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

Theoremmtest 20201* 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 20202* 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 20203 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 19441.) (Contributed by Mario Carneiro, 18-Mar-2015.)
MblFn              MblFn

Theoremiblulm 20204 A uniform limit of integrable functions is integrable. (Contributed by Mario Carneiro, 3-Mar-2015.)

Theoremitgulm 20205* A uniform limit of integrals of integrable functions converges to the integral of the limit function. (Contributed by Mario Carneiro, 18-Mar-2015.)

Theoremitgulm2 20206* 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 20207* Value of the function that gives the sequence of monomials of a power series. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theorempserval2 20208* Value of the function that gives the sequence of monomials of a power series. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theorempsergf 20209* The sequence of terms in the infinite sequence defining a power series for fixed . (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremradcnvlem1 20210* Lemma for radcnvlt1 20215, radcnvle 20217. 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 20211* Lemma for radcnvlt1 20215, radcnvle 20217. 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 20212* Lemma for radcnvlt1 20215, radcnvle 20217. 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 20213* Zero is always a convergent point for any power series. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremradcnvcl 20214* The radius of convergence of an infinite series is a nonnegative extended real number. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremradcnvlt1 20215* 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 20216* 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 20217* 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 20218* 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 20219* 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 20220* Since by pserulm 20219 the series converges uniformly, it is also continuous by ulmcn 20196. (Contributed by Mario Carneiro, 3-Mar-2015.)

Theorempsercnlem2 20221* Lemma for psercn 20223. (Contributed by Mario Carneiro, 18-Mar-2015.)

Theorempsercnlem1 20222* Lemma for psercn 20223. (Contributed by Mario Carneiro, 18-Mar-2015.)

Theorempsercn 20223* 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 20224* Lemma for pserdv 20226. (Contributed by Mario Carneiro, 7-May-2015.)

Theorempserdvlem2 20225* Lemma for pserdv 20226. (Contributed by Mario Carneiro, 7-May-2015.)

Theorempserdv 20226* The derivative of a power series on its region of convergence. (Contributed by Mario Carneiro, 31-Mar-2015.)

Theorempserdv2 20227* The derivative of a power series on its region of convergence. (Contributed by Mario Carneiro, 31-Mar-2015.)

Theoremabelthlem1 20228* Lemma for abelth 20238. (Contributed by Mario Carneiro, 1-Apr-2015.)

Theoremabelthlem2 20229* Lemma for abelth 20238. The peculiar region , known as a Stolz angle , is a teardrop-shaped subset of the closed unit ball containing . Indeed, except for itself, the rest of the Stolz angle is enclosed in the open unit ball. (Contributed by Mario Carneiro, 31-Mar-2015.)

Theoremabelthlem3 20230* Lemma for abelth 20238. (Contributed by Mario Carneiro, 31-Mar-2015.)

Theoremabelthlem4 20231* Lemma for abelth 20238. (Contributed by Mario Carneiro, 31-Mar-2015.)

Theoremabelthlem5 20232* Lemma for abelth 20238. (Contributed by Mario Carneiro, 1-Apr-2015.)

Theoremabelthlem6 20233* Lemma for abelth 20238. (Contributed by Mario Carneiro, 2-Apr-2015.)

Theoremabelthlem7a 20234* Lemma for abelth 20238. (Contributed by Mario Carneiro, 8-May-2015.)

Theoremabelthlem7 20235* Lemma for abelth 20238. (Contributed by Mario Carneiro, 2-Apr-2015.)

Theoremabelthlem8 20236* Lemma for abelth 20238. (Contributed by Mario Carneiro, 2-Apr-2015.)

Theoremabelthlem9 20237* Lemma for abelth 20238. By adjusting the constant term, we can assume that the entire series converges to . (Contributed by Mario Carneiro, 1-Apr-2015.)

Theoremabelth 20238* Abel's theorem. If the power series is convergent at , then it is equal to the limit from "below", along a Stolz angle (note that the case of a Stolz angle is the real line ). (Continuity on follows more generally from psercn 20223.) (Contributed by Mario Carneiro, 2-Apr-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)

Theoremabelth2 20239* Abel's theorem, restricted to the interval. (Contributed by Mario Carneiro, 2-Apr-2015.)

13.3  Basic trigonometry

13.3.1  The exponential, sine, and cosine functions (cont.)

Theoremefcn 20240 The exponential function is continuous. (Contributed by Paul Chapman, 15-Sep-2007.) (Revised by Mario Carneiro, 20-Jun-2015.)

Theoremsincn 20241 Sine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)

Theoremcoscn 20242 Cosine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)

Theoremreeff1olem 20243* Lemma for reeff1o 20244. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremreeff1o 20244 The real exponential function is one-to-one onto. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 10-Nov-2013.)

Theoremreefiso 20245 The exponential function on the reals determines an isomorphism from reals onto positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.) (Revised by Mario Carneiro, 11-Mar-2014.)

Theoremefcvx 20246 The exponential function on the reals is a strictly convex function. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremreefgim 20247 The exponential function is a group isomorphism from the group of reals under addition to the group of positive reals under multiplication. (Contributed by Mario Carneiro, 21-Jun-2015.)
flds        mulGrpflds        GrpIso

13.3.2  Properties of pi = 3.14159...

Theorempilem1 20248 Lemma for pire 20253, pigt2lt4 20251 and sinpi 20252. (Contributed by Mario Carneiro, 9-May-2014.)

Theorempilem2 20249 Lemma for pire 20253, pigt2lt4 20251 and sinpi 20252. (Contributed by Mario Carneiro, 12-Jun-2014.)

