| Product of primitive polynomials is primitive | Theorem | Commutative algebra |
| Existence of an eigenvalue over an algebraically closed field | Theorem | Linear and multilinear algebra; matrix theory |
| Product of cyclic groups is cyclic iff the orders are coprime | Theorem | Group theory and generalizations |
| Homomorphisms preserve identity and inverses | Proposition | Group theory and generalizations |
| Product of countable sets is countable | Proposition | Mathematical logic and foundations |
| Tutte's theorem | Theorem | Combinatorics |
| Distributive laws for sets | Theorem | Mathematical logic and foundations |
| Product formula for the order of HK | Proposition | Group theory and generalizations |
| Picard's great theorem | Theorem | Functions of a complex variable |
| Independence means the joint law is the product of the marginals | Proposition | Probability theory and stochastic processes |
| Number of functions between finite sets | Theorem | Combinatorics |
| Order of an element divides the order of the group | Corollary | Group theory and generalizations |
| Power series of cosine | Theorem | Special functions |
| Hall's marriage theorem | Theorem | Combinatorics |
| Cauchy–Binet formula | Theorem | Linear and multilinear algebra; matrix theory |
| Euler's four-square identity | Proposition | Number theory |
| An iterate of the Picard operator is a contraction | Lemma | Ordinary differential equations |
| Integration by parts | Theorem | Real functions |
| Subfields of a finite field | Theorem | Field theory and polynomials |
| Steinitz exchange lemma | Lemma | Linear and multilinear algebra; matrix theory |
| An injective endomorphism of a finite-dimensional space is surjective | Theorem | Linear and multilinear algebra; matrix theory |
| Euler's identity | Corollary | Special functions |
| Minimal polynomial divides the characteristic polynomial | Corollary | Linear and multilinear algebra; matrix theory |
| In a PID a nonzero element is irreducible iff it generates a maximal ideal | Theorem | Commutative algebra |
| Unique paths in trees | Theorem | Combinatorics |
| Geometric solutions come from roots of the characteristic polynomial | Theorem | Difference and functional equations |
| Linearly independent set of the right size is a basis | Proposition | Linear and multilinear algebra; matrix theory |
| Cramer's rule | Theorem | Linear and multilinear algebra; matrix theory |
| Riesz's lemma | Lemma | Functional analysis |
| Intermediate value theorem for connected spaces | Theorem | General topology |
| Fixed points of a p-group action | Lemma | Group theory and generalizations |
| Peano existence theorem | Theorem | Ordinary differential equations |
| Power series of the complex logarithm | Proposition | Functions of a complex variable |
| Area of a region as a line integral | Corollary | Real functions |
| Open sets characterised by neighbourhoods | Proposition | General topology |
| Coordinates in an orthonormal basis | Proposition | Linear and multilinear algebra; matrix theory |
| Trace of a transpose | Proposition | Linear and multilinear algebra; matrix theory |
| Epsilon-N definition of the limit of a sequence | Proposition | Sequences, series, summability |
| Second Sylow theorem | Theorem | Group theory and generalizations |
| Transpose of a product | Proposition | Linear and multilinear algebra; matrix theory |
| Closed sets contain their limits | Proposition | General topology |
| Completeness of the space of bounded continuous functions | Theorem | General topology |
| Euler's polyhedron formula | Theorem | Convex and discrete geometry |
| Sum of the totient over the divisors | Theorem | Number theory |
| Groups of order p squared are abelian | Theorem | Group theory and generalizations |
| Ring homomorphisms from a field are injective | Proposition | Field theory and polynomials |
| Finite intersection of open sets is open | Proposition | General topology |
| De Morgan's laws | Theorem | Mathematical logic and foundations |
| Mean of the normal distribution | Proposition | Probability theory and stochastic processes |
| Liouville's formula | Theorem | Ordinary differential equations |
| Cancellation in congruences | Proposition | Number theory |
| The rationals form an ordered field | Theorem | Field theory and polynomials |
| Every walk contains a path | Proposition | Combinatorics |
| Permuting rows multiplies the determinant by the sign | Proposition | Linear and multilinear algebra; matrix theory |
| Divisibility by 11 | Proposition | Number theory |
| Intersection of closed sets is closed | Proposition | General topology |
| The number of Sylow subgroups is the index of the normaliser | Proposition | Group theory and generalizations |
| Product of compact sets is compact | Theorem | General topology |
| Determinant of a matrix exponential | Theorem | Linear and multilinear algebra; matrix theory |
| Open sets of a metric space | Proposition | General topology |
| The reals are Archimedean | Theorem | Real functions |
| Derivative gives the directional derivatives | Proposition | Real functions |
| Archimedean property | Proposition | Field theory and polynomials |
| Baire category theorem | Theorem | General topology |
| Tower law | Theorem | Field theory and polynomials |
| Chain rule for a composition with a curve | Corollary | Real functions |
| Schröder–Bernstein theorem | Theorem | Mathematical logic and foundations |
| Finite-dimensional normed spaces are complete | Theorem | Functional analysis |
| Metrizable spaces are normal | Proposition | General topology |
| If G modulo its centre is cyclic then G is abelian | Proposition | Group theory and generalizations |
| Stokes' theorem | Theorem | Real functions |
| Gram matrices are positive semidefinite | Proposition | Linear and multilinear algebra; matrix theory |
| Cauchy product of absolutely convergent series | Theorem | Sequences, series, summability |
