| The integers are a Euclidean domain | Theorem | Number theory |
| Sequential characterization of limits | Theorem | Real functions |
| Singular value decomposition | Theorem | Linear and multilinear algebra; matrix theory |
| Power series are differentiable inside the radius of convergence | Theorem | Functions of a complex variable |
| A function with zero derivative is constant | Corollary | Real functions |
| Closure is the set of limits of sequences | Proposition | General topology |
| A differentiable function is convex iff its derivative is increasing | Theorem | Real functions |
| Taylor's theorem with the Cauchy remainder | Theorem | Approximations and expansions |
| The characteristic of an integral domain is zero or prime | Theorem | Commutative algebra |
| Dimension of the dual space | Proposition | Linear and multilinear algebra; matrix theory |
| Scaling a row scales the determinant | Proposition | Linear and multilinear algebra; matrix theory |
| Stars and bars | Theorem | Combinatorics |
| Picard–Lindelöf theorem in integral form | Theorem | Ordinary differential equations |
| Finite intersection property | Proposition | General topology |
| Addition formula for Fibonacci numbers | Theorem | Number theory |
| The logarithm grows slower than any power | Theorem | Real functions |
| Laws of exponents in a group | Proposition | Group theory and generalizations |
| −1 is a square modulo p if and only if p is not 3 modulo 4 | Corollary | Number theory |
| Congruence is compatible with addition | Proposition | Number theory |
| Intermediate value theorem | Theorem | Real functions |
| Modulus of the conjugate | Proposition | Functions of a complex variable |
| Every vector space has a basis | Theorem | Linear and multilinear algebra; matrix theory |
| Change of variables in multiple integrals | Theorem | Real functions |
| Derivative of sine | Theorem | Special functions |
| Divergence of the sum of the reciprocals of the primes | Theorem | Number theory |
| Ramsey's theorem | Theorem | Combinatorics |
| A finite group is nilpotent iff it is the product of its Sylow subgroups | Theorem | Group theory and generalizations |
| Volume of a ball in R^n | Theorem | Measure and integration |
| Additivity of line integrals under concatenation | Proposition | Real functions |
| Uniqueness of finite fields of a given order | Theorem | Field theory and polynomials |
| Hensel's lemma | Theorem | Number theory |
| Recurrence for Bell numbers | Theorem | Number theory |
| A convex function on an open interval is continuous | Theorem | Real functions |
| Jordan–Chevalley decomposition | Theorem | Linear and multilinear algebra; matrix theory |
| 1 + x is at most exp x | Proposition | Special functions |
| exp z = 1 iff z is an integer multiple of 2πi | Proposition | Special functions |
| Existence of splitting fields | Theorem | Field theory and polynomials |
| Chernoff bound | Theorem | Probability theory and stochastic processes |
| Cayley's theorem | Theorem | Group theory and generalizations |
| Integral equation form of an initial value problem | Proposition | Ordinary differential equations |
| Leibniz formula for determinants | Theorem | Linear and multilinear algebra; matrix theory |
| Riemann's criterion for integrability | Theorem | Real functions |
| Dyck words are counted by Catalan numbers | Theorem | Combinatorics |
| Rolle's theorem | Theorem | Real functions |
| Residue at a simple pole | Proposition | Functions of a complex variable |
| Derivative of the complex exponential | Proposition | Special functions |
| Integral test | Theorem | Sequences, series, summability |
| First derivative test | Proposition | Real functions |
| cos pi = -1 | Proposition | Special functions |
| Distance from a point to a line | Proposition | Geometry |
| Chapman–Kolmogorov equation | Theorem | Probability theory and stochastic processes |
| Well-ordering principle | Theorem | Mathematical logic and foundations |
| The complex numbers cannot be ordered | Theorem | Field theory and polynomials |
| Normality via left and right cosets | Proposition | Group theory and generalizations |
| Determinant of a scalar multiple | Proposition | Linear and multilinear algebra; matrix theory |
| Extensions of solvable groups are solvable | Proposition | Group theory and generalizations |
| Urysohn's lemma | Lemma | General topology |
| Composition of injections is injective | Proposition | Mathematical logic and foundations |
| Equivalence classes form a partition | Theorem | Mathematical logic and foundations |
| Vandermonde's identity | Theorem | Combinatorics |
| Triangle inequality for a metric | Axiom | General topology |
| Cauchy's integral theorem | Theorem | Functions of a complex variable |
| sin x / x tends to 1 | Proposition | Special functions |
| Sum of the first n natural numbers | Theorem | Combinatorics |
| A graph is bipartite iff it has no odd cycle | Theorem | Combinatorics |
| Symmetry of second derivatives | Theorem | Real functions |
| Bijective iff it has a two-sided inverse | Proposition | Mathematical logic and foundations |
| Uniqueness of the algebraic closure | Theorem | Field theory and polynomials |
| Cantor set is closed | Proposition | General topology |
| Arguments add under multiplication | Theorem | Functions of a complex variable |
| Legendre's formula | Theorem | Number theory |
| Cassini's identity | Theorem | Number theory |
| Adding a multiple of one row to another preserves the determinant | Proposition | Linear and multilinear algebra; matrix theory |
| Law of the unconscious statistician | Theorem | Probability theory and stochastic processes |
| Carathéodory's theorem | Theorem | Convex and discrete geometry |
