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Theorem List for Metamath Proof Explorer - 20201-20300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremaannen 20201 The algebraic numbers are countable. (Contributed by Stefan O'Rear, 16-Nov-2014.)

13.1.7  Liouville's approximation theorem

Theoremaalioulem1 20202 Lemma for aaliou 20208. An integer polynomial cannot inflate the denominator of a rational by more than its degree. (Contributed by Stefan O'Rear, 12-Nov-2014.)
Poly                     deg

Theoremaalioulem2 20203* Lemma for aaliou 20208. (Contributed by Stefan O'Rear, 15-Nov-2014.)
deg       Poly

Theoremaalioulem3 20204* Lemma for aaliou 20208. (Contributed by Stefan O'Rear, 15-Nov-2014.)
deg       Poly

Theoremaalioulem4 20205* Lemma for aaliou 20208. (Contributed by Stefan O'Rear, 16-Nov-2014.)
deg       Poly

Theoremaalioulem5 20206* Lemma for aaliou 20208. (Contributed by Stefan O'Rear, 16-Nov-2014.)
deg       Poly

Theoremaalioulem6 20207* Lemma for aaliou 20208. (Contributed by Stefan O'Rear, 16-Nov-2014.)
deg       Poly

Theoremaaliou 20208* Liouville's theorem on diophantine approximation: Any algebraic number, being a root of a polynomial in integer coefficients, is not approximable beyond order deg by rational numbers. In this form, it also applies to rational numbers themselves, which are not well approximable by other rational numbers. (Contributed by Stefan O'Rear, 16-Nov-2014.)
deg       Poly

Theoremgeolim3 20209* Geometric series convergence with arbitrary shift, radix, and multiplicative constant. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremaaliou2 20210* Liouville's approximation theorem for algebraic numbers per se. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremaaliou2b 20211* Liouville's approximation theorem extended to complex . (Contributed by Stefan O'Rear, 20-Nov-2014.)

Theoremaaliou3lem1 20212* Lemma for aaliou3 20221. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremaaliou3lem2 20213* Lemma for aaliou3 20221. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremaaliou3lem3 20214* Lemma for aaliou3 20221. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremaaliou3lem8 20215* Lemma for aaliou3 20221. (Contributed by Stefan O'Rear, 20-Nov-2014.)

Theoremaaliou3lem4 20216* Lemma for aaliou3 20221. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremaaliou3lem5 20217* Lemma for aaliou3 20221. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremaaliou3lem6 20218* Lemma for aaliou3 20221. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremaaliou3lem7 20219* Lemma for aaliou3 20221. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Theoremaaliou3lem9 20220* Example of a "Liouville number", a very simple definable transcendental real. (Contributed by Stefan O'Rear, 20-Nov-2014.)

Theoremaaliou3 20221 Example of a "Liouville number", a very simple definable transcendental real. (Contributed by Stefan O'Rear, 23-Nov-2014.)

13.2  Sequences and series

13.2.1  Taylor polynomials and Taylor's theorem

Syntaxctayl 20222 Taylor polynomial of a function.
Tayl

Syntaxcana 20223 The class of analytic functions.
Ana

Definitiondf-tayl 20224* Define the Taylor polynomial or Taylor series of a function. (Contributed by Mario Carneiro, 30-Dec-2016.)
Tayl fld tsums

Definitiondf-ana 20225* Define the set of analytic functions, which are functions such that the Taylor series of the function at each point converges to the function in some neighborhood of the point. (Contributed by Mario Carneiro, 31-Dec-2016.)
Ana fldt Tayl

Theoremtaylfvallem1 20226* Lemma for taylfval 20228. (Contributed by Mario Carneiro, 30-Dec-2016.)

Theoremtaylfvallem 20227* Lemma for taylfval 20228. (Contributed by Mario Carneiro, 30-Dec-2016.)
fld tsums

Theoremtaylfval 20228* Define the Taylor polynomial of a function. The constant Tayl is a function of five arguments: is the base set with respect to evaluate the derivatives (generally or ), is the function we are approximating, at point , to order . The result is a polynomial function of .

