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In mathematics, a corollary is a theorem connected by a short proof to an existing theorem. The use of the term corollary, rather than proposition or theorem, is intrinsically subjective. 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.
Although the theorem is named after Michel Rolle, Rolle's 1691 proof covered only the case of polynomial functions. His proof did not use the methods of differential calculus, which at that point in his life he considered to be fallacious. The theorem was first proved by Cauchy in 1823 as a corollary of a proof of the mean value theorem. [1]
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 ...
1896 Schröder announces a proof (as a corollary of a theorem by Jevons). [11] 1897 Bernstein, a 19-year-old student in Cantor's Seminar, presents his proof. [12] [13] 1897 Almost simultaneously, but independently, Schröder finds a proof. [12] [13] 1897 After a visit by Bernstein, Dedekind independently proves the theorem a second time.
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.
An immediate corollary of Löb's theorem is that, if P is not provable in PA, then "if P is provable in PA, then P is true" is not provable in PA. Given we know PA is consistent (but PA does not know PA is consistent), here are some simple examples:
For example, if the matrix is ... This corollary can also be proved directly by using the intermediate value theorem. Proof. One proof of the theorem is as follows: [2]
This does not follow from Abel's statement of the theorem, but is a corollary of his proof, as his proof is based on the fact that some polynomials in the coefficients of the equation are not the zero polynomial. This improved statement follows directly from Galois theory § A non-solvable quintic example. Galois theory implies also that