| Euler's theorem on Eulerian circuits | Theorem | Combinatorics |
| Existence of prime factorization | Theorem | Number theory |
| Darboux and Riemann integrals agree | Theorem | Real functions |
| Uniqueness of solutions for Lipschitz vector fields | Theorem | Ordinary differential equations |
| A function with nonnegative derivative is increasing | Corollary | Real functions |
| Uniformly continuous functions map Cauchy sequences to Cauchy sequences | Proposition | Real functions |
| Parallelogram law | Proposition | Functional analysis |
| Quotients of solvable groups are solvable | Proposition | Group theory and generalizations |
| Angle addition formulas | Theorem | Special functions |
| Euclid's theorem | Theorem | Number theory |
| Holomorphic functions on a disc have primitives | Theorem | Functions of a complex variable |
| Eisenstein's criterion | Theorem | Field theory and polynomials |
| Mean value inequality | Theorem | Real functions |
| Open balls form a basis of the metric topology | Proposition | General topology |
| Rank of a product of matrices | Proposition | Linear and multilinear algebra; matrix theory |
| Casorati–Weierstrass theorem | Theorem | Functions of a complex variable |
| Existence of maximal ideals | Theorem | Commutative algebra |
| Chebyshev's inequality | Theorem | Probability theory and stochastic processes |
| Isometries are injective | Proposition | General topology |
| Characteristic function of a sum of independent random variables | Proposition | Probability theory and stochastic processes |
| Power series of sine | Theorem | Special functions |
| The complex exponential is 2πi-periodic | Proposition | Special functions |
| Sum of the roots of unity | Proposition | Number theory |
| Cauchy mean value theorem | Theorem | Real functions |
| Closure is closed | Proposition | General topology |
| Decreasing sequence bounded below converges | Corollary | Sequences, series, summability |
| Riemann mapping theorem | Theorem | Functions of a complex variable |
| K5 and K3,3 are not planar | Proposition | Combinatorics |
| The real part of a holomorphic function is harmonic | Proposition | Potential theory |
| A tangent is perpendicular to the radius | Theorem | Geometry |
| Order of a cyclic subgroup equals the order of its generator | Proposition | Group theory and generalizations |
| Residue theorem | Theorem | Functions of a complex variable |
| Factorial formula for binomial coefficients | Theorem | Combinatorics |
| Formula for the modulus | Proposition | Functions of a complex variable |
| Triangularisation | Theorem | Linear and multilinear algebra; matrix theory |
| Gauss's lemma on quadratic residues | Lemma | Number theory |
| Leibniz formula for pi | Theorem | Sequences, series, summability |
| Arzelà–Ascoli theorem | Theorem | General topology |
| Freshman's dream | Theorem | Commutative algebra |
| Taylor's theorem with the Peano remainder | Corollary | Approximations and expansions |
| Generalized Euclid's lemma | Proposition | Number theory |
| Law of total variance | Theorem | Probability theory and stochastic processes |
| Variance of the normal distribution | Proposition | Probability theory and stochastic processes |
| Vector triple product | Proposition | Linear and multilinear algebra; matrix theory |
| Chain rule for Jacobian matrices | Corollary | Real functions |
| Pythagorean trigonometric identity | Theorem | Special functions |
| Triangle inequality | Theorem | Real functions |
| Mean value theorem in several variables | Theorem | Real functions |
| Wilson's theorem | Theorem | Number theory |
| Equivalence of norms in finite dimension | Theorem | Functional analysis |
| de Moivre's formula | Theorem | Special functions |
| Vectors are perpendicular if and only if their dot product is zero | Proposition | Geometry |
| Cauchy–Schwarz inequality for expectations | Theorem | Probability theory and stochastic processes |
| A matrix is invertible iff its determinant is a unit | Theorem | Linear and multilinear algebra; matrix theory |
| Complement principle | Proposition | Combinatorics |
| Taylor's theorem | Theorem | Approximations and expansions |
| Formula for the number of derangements | Theorem | Combinatorics |
| Limit of a quotient of sequences | Proposition | Sequences, series, summability |
| Sum of a finite geometric progression | Theorem | Combinatorics |
| Open mapping theorem | Theorem | Functions of a complex variable |
| C^1 vector fields satisfy the Picard–Lindelöf hypotheses | Proposition | Ordinary differential equations |
| Binet's formula | Theorem | Number theory |
| Lebesgue number lemma | Lemma | General topology |
| Subgroups of a finite cyclic group correspond to divisors of its order | Theorem | Group theory and generalizations |
| Rational canonical form | Theorem | Linear and multilinear algebra; matrix theory |
| Continuous bijection from a compact space to a Hausdorff space is a homeomorphism | Theorem | General topology |
| Uniform limits of holomorphic functions are holomorphic | Theorem | Functions of a complex variable |
| Fermat's theorem on sums of two squares | Theorem | Number theory |
| The derangement numbers count derangements | Theorem | Combinatorics |
| Monotonicity of probability | Proposition | Probability theory and stochastic processes |
| Every real number has a decimal expansion | Theorem | Number theory |
| Parseval's identity | Theorem | Functional analysis |
| Maximum modulus principle | Theorem | Functions of a complex variable |
| Lagrange multiplier with one constraint | Corollary | Operations research, mathematical programming |
| Exponential of a sum of commuting matrices | Theorem | Linear and multilinear algebra; matrix theory |
