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More formally, proposition B is a corollary of proposition A, if B can be readily deduced from A or is self-evident from its proof. In many cases, a corollary corresponds to a special case of a larger theorem, [4] which makes the theorem easier to use and apply, [5] even though its importance is generally considered to be secondary to that of ...
A set of rules can be used to infer any valid conclusion if it is complete, while never inferring an invalid conclusion, if it is sound. A sound and complete set of rules need not include every rule in the following list, as many of the rules are redundant, and can be proven with the other rules.
The following corollary is also known as Nakayama's lemma, and it is in this form that it most often appears. [ 4 ] Statement 3 : If M {\displaystyle M} is a finitely generated module over R {\displaystyle R} , J ( R ) {\displaystyle J(R)} is the Jacobson radical of R {\displaystyle R} , and J ( R ) M = M {\displaystyle J(R)M=M} , then M = 0 ...
A porism is a mathematical proposition or corollary. It has been used to refer to a direct consequence of a proof, analogous to how a corollary refers to a direct consequence of a theorem. In modern usage, it is a relationship that holds for an infinite range of values but only if a certain condition is assumed, such as Steiner's porism. [1]
In mathematics and other fields, [a] a lemma (pl.: lemmas or lemmata) is a generally minor, proven proposition which is used to prove a larger statement. For that reason, it is also known as a "helping theorem" or an "auxiliary theorem".
Ptolemy's Theorem yields as a corollary a pretty theorem [2] regarding an equilateral triangle inscribed in a circle. Given An equilateral triangle inscribed on a circle and a point on the circle. The distance from the point to the most distant vertex of the triangle is the sum of the distances from the point to the two nearer vertices.
Berger–Kazdan comparison theorem (Riemannian geometry) Bernstein's theorem (approximation theory) Bernstein's theorem (functional analysis) Berry–Esséen theorem (probability theory) Bertini's theorem (algebraic geometry) Bertrand–Diquet–Puiseux theorem (differential geometry) Bertrand's ballot theorem (probability theory, combinatorics)
In Euclid's Elements, the first 28 Propositions and Proposition 31 avoid using the parallel postulate, and therefore are valid in absolute geometry.One can also prove in absolute geometry the exterior angle theorem (an exterior angle of a triangle is larger than either of the remote angles), as well as the Saccheri–Legendre theorem, which states that the sum of the measures of the angles in ...