This "extended" version of taylpfval 20234 additionally handles the case , in which case this is not a polynomial but an infinite series, the Taylor series of the function. (Contributed by Mario Carneiro, 30-Dec-2016.)

Tayl        fld tsums

Theoremeltayl 20229* Value of the Taylor series as a relation (elementhood in the domain here expresses that the series is convergent). (Contributed by Mario Carneiro, 30-Dec-2016.)
Tayl        fld tsums

Theoremtaylf 20230* The Taylor series defines a function on a subset of the complexes. (Contributed by Mario Carneiro, 30-Dec-2016.)
Tayl

Theoremtayl0 20231* The Taylor series is always defined at the basepoint, with value equal to the value of the function. (Contributed by Mario Carneiro, 30-Dec-2016.)
Tayl

Theoremtaylplem1 20232* Lemma for taylpfval 20234 and similar theorems. (Contributed by Mario Carneiro, 31-Dec-2016.)

Theoremtaylplem2 20233* Lemma for taylpfval 20234 and similar theorems. (Contributed by Mario Carneiro, 31-Dec-2016.)

Theoremtaylpfval 20234* Define the Taylor polynomial of a function. The constant Tayl is a function of five arguments: is the base set with respect to evaluate the derivatives (generally or ), is the function we are approximating, at point , to order . The result is a polynomial function of . (Contributed by Mario Carneiro, 31-Dec-2016.)
Tayl

Theoremtaylpf 20235 The Taylor polynomial is a function on the complexes (even if the base set of the original function is the reals). (Contributed by Mario Carneiro, 31-Dec-2016.)
Tayl

Theoremtaylpval 20236* Value of the Taylor polynomial. (Contributed by Mario Carneiro, 31-Dec-2016.)
Tayl

Theoremtaylply2 20237* The Taylor polynomial is a polynomial of degree (at most) . This version of taylply 20238 shows that the coefficients of are in a subring of the complexes. (Contributed by Mario Carneiro, 1-Jan-2017.)
Tayl        SubRingfld                     Poly deg

Theoremtaylply 20238 The Taylor polynomial is a polynomial of degree (at most) . (Contributed by Mario Carneiro, 31-Dec-2016.)
Tayl        Poly deg

Theoremdvtaylp 20239 The derivative of the Taylor polynomial is the Taylor polynomial of the derivative of the function. (Contributed by Mario Carneiro, 31-Dec-2016.)
Tayl Tayl

Theoremdvntaylp 20240 The -th derivative of the Taylor polynomial is the Taylor polynomial of the -th derivative of the function. (Contributed by Mario Carneiro, 1-Jan-2017.)
Tayl Tayl

Theoremdvntaylp0 20241 The first derivatives of the Taylor polynomial at match the derivatives of the function from which it is derived. (Contributed by Mario Carneiro, 1-Jan-2017.)
Tayl

Theoremtaylthlem1 20242* Lemma for taylth 20244. This is the main part of Taylor's theorem, except for the induction step, which is supposed to be proven using L'Hôpital's rule. However, since our proof of L'Hôpital assumes that , we can only do this part generically, and for taylth 20244 itself we must restrict to . (Contributed by Mario Carneiro, 1-Jan-2017.)
Tayl               ..^ lim lim        lim

Theoremtaylthlem2 20243* Lemma for taylth 20244. (Contributed by Mario Carneiro, 1-Jan-2017.)
Tayl        ..^       lim        lim

Theoremtaylth 20244* Taylor's theorem. The Taylor polynomial of a -times differentiable function is such that the error term goes to zero faster than . (Contributed by Mario Carneiro, 1-Jan-2017.)
Tayl               lim

13.2.2  Uniform convergence

Syntaxculm 20245 Extend class notation to include the uniform convergence predicate.