| Second isomorphism theorem for vector spaces | Theorem | Linear and multilinear algebra; matrix theory |
| The symmetric group is generated by transpositions | Proposition | Group theory and generalizations |
| Differentiation under the integral sign | Theorem | Real functions |
| Simplicity of the alternating group | Theorem | Group theory and generalizations |
| The Möbius function is the inverse of the constant function 1 | Theorem | Number theory |
| Boundedness theorem | Theorem | Real functions |
| R/P is an integral domain iff P is prime | Theorem | Commutative algebra |
| Totient of a prime power | Proposition | Number theory |
| K[x]/(f) is a field iff f is irreducible | Theorem | Field theory and polynomials |
| Completeness axiom | Axiom | Real functions |
| Moment-generating function of the normal distribution | Proposition | Probability theory and stochastic processes |
| Curl-free vector field on a convex set is conservative | Theorem | Real functions |
| Open balls are open | Proposition | General topology |
| R[x] is an integral domain iff R is | Theorem | Commutative algebra |
| Legendre's three-square theorem | Theorem | Number theory |
| Continuity via closed sets | Proposition | General topology |
| Cauchy condensation test | Theorem | Sequences, series, summability |
| Connected subsets of the real line are intervals | Theorem | General topology |
| Mean value theorem | Theorem | Real functions |
| Matrix determinant lemma | Theorem | Linear and multilinear algebra; matrix theory |
| Order of the Galois group of a Galois extension equals its degree | Theorem | Field theory and polynomials |
| Solutions for complex conjugate roots | Theorem | Ordinary differential equations |
| Global existence for globally Lipschitz vector fields | Theorem | Ordinary differential equations |
| Weak law of large numbers | Theorem | Probability theory and stochastic processes |
| Number of k-element subsets | Theorem | Combinatorics |
| Path independence of line integrals | Theorem | Real functions |
| A linear map is injective iff its kernel is trivial | Proposition | Linear and multilinear algebra; matrix theory |
| Mertens' theorem on Cauchy products | Theorem | Sequences, series, summability |
| Hölder's inequality for sums | Theorem | Real functions |
| Fermat's little theorem in the form a^p = a | Corollary | Number theory |
| Trees are bipartite | Proposition | Combinatorics |
| Side-side-side congruence | Theorem | Geometry |
| Goursat's theorem | Theorem | Functions of a complex variable |
| Polar form | Theorem | Functions of a complex variable |
| Quotient rule | Theorem | Real functions |
| Multiplication rule | Proposition | Probability theory and stochastic processes |
| Riemann's theorem on removable singularities | Theorem | Functions of a complex variable |
| Pythagorean theorem in an inner product space | Theorem | Functional analysis |
| Uniform limits commute with integration | Theorem | Sequences, series, summability |
| Power series converge uniformly on smaller discs inside the radius of convergence | Theorem | Functions of a complex variable |
| Ring of integers | Theorem | Commutative algebra |
| Implicit function theorem | Theorem | Real functions |
| Slopes of a convex function are increasing | Proposition | Real functions |
| Generalized pigeonhole principle | Theorem | Combinatorics |
| Zero derivative on a connected open set implies constant | Theorem | Real functions |
| Addition rule for two events | Proposition | Probability theory and stochastic processes |
| Order of the dihedral group | Proposition | Group theory and generalizations |
| Recurrence for Stirling numbers of the second kind | Theorem | Number theory |
| Laplace expansion along a row | Theorem | Linear and multilinear algebra; matrix theory |
| Linear maps on finite-dimensional spaces are continuous | Theorem | Operator theory |
| Determinant is the product of the eigenvalues | Theorem | Linear and multilinear algebra; matrix theory |
| Associativity of addition | Theorem | Mathematical logic and foundations |
| Euler product formula | Theorem | Number theory |
| Eigenvalues of a self-adjoint operator are real | Theorem | Operator theory |
| Inclusion–exclusion for two sets | Theorem | Combinatorics |
| Recurrence for Stirling numbers of the first kind | Theorem | Number theory |
| Number of subsets of a finite set | Theorem | Combinatorics |
| Polynomial functions are continuous | Proposition | Real functions |
| Reverse triangle inequality | Corollary | Real functions |
| The kernel is a normal subgroup | Proposition | Group theory and generalizations |
| A finite field of order q is the splitting field of x^q - x | Theorem | Field theory and polynomials |
| Degrees along an Eulerian trail | Theorem | Combinatorics |
| Universal property of the free group | Theorem | Group theory and generalizations |
| Number of permutations | Theorem | Combinatorics |
| Classification of complex quadratic forms | Theorem | Number theory |
| exp(2 pi i/n) is a primitive root of unity | Proposition | Number theory |
| Division algorithm | Theorem | Number theory |
| Finite subgroups of the multiplicative group of a field are cyclic | Theorem | Field theory and polynomials |
| Abel's test | Theorem | Sequences, series, summability |
| Impossibility of doubling the cube | Theorem | Field theory and polynomials |
| Uniform convergence of derivatives allows term-by-term differentiation | Theorem | Sequences, series, summability |
| Exponential growth beats polynomial growth | Theorem | Real functions |
| First isomorphism theorem | Theorem | Group theory and generalizations |
| Linearity of the derivative | Proposition | Real functions |
| Derivative of a continuous linear map | Proposition | Real functions |
| Fibonacci recurrence | Fact | Number theory |
| Correspondence theorem for rings | Theorem | Associative rings and algebras |