Theorempilem3 20250 Lemma for pire 20253, pigt2lt4 20251 and sinpi 20252. Existence part. (Contributed by Paul Chapman, 23-Jan-2008.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)

Theorempigt2lt4 20251 is between 2 and 4. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)

Theoremsinpi 20252 The sine of is 0. (Contributed by Paul Chapman, 23-Jan-2008.)

Theorempire 20253 is a real number. (Contributed by Paul Chapman, 23-Jan-2008.)

Theorempipos 20254 is positive. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)

Theoremsinhalfpilem 20255 Lemma for sinhalfpi 20257 and coshalfpi 20258. (Contributed by Paul Chapman, 23-Jan-2008.)

Theoremhalfpire 20256 is real. (Contributed by David Moews, 28-Feb-2017.)

Theoremsinhalfpi 20257 The sine of is 1. (Contributed by Paul Chapman, 23-Jan-2008.)

Theoremcoshalfpi 20258 The cosine of is 0. (Contributed by Paul Chapman, 23-Jan-2008.)

Theoremcosneghalfpi 20259 The cosine of is zero. (Contributed by David Moews, 28-Feb-2017.)

Theoremefhalfpi 20260 The exponential of is . (Contributed by Mario Carneiro, 9-May-2014.)

Theoremcospi 20261 The cosine of is . (Contributed by Paul Chapman, 23-Jan-2008.)

Theoremefipi 20262 The exponential of . (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremeulerid 20263 Euler's identity. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)

Theoremsin2pi 20264 The sine of is 0. (Contributed by Paul Chapman, 23-Jan-2008.)

Theoremcos2pi 20265 The cosine of is 1. (Contributed by Paul Chapman, 23-Jan-2008.)

Theoremef2pi 20266 The exponential of is . (Contributed by Mario Carneiro, 9-May-2014.)

Theoremef2kpi 20267 The exponential of is . (Contributed by Mario Carneiro, 9-May-2014.)

Theoremefper 20268 The exponential function is periodic. (Contributed by Paul Chapman, 21-Apr-2008.) (Proof shortened by Mario Carneiro, 10-May-2014.)

Theoremsinperlem 20269 Lemma for sinper 20270 and cosper 20271. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremsinper 20270 The sine function is periodic. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremcosper 20271 The cosine function is periodic. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremsin2kpi 20272 If is an integer, the sine of is 0. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremcos2kpi 20273 If is an integer, the cosine of is 1. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremsin2pim 20274 Sine of a number subtracted from . (Contributed by Paul Chapman, 15-Mar-2008.)

Theoremcos2pim 20275 Cosine of a number subtracted from . (Contributed by Paul Chapman, 15-Mar-2008.)

Theoremsinmpi 20276 Sine of a number less . (Contributed by Paul Chapman, 15-Mar-2008.)

Theoremcosmpi 20277 Cosine of a number less . (Contributed by Paul Chapman, 15-Mar-2008.)

Theoremsinppi 20278 Sine of a number plus . (Contributed by NM, 10-Aug-2008.)

Theoremcosppi 20279 Cosine of a complex number plus . (Contributed by NM, 18-Aug-2008.)

Theoremefimpi 20280 The exponential function of times a real number less . (Contributed by Paul Chapman, 15-Mar-2008.)

Theoremsinhalfpip 20281 The sine of plus a number. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremsinhalfpim 20282 The sine of minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremcoshalfpip 20283 The cosine of plus a number. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremcoshalfpim 20284 The cosine of minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremptolemy 20285 Ptolemy's Theorem. This theorem is named after the Greek astronomer and mathematician Ptolemy (Claudius Ptolemaeus). This particular version is expressed using the sine function. It is proved by expanding all the multiplication of sines to a product of cosines of differences using sinmul 12714, then using algebraic simplification to show that both sides are equal. This formalization is based on the proof in "Trigonometry" by Gelfand and Saul. (Contributed by David A. Wheeler, 31-May-2015.)

Theoremsincosq1lem 20286 Lemma for sincosq1sgn 20287. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremsincosq1sgn 20287 The signs of the sine and cosine functions in the first quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremsincosq2sgn 20288 The signs of the sine and cosine functions in the second quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremsincosq3sgn 20289 The signs of the sine and cosine functions in the third quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremsincosq4sgn 20290 The signs of the sine and cosine functions in the fourth quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremcoseq00topi 20291 Location of the zeroes of cosine in . (Contributed by David Moews, 28-Feb-2017.)

Theoremcoseq0negpitopi 20292 Location of the zeroes of cosine in . (Contributed by David Moews, 28-Feb-2017.)

Theoremtanrpcl 20293 Positive real closure of the tangent function. (Contributed by Mario Carneiro, 29-Jul-2014.)

Theoremtangtx 20294 The tangent function is greater than its argument on positive reals in its principal domain. (Contributed by Mario Carneiro, 29-Jul-2014.)

Theoremtanabsge 20295 The tangent function is greater than or equal to its argument in absolute value. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremsinq12gt0 20296 The sine of a number strictly between and is positive. (Contributed by Paul Chapman, 15-Mar-2008.)

Theoremsinq12ge0 20297 The sine of a number between and is nonnegative. (Contributed by Mario Carneiro, 13-May-2014.)

Theoremsinq34lt0t 20298 The sine of a number strictly between and is negative. (Contributed by NM, 17-Aug-2008.)

Theoremcosq14gt0 20299 The cosine of a number strictly between and is positive. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremcosq14ge0 20300 The cosine of a number between and is nonnegative. (Contributed by Mario Carneiro, 13-May-2014.)

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