Definitiondf-ulm 20246* Define the uniform convergence of a sequence of functions. Here if is a sequence of functions defined on and is a function on , and for every there is a such that the functions for are all uniformly within of on the domain . Compare with df-clim 12237. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremulmrel 20247 The uniform limit relation is a relation. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremulmscl 20248 Closure of the base set in a uniform limit. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremulmval 20249* Express the predicate: The sequence of functions converges uniformly to on . (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremulmcl 20250 Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremulmf 20251* Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremulmpm 20252 Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremulmf2 20253 Closure of a uniform limit of functions. (Contributed by Mario Carneiro, 18-Mar-2015.)

Theoremulm2 20254* Simplify ulmval 20249 when and are known to be functions. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremulmi 20255* The uniform limit property. (Contributed by Mario Carneiro, 27-Feb-2015.)

Theoremulmclm 20256* A uniform limit of functions converges pointwise. (Contributed by Mario Carneiro, 27-Feb-2015.)

Theoremulmres 20257 A sequence of functions converges iff the tail of the sequence converges (for any finite cutoff). (Contributed by Mario Carneiro, 24-Mar-2015.)

Theoremulmshftlem 20258* Lemma for ulmshft 20259. (Contributed by Mario Carneiro, 24-Mar-2015.)

Theoremulmshft 20259* A sequence of functions converges iff the shifted sequence converges. (Contributed by Mario Carneiro, 24-Mar-2015.)

Theoremulm0 20260 Every function converges uniformly on the empty set. (Contributed by Mario Carneiro, 3-Mar-2015.)

Theoremulmuni 20261 An sequence of functions uniformly converges to at most one limit. (Contributed by Mario Carneiro, 5-Jul-2017.)

Theoremulmdm 20262 Two ways to express that a function has a limit. (The expression is sometimes useful as a shorthand for "the unique limit of the function "). (Contributed by Mario Carneiro, 5-Jul-2017.)

Theoremulmcaulem 20263* Lemma for ulmcau 20264 and ulmcau2 20265: show the equivalence of the four- and five-quantifier forms of the Cauchy convergence condition. Compare cau3 12114. (Contributed by Mario Carneiro, 1-Mar-2015.)

Theoremulmcau 20264* A sequence of functions converges uniformly iff it is uniformly Cauchy, which is to say that for every there is a such that for all the functions and are uniformly within of each other on . This is the four-quantifier version, see ulmcau2 20265 for the more conventional five-quantifier version. (Contributed by Mario Carneiro, 1-Mar-2015.)

Theoremulmcau2 20265* A sequence of functions converges uniformly iff it is uniformly Cauchy, which is to say that for every there is a such that for all the functions and are uniformly within of each other on . (Contributed by Mario Carneiro, 1-Mar-2015.)

Theoremulmss 20266* A uniform limit of functions is still a uniform limit if restricted to a subset. (Contributed by Mario Carneiro, 3-Mar-2015.)

Theoremulmbdd 20267* A uniform limit of bounded functions is bounded. (Contributed by Mario Carneiro, 27-Feb-2015.)

Theoremulmcn 20268 A uniform limit of continuous functions is continuous. (Contributed by Mario Carneiro, 27-Feb-2015.)

Theoremulmdvlem1 20269* Lemma for ulmdv 20272. (Contributed by Mario Carneiro, 3-Mar-2015.)

Theoremulmdvlem2 20270* Lemma for ulmdv 20272. (Contributed by Mario Carneiro, 8-May-2015.)

Theoremulmdvlem3 20271* Lemma for ulmdv 20272. (Contributed by Mario Carneiro, 8-May-2015.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)

Theoremulmdv 20272* If is a sequence of differentiable functions on which converge pointwise to , and the derivatives of converge uniformly to , then is differentiable with derivative . (Contributed by Mario Carneiro, 27-Feb-2015.)

Theoremmtest 20273* The Weierstrass M-test. If is a sequence of functions which are uniformly bounded by the convergent sequence , then the series generated by the sequence converges uniformly. (Contributed by Mario Carneiro, 3-Mar-2015.)

Theoremmtestbdd 20274* Given the hypotheses of the Weierstrass M-test, the convergent function of the sequence is uniformly bounded. (Contributed by Mario Carneiro, 9-Jul-2017.)

