Statements
A theorem, a lemma, a proposition, a corollary, a fact or an axiom.
- Concepts 2778
- Definitions 2634
- Statements 1926
- Conjectures 38
- Problems 54
- Examples 213
- Counterexamples 52
- Objects 99
- Proofs 10
- Computations 1
- Topics 1
Ideas 451–600 of 1926.
| Name | Kind | Subject |
|---|---|---|
| A set is closed iff it equals its closure | Corollary | General topology |
| Integration by substitution | Theorem | Real functions |
| Consecutive Fibonacci numbers are coprime | Theorem | Number theory |
| Rouché's theorem | Theorem | Functions of a complex variable |
| A continuous injective function on an interval is strictly monotone | Theorem | Real functions |
| Polynomial ring over a UFD is a UFD | Theorem | Commutative algebra |
| Zero is not a successor | Axiom | Mathematical logic and foundations |
| The exponential function is convex | Proposition | Real functions |
| Variance of a sum | Proposition | Probability theory and stochastic processes |
| Limits are unique in Hausdorff spaces | Proposition | General topology |
| Uniqueness of inverses | Proposition | Group theory and generalizations |
| Mean of the geometric distribution | Proposition | Probability theory and stochastic processes |
| Matrix of the adjoint is the conjugate transpose | Proposition | Linear and multilinear algebra; matrix theory |
| Strong law of large numbers | Theorem | Probability theory and stochastic processes |
| Finite extensions are algebraic | Proposition | Field theory and polynomials |
| Commutativity of addition | Theorem | Mathematical logic and foundations |
| Cauchy's integral formula | Theorem | Functions of a complex variable |
| Spectral theorem for Hermitian matrices | Theorem | Linear and multilinear algebra; matrix theory |
| Order of a permutation is the lcm of its cycle lengths | Proposition | Group theory and generalizations |
| Estimation lemma | Lemma | Functions of a complex variable |
| Cosets have the same cardinality as the subgroup | Proposition | Group theory and generalizations |
| Basel problem | Theorem | Sequences, series, summability |
| K[x] is a Euclidean domain | Theorem | Commutative algebra |
| Universal property of the polynomial ring | Theorem | Commutative algebra |
| Bézout's identity | Theorem | Number theory |
| Inequality of arithmetic and geometric means | Theorem | Real functions |
| The rationals are countable | Theorem | Mathematical logic and foundations |
| Stone's representation theorem for finite Boolean algebras | Theorem | Order, lattices, ordered algebraic structures |
| Irrationality of the square root of 2 | Theorem | Number theory |
| Number of nth roots of unity | Theorem | Number theory |
| Jordan–Hölder theorem | Theorem | Group theory and generalizations |
| Primitive root theorem | Theorem | Number theory |
| Orbit-stabiliser theorem | Theorem | Group theory and generalizations |
| Dimension of the orthogonal complement | Corollary | Linear and multilinear algebra; matrix theory |
| Law of total probability | Theorem | Probability theory and stochastic processes |
| Convergents converge to the number | Theorem | Number theory |
| Fermat's Last Theorem for n = 4 | Theorem | Number theory |
| Riemann rearrangement theorem | Theorem | Sequences, series, summability |
| Exponential of a sum | Theorem | Special functions |
| Harmonic series diverges | Theorem | Sequences, series, summability |
| Chain rule in several variables | Theorem | Real functions |
| A composite number has a prime factor at most its square root | Proposition | Number theory |
| A finite integral domain is a field | Theorem | Commutative algebra |
| The multiplicative group of a finite field is cyclic | Corollary | Field theory and polynomials |
| Existence of the infimum | Theorem | Real functions |
| Neyman–Pearson lemma | Lemma | Statistics |
| Rule of sum | Theorem | Combinatorics |
| Boole's inequality | Theorem | Probability theory and stochastic processes |
| Prime number theorem | Theorem | Number theory |
| Liouville's theorem | Theorem | Functions of a complex variable |
| Binomial theorem | Theorem | Combinatorics |
| Universal property of the abelianisation | Theorem | Group theory and generalizations |
| Epsilon-delta continuity in metric spaces | Proposition | General topology |
| Existence of nontrivial solutions of Pell's equation | Theorem | Number theory |
| Inverse in terms of the adjugate | Corollary | Linear and multilinear algebra; matrix theory |
| Continuous image of a connected set is connected | Proposition | General topology |
| Convergence in probability implies convergence in distribution | Theorem | Probability theory and stochastic processes |
| Central limit theorem | Theorem | Probability theory and stochastic processes |
| QR decomposition | Theorem | Linear and multilinear algebra; matrix theory |
| Divisibility by 9 | Proposition | Number theory |
| Existence of an algebraic closure | Theorem | Field theory and polynomials |
| Gershgorin circle theorem | Theorem | Linear and multilinear algebra; matrix theory |
| Poincaré–Bendixson theorem | Theorem | Ordinary differential equations |
| Impossibility of trisecting the angle | Theorem | Field theory and polynomials |
| The exponential function is its own derivative | Theorem | Special functions |
| Cantor's intersection theorem | Theorem | General topology |
| Cauchy's integral formula for derivatives | Theorem | Functions of a complex variable |
| Prime elements are irreducible in an integral domain | Proposition | Commutative algebra |
| Superposition principle | Proposition | Ordinary differential equations |
| Bounded linear maps are continuous | Theorem | Operator theory |
| Derivative vanishes at a local extremum | Theorem | Real functions |
| Divergence of a curl is zero | Proposition | Real functions |
| g to the n equals 1 iff the order divides n | Proposition | Group theory and generalizations |
