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 901–1050 of 1926.
| Name | Kind | Subject |
|---|---|---|
| Degree of a product | Proposition | Commutative algebra |
| Power series of arctangent | Theorem | Functions of a complex variable |
| The cross product is perpendicular to its factors | Proposition | Linear and multilinear algebra; matrix theory |
| First Borel–Cantelli lemma | Lemma | Probability theory and stochastic processes |
| Cauchy–Riemann equations | Theorem | Functions of a complex variable |
| Integration in polar coordinates | Theorem | Real functions |
| The characteristic function determines the distribution | Theorem | Probability theory and stochastic processes |
| Existence of finite fields of every prime power order | Theorem | Field theory and polynomials |
| Axiom of extensionality | Axiom | Mathematical logic and foundations |
| The successor function is injective | Axiom | Mathematical logic and foundations |
| Dirac's theorem on Hamiltonian cycles | Theorem | Combinatorics |
| C^k function characterised by continuous derivatives | Proposition | Real functions |
| Product of the gcd and the lcm | Proposition | Number theory |
| Remainder theorem | Theorem | Commutative algebra |
| Reduction of a higher-order equation to a first-order system | Proposition | Ordinary differential equations |
| Third isomorphism theorem | Theorem | Group theory and generalizations |
| Term-by-term differentiation | Theorem | Sequences, series, summability |
| Bounded derivative on a convex set implies Lipschitz | Corollary | Real functions |
| An algebraic extension of an algebraic extension is algebraic | Theorem | Field theory and polynomials |
| Product of a subgroup and a normal subgroup is a subgroup | Proposition | Group theory and generalizations |
| Characteristic polynomial does not depend on the basis | Proposition | Linear and multilinear algebra; matrix theory |
| Surjective iff it has a right inverse | Proposition | Mathematical logic and foundations |
| Row rank equals column rank | Theorem | Linear and multilinear algebra; matrix theory |
| Variance of a sum of independent random variables | Theorem | Probability theory and stochastic processes |
| Sign of a cycle | Proposition | Group theory and generalizations |
| Frobenius coin problem | Theorem | Number theory |
| Derivative of cosine | Theorem | Special functions |
| Intersection of subspaces | Proposition | Linear and multilinear algebra; matrix theory |
| Continuous real function on a compact set attains its bounds | Theorem | General topology |
| Fibonacci numbers as sums of binomial coefficients | Theorem | Number theory |
| Subgroups of index two are normal | Proposition | Group theory and generalizations |
| Logarithm of a power | Corollary | Special functions |
| First Sylow theorem | Theorem | Group theory and generalizations |
| Product rule | Theorem | Real functions |
| Dirichlet's test | Theorem | Sequences, series, summability |
| Multiplicativity of the Möbius function | Proposition | Number theory |
| Kummer's theorem | Theorem | Number theory |
| Euler's theorem | Theorem | Number theory |
| Morera's theorem | Theorem | Functions of a complex variable |
| Volume of a box | Proposition | Measure and integration |
| Every polynomial has a root in some extension field | Theorem | Field theory and polynomials |
| Product of continuous functions is continuous | Proposition | Real functions |
| Continuity is equivalent to sequential continuity in metric spaces | Theorem | General topology |
| Abel's identity | Theorem | Ordinary differential equations |
| Subsequences of a convergent sequence converge to the same limit | Proposition | Sequences, series, summability |
| Finite-dimensional spaces are isomorphic iff they have equal dimension | Theorem | Linear and multilinear algebra; matrix theory |
| De Morgan's laws for sets | Theorem | Mathematical logic and foundations |
| Size of a conjugacy class is the index of the centraliser | Proposition | Group theory and generalizations |
| Banach fixed point theorem | Theorem | Operator theory |
| Lagrange multipliers | Theorem | Operations research, mathematical programming |
| Cartan–Dieudonné theorem | Theorem | Geometry |
| If 2^n + 1 is prime then n is a power of 2 | Proposition | Number theory |
| Polynomials of degree two or three are irreducible iff they have no roots | Proposition | Field theory and polynomials |
| Pythagorean theorem | Theorem | Geometry |
| Rule of product | Theorem | Combinatorics |
| A tree on n vertices has n − 1 edges | Theorem | Combinatorics |
| Cancellation laws | Proposition | Group theory and generalizations |
| Minimal polynomial divides every annihilating polynomial | Proposition | Linear and multilinear algebra; matrix theory |
| Order of a power | Proposition | Group theory and generalizations |
| Term test | Proposition | Sequences, series, summability |
| Euler line | Theorem | Geometry |
| Gradient is orthogonal to level sets | Proposition | Real functions |
| Bézout's identity in a Euclidean domain | Theorem | Commutative algebra |
| Compact closed unit ball implies finite dimension | Theorem | Functional analysis |
| Continuous functions on a closed bounded interval are uniformly continuous | Theorem | Real functions |
| Euler's formula for planar graphs | Theorem | Combinatorics |
| Line integral as an integral over the parameter interval | Theorem | Real functions |
| Complete subspaces are closed | Proposition | General topology |
| Rank–nullity theorem | Theorem | Linear and multilinear algebra; matrix theory |
| Irrationality of e | Theorem | Number theory |
| Law of total probability for an event and its complement | Corollary | Probability theory and stochastic processes |
| Epsilon-delta definition of continuity | Proposition | Real functions |
| Chinese remainder theorem for rings | Theorem | Commutative algebra |
| Time translation of solutions of an autonomous system | Proposition | Ordinary differential equations |
