Home Metamath Proof ExplorerTheorem List (p. 197 of 309) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-21328) Hilbert Space Explorer (21329-22851) Users' Mathboxes (22852-30843)

Theorem List for Metamath Proof Explorer - 19601-19700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremulmshft 19601* A sequence of functions converges iff the shifted sequence converges. (Contributed by Mario Carneiro, 24-Mar-2015.)

Theoremulm0 19602 Every function converges uniformly on the empty set. (Contributed by Mario Carneiro, 3-Mar-2015.)

Theoremulmcaulem 19603* Lemma for ulmcau 19604 and ulmcau2 19605: show the equivalence of the four- and five-quantifier forms of the Cauchy convergence condition. Compare cau3 11716. (Contributed by Mario Carneiro, 1-Mar-2015.)

Theoremulmcau 19604* 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 19605 for the more conventional five-quantifier version. (Contributed by Mario Carneiro, 1-Mar-2015.)

Theoremulmcau2 19605* 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 19606* A uniform limit of functions is still a uniform limit if restricted to a subset. (Contributed by Mario Carneiro, 3-Mar-2015.)

Theoremulmbdd 19607* A uniform limit of bounded functions is bounded. (Contributed by Mario Carneiro, 27-Feb-2015.)

Theoremulmcn 19608 A uniform limit of continuous functions is continuous. (Contributed by Mario Carneiro, 27-Feb-2015.)

Theoremulmdvlem1 19609* Lemma for ulmdv 19612. (Contributed by Mario Carneiro, 3-Mar-2015.)

Theoremulmdvlem2 19610* Lemma for ulmdv 19612. (Contributed by Mario Carneiro, 8-May-2015.)

Theoremulmdvlem3 19611* Lemma for ulmdv 19612. (Contributed by Mario Carneiro, 8-May-2015.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)

Theoremulmdv 19612* 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 19613* 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.)

Theoremmbfulm 19614 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 18855.) (Contributed by Mario Carneiro, 18-Mar-2015.)
MblFn              MblFn

Theoremiblulm 19615 A uniform limit of integrable functions is integrable. (Contributed by Mario Carneiro, 3-Mar-2015.)

Theoremitgulm 19616* A uniform limit of integrals of integrable functions converges to the integral of the limit function. (Contributed by Mario Carneiro, 18-Mar-2015.)

Theoremitgulm2 19617* 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 19618* Value of the function that gives the sequence of monomials of a power series. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theorempserval2 19619* Value of the function that gives the sequence of monomials of a power series. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theorempsergf 19620* The sequence of terms in the infinite sequence defining a power series for fixed . (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremradcnvlem1 19621* Lemma for radcnvlt1 19626, radcnvle 19628. 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 19622* Lemma for radcnvlt1 19626, radcnvle 19628. 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 19623* Lemma for radcnvlt1 19626, radcnvle 19628. 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 19624* Zero is always a convergent point for any power series. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremradcnvcl 19625* The radius of convergence of an infinite series is a nonnegative extended real number. (Contributed by Mario Carneiro, 26-Feb-2015.)

Theoremradcnvlt1 19626* 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 19627* 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 19628* 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 19629* 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 19630* 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 19631* Since by pserulm 19630 the series converges uniformly, it is also continuous by ulmcn 19608. (Contributed by Mario Carneiro, 3-Mar-2015.)

Theorempsercnlem2 19632* Lemma for psercn 19634. (Contributed by Mario Carneiro, 18-Mar-2015.)

Theorempsercnlem1 19633* Lemma for psercn 19634. (Contributed by Mario Carneiro, 18-Mar-2015.)

Theorempsercn 19634* 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 19635* Lemma for pserdv 19637. (Contributed by Mario Carneiro, 7-May-2015.)

Theorempserdvlem2 19636* Lemma for pserdv 19637. (Contributed by Mario Carneiro, 7-May-2015.)

Theorempserdv 19637* The derivative of a power series on its region of convergence. (Contributed by Mario Carneiro, 31-Mar-2015.)

Theorempserdv2 19638* The derivative of a power series on its region of convergence. (Contributed by Mario Carneiro, 31-Mar-2015.)

Theoremabelthlem1 19639* Lemma for abelth 19649. (Contributed by Mario Carneiro, 1-Apr-2015.)

Theoremabelthlem2 19640* Lemma for abelth 19649. 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 19641* Lemma for abelth 19649. (Contributed by Mario Carneiro, 31-Mar-2015.)

Theoremabelthlem4 19642* Lemma for abelth 19649. (Contributed by Mario Carneiro, 31-Mar-2015.)

Theoremabelthlem5 19643* Lemma for abelth 19649. (Contributed by Mario Carneiro, 1-Apr-2015.)

Theoremabelthlem6 19644* Lemma for abelth 19649. (Contributed by Mario Carneiro, 2-Apr-2015.)

Theoremabelthlem7a 19645* Lemma for abelth 19649. (Contributed by Mario Carneiro, 8-May-2015.)

Theoremabelthlem7 19646* Lemma for abelth 19649. (Contributed by Mario Carneiro, 2-Apr-2015.)

Theoremabelthlem8 19647* Lemma for abelth 19649. (Contributed by Mario Carneiro, 2-Apr-2015.)

