergodic theory)

## Geometry

* Shadowing lemma
* Big-little-big lemma (mathematics of paper folding)
* Gordan's lemma
* Hilbert's lemma

### Euclidean geometry

* Archimedes's lemmas
* Johnson–Lindenstrauss lemma (Euclidean geometry)

### Hyperbolic geometry

* Margulis lemma

### Metric spaces

* Lebesgue's number lemma (dimension theory)

### Riemannian geometry

* Gauss's lemma (Riemannian geometry)

## Mathematical logic

* Craig interpolation lemma
* Diagonal lemma
* Lindenbaum's lemma
* Mostowski collapse lemma
* Teichmüller–Tukey lemma also known as Tukey's lemma
* Zorn's lemma; equivalent to the axiom of choice

### Set theory

* Covering lemma
* Delta lemma
* Dynkin lemma
* Fodor's lemma
* Fixed-point lemma for normal functions (axiomatic set theory)
* Moschovakis coding lemma
* Rasiowa–Sikorski lemma

## Number theory

* Bézout's lemma
* Dwork's lemma
* Euclid's lemma
* Gauss's lemma
* Hensel's lemma
* Zolotarev's lemma
* Siegel's lemma (Diophantine approximation)

### Analytic number theory

* Hua's lemma
* Vaughan's lemma

### Diophantine equations

* Bhaskara's lemma

### Sieve theory

* Fundamental lemma of sieve theory

## Probability theory

* Borel–Cantelli lemma
* Doob–Dynkin lemma
* Itô's lemma (stochastic calculus)
* Lovász local lemma
* Stein's lemma
* Wald's lemma

### Statistics

* Glivenko–Cantelli lemma
* Neyman–Pearson lemma
* Robbins lemma

### Measure theory

* Factorization lemma
* Fatou's lemma
* Frostman's lemma (geometric measure theory)
* Malliavin's absolute continuity lemma

## Topology

* Lindelöf's lemma
* Urysohn's lemma
* Tube lemma

### Differential topology

* Morse lemma

### Fixed-point theory

* Knaster–Kuratowski–Mazurkiewicz lemma

### Geometric topology

* Dehn's lemma

### Topological groups and semigroups

* Ellis–Numakura lemma (topological semigroups)