Theorem

A theorem is a mathematical statement established by a proof from definitions, axioms, and previously established results. Its proof shows why the statement follows from those assumptions.

Mathematical proof: A deductive argument that establishes a mathematical statement from accepted assumptions and rules of inference. A theorem is established by a proof that derives its conclusion from accepted premises.
Direct proof: A proof that derives a conclusion through a sequence of valid steps from its assumptions. It establishes a theorem by following the assumptions straight to the conclusion.
Proposition (mathematics): A proven mathematical statement, often presented as a result of limited scope. Like a theorem, it is proved; the distinction often reflects presentation rather than logic.
Pythagorean theorem: A theorem stating that a right triangle's hypotenuse squared equals the sum of its legs squared. Its many proofs make a precise geometric relation a classic example of a theorem.
Axiom: A foundational assumption adopted as a starting point in a mathematical theory. Axioms supply premises from which theorems in a theory can be derived.
Proof by contradiction: A proof that assumes a claim is false and derives an inconsistency. It proves a theorem by showing that denying its conclusion conflicts with established facts.
Conjecture: A mathematical statement proposed as true but not yet established by proof. A conjecture becomes a theorem only after a valid proof establishes it.
Fundamental theorem of arithmetic: The theorem that every integer greater than one has a unique prime factorization, up to order. It illustrates how a theorem can establish a structural fact about all positive integers.
Definition: A precise specification of the meaning of a mathematical term or object. Definitions fix the objects and conditions a theorem's statement concerns.
Mathematical induction: A proof method for statements indexed by natural numbers, using a base case and an inductive step. Induction establishes whole families of theorems about natural-number cases.

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