| Trace of a linear map is the trace of its matrix | Proposition | Linear and multilinear algebra; matrix theory |
| R/M is a field iff M is maximal | Theorem | Commutative algebra |
| Euclid–Euler theorem | Theorem | Number theory |
| Sylvester's law of inertia | Theorem | Linear and multilinear algebra; matrix theory |
| Newton's identities | Theorem | Combinatorics |
| A subspace of full dimension is the whole space | Proposition | Linear and multilinear algebra; matrix theory |
| There is a natural number n with 1/n below any positive epsilon | Corollary | Real functions |
| Taylor's theorem with the integral remainder | Theorem | Approximations and expansions |
| Dimension theorem for vector spaces | Theorem | Linear and multilinear algebra; matrix theory |
| Sylvester's criterion | Theorem | Linear and multilinear algebra; matrix theory |
| Cholesky decomposition | Theorem | Linear and multilinear algebra; matrix theory |
| Product of exponentials | Theorem | Special functions |
| Volume of a ball in R^3 | Proposition | Real functions |
| A prime is a Gaussian prime if and only if it is 3 modulo 4 | Theorem | Number theory |
| Hoeffding's inequality | Theorem | Probability theory and stochastic processes |
| Moment-generating function of a sum of independent random variables | Proposition | Probability theory and stochastic processes |
| Extreme value theorem | Theorem | Real functions |
| Cantor's theorem | Theorem | Mathematical logic and foundations |
| Gram–Schmidt vectors are orthogonal | Proposition | Linear and multilinear algebra; matrix theory |
| A monotone function has countably many discontinuities | Theorem | Real functions |
| Number of injections | Theorem | Combinatorics |
| Hall's theorem for bipartite graphs | Theorem | Combinatorics |
| Countable intersection of dense open sets is dense | Proposition | General topology |
| The exterior derivative squares to zero | Theorem | Global analysis, analysis on manifolds |
| Tietze extension theorem | Theorem | General topology |
| Impossibility of squaring the circle | Theorem | Field theory and polynomials |
| Continuity of probability measures | Proposition | Probability theory and stochastic processes |
| Uniform limit theorem | Theorem | Sequences, series, summability |
| Handshaking lemma | Theorem | Combinatorics |
| Wallis product | Theorem | Number theory |
| Number of elements of each order in a cyclic group | Theorem | Group theory and generalizations |
| Monotonicity of the cumulative distribution function | Proposition | Probability theory and stochastic processes |
| Multinomial theorem | Theorem | Combinatorics |
| Euler's partition theorem | Theorem | Combinatorics |
| Decomposition of a permutation into transpositions | Theorem | Group theory and generalizations |
| Poisson limit theorem | Theorem | Probability theory and stochastic processes |
| Squeeze theorem | Theorem | Sequences, series, summability |
| Dini's theorem | Theorem | Sequences, series, summability |
| Eisenstein's lemma | Lemma | Number theory |
| The principal logarithm is holomorphic off the branch cut | Proposition | Special functions |
| A root of an integer is an integer or irrational | Theorem | Number theory |
| Second fundamental theorem of calculus | Theorem | Real functions |
| De Morgan's laws in a Boolean algebra | Proposition | Order, lattices, ordered algebraic structures |
| Absolutely convergent series converge | Theorem | Sequences, series, summability |
| Catalan recurrence | Theorem | Combinatorics |
| Gauss–Wantzel theorem | Theorem | Field theory and polynomials |
| The order modulo p divides p − 1 | Corollary | Number theory |
| Trace of a product is symmetric | Proposition | Linear and multilinear algebra; matrix theory |
| Brooks' theorem | Theorem | Combinatorics |
| Fermat's Last Theorem | Theorem | Number theory |
| Tail sum formula for expectation | Proposition | Probability theory and stochastic processes |
| Existence of an orthonormal basis | Theorem | Linear and multilinear algebra; matrix theory |
| Distinct eigenvalues imply diagonalisable | Corollary | Linear and multilinear algebra; matrix theory |
| Binary trees are counted by Catalan numbers | Theorem | Combinatorics |
| Comparability of cardinals | Theorem | Mathematical logic and foundations |
| Identity theorem | Theorem | Functions of a complex variable |
| Term-by-term integration | Theorem | Sequences, series, summability |
| Matrix exponential as the solution of a linear system | Theorem | Ordinary differential equations |
| Universal property of the tensor product | Theorem | Linear and multilinear algebra; matrix theory |
| Continuity via open sets | Theorem | General topology |
| Sum of independent Poisson random variables | Proposition | Probability theory and stochastic processes |
| Irrationality of the square root of a prime | Theorem | Number theory |
| Hilbert's basis theorem | Theorem | Commutative algebra |
| Existence of nth roots of complex numbers | Theorem | Field theory and polynomials |
| Power series are holomorphic in the disc of convergence | Theorem | Functions of a complex variable |
| Fundamental theorem of contour integration | Theorem | Functions of a complex variable |
| Intersecting chords theorem | Theorem | Geometry |
| Pascal's rule | Theorem | Combinatorics |
| Closed balls are closed | Proposition | General topology |
| Homotopy invariance of line integrals of closed forms | Theorem | Real functions |
| Decomposition of a permutation into disjoint cycles | Theorem | Group theory and generalizations |
| Quadratic reciprocity for the Jacobi symbol | Theorem | Number theory |
| Mercator series | Theorem | Functions of a complex variable |
| Power series of complex cosine | Proposition | Special functions |
| Diagonalisation of a quadratic form | Theorem | Number theory |
| Decomposition into generalised eigenspaces | Theorem | Linear and multilinear algebra; matrix theory |