Theoremabelthlem9 19648* Lemma for abelth 19649. By adjusting the constant term, we can assume that the entire series converges to . (Contributed by Mario Carneiro, 1-Apr-2015.)

Theoremabelth 19649* 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 19634.) (Contributed by Mario Carneiro, 2-Apr-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)

Theoremabelth2 19650* 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 19651 The exponential function is continuous. (Contributed by Paul Chapman, 15-Sep-2007.) (Revised by Mario Carneiro, 20-Jun-2015.)

Theoremsincn 19652 Sine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)

Theoremcoscn 19653 Cosine is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 3-Sep-2014.)

Theoremreeff1olem 19654* Lemma for reeff1o 19655. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)

Theoremreeff1o 19655 The real exponential function is one-to-one onto. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 10-Nov-2013.)

Theoremreefiso 19656 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 19657 The exponential function on the reals is a strictly convex function. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremreefgim 19658 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 19659 Lemma for pire 19664, pigt2lt4 19662 and sinpi 19663. (Contributed by Mario Carneiro, 9-May-2014.)

Theorempilem2 19660 Lemma for pire 19664, pigt2lt4 19662 and sinpi 19663. (Contributed by Mario Carneiro, 12-Jun-2014.)

Theorempilem3 19661 Lemma for pire 19664, pigt2lt4 19662 and sinpi 19663. Existence part. (Contributed by Paul Chapman, 23-Jan-2008.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)

Theorempigt2lt4 19662 is between 2 and 4. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)

Theoremsinpi 19663 The sine of is 0. (Contributed by Paul Chapman, 23-Jan-2008.)

Theorempire 19664 is a real number. (Contributed by Paul Chapman, 23-Jan-2008.)

Theorempipos 19665 is positive. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)

Theoremsinhalfpilem 19666 Lemma for sinhalfpi 19668 and coshalfpi 19669. (Contributed by Paul Chapman, 23-Jan-2008.)

Theoremhalfpire 19667 is real. (Contributed by David Moews, 28-Feb-2017.)

Theoremsinhalfpi 19668 The sine of is 1. (Contributed by Paul Chapman, 23-Jan-2008.)

Theoremcoshalfpi 19669 The cosine of is 0. (Contributed by Paul Chapman, 23-Jan-2008.)

Theoremcosneghalfpi 19670 The cosine of is zero. (Contributed by David Moews, 28-Feb-2017.)

Theoremefhalfpi 19671 The exponential of is . (Contributed by Mario Carneiro, 9-May-2014.)

Theoremcospi 19672 The cosine of is . (Contributed by Paul Chapman, 23-Jan-2008.)

Theoremefipi 19673 The exponential of . (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremeulerid 19674 Euler's identity. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 9-May-2014.)

Theoremsin2pi 19675 The sine of is 0. (Contributed by Paul Chapman, 23-Jan-2008.)

Theoremcos2pi 19676 The cosine of is 1. (Contributed by Paul Chapman, 23-Jan-2008.)

Theoremef2pi 19677 The exponential of is . (Contributed by Mario Carneiro, 9-May-2014.)

Theoremef2kpi 19678 The exponential of is . (Contributed by Mario Carneiro, 9-May-2014.)

Theoremefper 19679 The exponential function is periodic. (Contributed by Paul Chapman, 21-Apr-2008.) (Proof shortened by Mario Carneiro, 10-May-2014.)

Theoremsinperlem 19680 Lemma for sinper 19681 and cosper 19682. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremsinper 19681 The sine function is periodic. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremcosper 19682 The cosine function is periodic. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremsin2kpi 19683 If is an integer, the sine of is 0. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremcos2kpi 19684 If is an integer, the cosine of is 1. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)

Theoremsin2pim 19685 Sine of a number subtracted from . (Contributed by Paul Chapman, 15-Mar-2008.)

Theoremcos2pim 19686 Cosine of a number subtracted from . (Contributed by Paul Chapman, 15-Mar-2008.)

Theoremsinmpi 19687 Sine of a number less . (Contributed by Paul Chapman, 15-Mar-2008.)

Theoremcosmpi 19688 Cosine of a number less . (Contributed by Paul Chapman, 15-Mar-2008.)

Theoremsinppi 19689 Sine of a number plus . (Contributed by NM, 10-Aug-2008.)

Theoremcosppi 19690 Cosine of a complex number plus . (Contributed by NM, 18-Aug-2008.)

Theoremefimpi 19691 The exponential function of times a real number less . (Contributed by Paul Chapman, 15-Mar-2008.)

Theoremsinhalfpip 19692 The sine of plus a number. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremsinhalfpim 19693 The sine of minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremcoshalfpip 19694 The cosine of plus a number. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremcoshalfpim 19695 The cosine of minus a number. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremptolemy 19696 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 12326, 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 19697 Lemma for sincosq1sgn 19698. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremsincosq1sgn 19698 The signs of the sine and cosine functions in the first quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremsincosq2sgn 19699 The signs of the sine and cosine functions in the second quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)

Theoremsincosq3sgn 19700 The signs of the sine and cosine functions in the third quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30843
 Copyright terms: Public domain < Previous  Next >