| Cayley–Hamilton theorem | Theorem | Linear and multilinear algebra; matrix theory |
| Closed subsets of complete spaces are complete | Proposition | General topology |
| The gamma function extends the factorial | Theorem | Special functions |
| A subgroup whose index is the smallest prime divisor of the order is normal | Theorem | Group theory and generalizations |
| e as the limit of (1 + 1/n)^n | Proposition | Real functions |
| Universal property of the quotient topology | Theorem | General topology |
| Inclusion–exclusion principle | Theorem | Combinatorics |
| The reals form a field | Theorem | Field theory and polynomials |
| Characterization of bridges | Theorem | Combinatorics |
| Epsilon-delta uniform continuity | Proposition | General topology |
| Chromatic number of a complete bipartite graph | Proposition | Combinatorics |
| Bertrand's postulate | Theorem | Number theory |
| Triangle inequality for norms | Proposition | Functional analysis |
| Simultaneous diagonalisation | Theorem | Linear and multilinear algebra; matrix theory |
| Monotone convergence theorem for sequences | Theorem | Sequences, series, summability |
| Burnside's p^a q^b theorem | Theorem | Group theory and generalizations |
| Chinese remainder theorem | Theorem | Number theory |
| Lévy's continuity theorem | Theorem | Probability theory and stochastic processes |
| Inverse of a complex number | Proposition | Field theory and polynomials |
| Comparison test for improper integrals | Theorem | Sequences, series, summability |
| Heron's formula | Theorem | Geometry |
| The complex numbers form a field | Theorem | Field theory and polynomials |
| Trace is a coefficient of the characteristic polynomial | Proposition | Linear and multilinear algebra; matrix theory |
| Infinite pigeonhole principle | Theorem | Combinatorics |
| Determinant of a matrix with a zero row | Proposition | Linear and multilinear algebra; matrix theory |
| Bertrand's ballot theorem | Theorem | Combinatorics |
| Reversing a path changes the sign of a line integral | Proposition | Real functions |
| Multiplicativity of the totient function | Theorem | Number theory |
| Second derivative test | Proposition | Real functions |
| Transcendence of pi | Theorem | Number theory |
| Sylvester's determinant theorem | Theorem | Linear and multilinear algebra; matrix theory |
| The complex exponential is entire | Proposition | Special functions |
| Continuous dependence on initial data | Theorem | Ordinary differential equations |
| Automorphism group of a cyclic group | Theorem | Group theory and generalizations |
| Derivative of the inverse function | Proposition | Real functions |
| Cauchy sequences are bounded | Proposition | Sequences, series, summability |
| Uniqueness of the complete ordered field | Theorem | Field theory and polynomials |
| Inscribed angle theorem | Theorem | Geometry |
| Derivative of arcsine | Proposition | Special functions |
| Epsilon-delta definition of a limit | Proposition | Real functions |
| Induction from an arbitrary base case | Theorem | Mathematical logic and foundations |
| General solution of an inhomogeneous linear equation | Theorem | Ordinary differential equations |
| Universal property of the quotient group | Theorem | Group theory and generalizations |
| Cancellation in an integral domain | Proposition | Commutative algebra |
| Trace is linear | Proposition | Linear and multilinear algebra; matrix theory |
| Addition formulas for hyperbolic functions | Proposition | Special functions |
| Solution space of a homogeneous linear equation has dimension n | Theorem | Ordinary differential equations |
| Absolute value is multiplicative | Proposition | Real functions |
| Nilpotent groups are solvable | Proposition | Group theory and generalizations |
| Dirichlet's theorem on arithmetic progressions | Theorem | Number theory |
| Cardinality of the power set | Proposition | Mathematical logic and foundations |
| The medians meet at the centroid | Theorem | Geometry |
| Quotient of continuous functions is continuous | Proposition | Real functions |
| Angles in the same segment are equal | Corollary | Geometry |
| Thales's theorem | Theorem | Geometry |
| Cavalieri's principle | Theorem | Measure and integration |
| Bijection principle | Theorem | Combinatorics |
| Fundamental theorem of symmetric polynomials | Theorem | Combinatorics |
| Integral of sine | Proposition | Real functions |
| Units of Z/nZ are the residues coprime to n | Proposition | Number theory |
| Second derivative test in several variables | Theorem | Calculus of variations and optimal control; optimization |
| Divergence theorem on a rectangle in the plane | Proposition | Real functions |
| The order of a finite field is a prime power | Theorem | Field theory and polynomials |
| Taylor's theorem with the Lagrange remainder | Theorem | Approximations and expansions |
| Changing the order of integration | Proposition | Real functions |
| Product of power series | Theorem | Commutative algebra |
| Every finite tree with at least two vertices has a leaf | Lemma | Combinatorics |
| Rule of Sarrus | Proposition | Linear and multilinear algebra; matrix theory |
| Interior is the largest open subset | Proposition | General topology |
| Induction on the integers | Theorem | Number theory |
| Union of open sets is open | Proposition | General topology |
| Russell's paradox | Theorem | Mathematical logic and foundations |
| Pigeonhole principle | Theorem | Combinatorics |
| Solutions for distinct real roots | Theorem | Ordinary differential equations |
| Second isomorphism theorem for rings | Theorem | Associative rings and algebras |
| A sequence converges iff its liminf and limsup agree | Proposition | Sequences, series, summability |
| Eigenvectors with distinct eigenvalues are linearly independent | Theorem | Linear and multilinear algebra; matrix theory |