Theoremmbfulm 20275 A uniform limit of measurable functions is measurable. (This is just a corollary of the fact that a pointwise limit of measurable functions is measurable, see mbflim 19513.) (Contributed by Mario Carneiro, 18-Mar-2015.)
MblFn              MblFn

Theoremiblulm 20276 A uniform limit of integrable functions is integrable. (Contributed by Mario Carneiro, 3-Mar-2015.)

Theoremitgulm 20277* A uniform limit of integrals of integrable functions converges to the integral of the limit function. (Contributed by Mario Carneiro, 18-Mar-2015.)

Theoremitgulm2 20278* A uniform limit of integrals of integrable functions converges to the integral of the limit function. (Contributed by Mario Carneiro, 18-Mar-2015.)

13.2.3  Power series

Theorempserval 20279* Value of the function that gives the sequence of monomials of a power series. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theorempserval2 20280* Value of the function that gives the sequence of monomials of a power series. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theorempsergf 20281* The sequence of terms in the infinite sequence defining a power series for fixed . (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremradcnvlem1 20282* Lemma for radcnvlt1 20287, radcnvle 20289. If is a point closer to zero than and the power series converges at , then it converges absolutely at , even if the terms in the sequence are multiplied by . (Contributed by Mario Carneiro, 31-Mar-2015.)

Theoremradcnvlem2 20283* Lemma for radcnvlt1 20287, radcnvle 20289. If is a point closer to zero than and the power series converges at , then it converges absolutely at . (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremradcnvlem3 20284* Lemma for radcnvlt1 20287, radcnvle 20289. If is a point closer to zero than and the power series converges at , then it converges at . (Contributed by Mario Carneiro, 31-Mar-2015.)

Theoremradcnv0 20285* Zero is always a convergent point for any power series. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremradcnvcl 20286* The radius of convergence of an infinite series is a nonnegative extended real number. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremradcnvlt1 20287* If is within the open disk of radius centered at zero, then the infinite series converges absolutely at , and also converges when the series is multiplied by . (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremradcnvlt2 20288* If is within the open disk of radius centered at zero, then the infinite series converges at . (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremradcnvle 20289* If is a convergent point of the infinite series, then is within the closed disk of radius centered at zero. Or, by contraposition, the series divergers at any point strictly more than from the origin. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremdvradcnv 20290* The radius of convergence of the (formal) derivative of the power series is at least as large as the radius of convergence of . (In fact they are equal, but we don't have as much use for the negative side of this claim.) (Contributed by Mario Carneiro, 31-Mar-2015.)

Theorempserulm 20291* If is a region contained in a circle of radius , then the sequence of partial sums of the infinite series converges uniformly on . (Contributed by Mario Carneiro, 26-Feb-2015.)

Theorempsercn2 20292* Since by pserulm 20291 the series converges uniformly, it is also continuous by ulmcn 20268. (Contributed by Mario Carneiro, 3-Mar-2015.)

Theorempsercnlem2 20293* Lemma for psercn 20295. (Contributed by Mario Carneiro, 18-Mar-2015.)

Theorempsercnlem1 20294* Lemma for psercn 20295. (Contributed by Mario Carneiro, 18-Mar-2015.)

Theorempsercn 20295* An infinite series converges to a continuous function on the open disk of radius , where is the radius of convergence of the series. (Contributed by Mario Carneiro, 4-Mar-2015.)

Theorempserdvlem1 20296* Lemma for pserdv 20298. (Contributed by Mario Carneiro, 7-May-2015.)

Theorempserdvlem2 20297* Lemma for pserdv 20298. (Contributed by Mario Carneiro, 7-May-2015.)

Theorempserdv 20298* The derivative of a power series on its region of convergence. (Contributed by Mario Carneiro, 31-Mar-2015.)

Theorempserdv2 20299* The derivative of a power series on its region of convergence. (Contributed by Mario Carneiro, 31-Mar-2015.)

Theoremabelthlem1 20300* Lemma for abelth 20310. (Contributed by Mario Carneiro, 1-Apr-2015.)

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