Home Metamath Proof ExplorerTheorem List (p. 206 of 327) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-22411) Hilbert Space Explorer (22412-23934) Users' Mathboxes (23935-32663)

Theorem List for Metamath Proof Explorer - 20501-20600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlogmul2 20501 Generalization of relogmul 20476 to a complex left argument. (Contributed by Mario Carneiro, 9-Jul-2017.)

Theoremlogdiv2 20502 Generalization of relogdiv 20477 to a complex left argument. (Contributed by Mario Carneiro, 8-Jul-2017.)

Theoremabslogle 20503 Bound on the magnitude of the complex logarithm function. (Contributed by Mario Carneiro, 3-Jul-2017.)

Theoremtanarg 20504 The basic relation between the "arg" function and the arctangent. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremlogdivlti 20505 The function is strictly decreasing on the reals greater than . (Contributed by Mario Carneiro, 14-Mar-2014.)

Theoremlogdivlt 20506 The function is strictly decreasing on the reals greater than . (Contributed by Mario Carneiro, 14-Mar-2014.)

Theoremlogdivle 20507 The function is strictly decreasing on the reals greater than . (Contributed by Mario Carneiro, 3-May-2016.)

Theoremrelogcld 20508 Closure of the natural logarithm function. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremreeflogd 20509 Relationship between the natural logarithm function and the exponential function. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremrelogmuld 20510 The natural logarithm of the product of two positive real numbers is the sum of natural logarithms. Property 2 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremrelogdivd 20511 The natural logarithm of the quotient of two positive real numbers is the difference of natural logarithms. Exercise 72(a) and Property 3 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremlogled 20512 Natural logarithm preserves . (Contributed by Mario Carneiro, 29-May-2016.)

Theoremrelogefd 20513 Relationship between the natural logarithm function and the exponential function. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremrplogcld 20514 Closure of the logarithm function in the positive reals. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremlogge0d 20515 The logarithm of a number greater than 1 is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremdivlogrlim 20516 The inverse logarithm function converges to zero. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlogno1 20517 The logarithm function is not eventually bounded. (Contributed by Mario Carneiro, 30-Apr-2016.) (Proof shortened by Mario Carneiro, 30-May-2016.)

Theoremdvrelog 20518 The derivative of the real logarithm function. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremrelogcn 20519 The real logarithm function is continuous. (Contributed by Mario Carneiro, 17-Feb-2015.)

Theoremellogdm 20520 Elementhood in the "continuous domain" of the complex logarithm. (Contributed by Mario Carneiro, 18-Feb-2015.)

Theoremlogdmn0 20521 A number in the continuous domain of is nonzero. (Contributed by Mario Carneiro, 18-Feb-2015.)

Theoremlogdmnrp 20522 A number in the continuous domain of is not a strictly negative number. (Contributed by Mario Carneiro, 18-Feb-2015.)

Theoremlogdmss 20523 The continuity domain of is a subset of the regular domain of . (Contributed by Mario Carneiro, 1-Mar-2015.)

Theoremlogcnlem2 20524 Lemma for logcn 20528. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremlogcnlem3 20525 Lemma for logcn 20528. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremlogcnlem4 20526 Lemma for logcn 20528. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremlogcnlem5 20527* Lemma for logcn 20528. (Contributed by Mario Carneiro, 18-Feb-2015.)

Theoremlogcn 20528 The logarithm function is continuous away from the branch cut at negative reals. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremdvloglem 20529 Lemma for dvlog 20532. (Contributed by Mario Carneiro, 24-Feb-2015.)
fld

Theoremlogdmopn 20530 The "continuous domain" of is an open set. (Contributed by Mario Carneiro, 7-Apr-2015.)
fld

Theoremlogf1o2 20531 The logarithm maps its continuous domain bijectively onto the set of numbers with imaginary part . The negative reals are mapped to the numbers with imaginary part equal to . (Contributed by Mario Carneiro, 2-May-2015.)

Theoremdvlog 20532* The derivative of the complex logarithm function. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremdvlog2lem 20533 Lemma for dvlog2 20534. (Contributed by Mario Carneiro, 1-Mar-2015.)

Theoremdvlog2 20534* The derivative of the complex logarithm function on the open unit ball centered at , a sometimes easier region to work with than the of dvlog 20532. (Contributed by Mario Carneiro, 1-Mar-2015.)

Theoremadvlog 20535 The antiderivative of the logarithm. (Contributed by Mario Carneiro, 21-May-2016.)

Theoremadvlogexp 20536* The antiderivative of a power of the logarithm. (Set and multiply by to get the antiderivative of itself.) (Contributed by Mario Carneiro, 22-May-2016.)

Theoremefopnlem1 20537 Lemma for efopn 20539. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)

Theoremefopnlem2 20538 Lemma for efopn 20539. (Contributed by Mario Carneiro, 2-May-2015.)
fld

Theoremefopn 20539 The exponential map is an open map. (Contributed by Mario Carneiro, 23-Apr-2015.)
fld

Theoremlogtayllem 20540* Lemma for logtayl 20541. (Contributed by Mario Carneiro, 1-Apr-2015.)

Theoremlogtayl 20541* The Taylor series for . (Contributed by Mario Carneiro, 1-Apr-2015.)

Theoremlogtaylsum 20542* The Taylor series for , as an infinite sum. (Contributed by Mario Carneiro, 31-Mar-2015.)

Theoremlogtayl2 20543* Power series expression for the logarithm. (Contributed by Mario Carneiro, 31-Mar-2015.)

Theoremlogccv 20544 The natural logarithm function on the reals is a strictly concave function. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremcxpval 20545 Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxpef 20546 Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theorem0cxp 20547 Value of the complex power function when the first argument is zero. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxpexpz 20548 Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxpexp 20549 Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremlogcxp 20550 Logarithm of a complex power. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxp0 20551 Value of the complex power function when the second argument is zero. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxp1 20552 Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theorem1cxp 20553 Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremecxp 20554 Write the exponential function as an exponent to the power . (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxpcl 20555 Closure of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremrecxpcl 20556 Real closure of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremrpcxpcl 20557 Positive real closure of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxpne0 20558 Complex exponentiation is nonzero if its mantissa is nonzero. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxpeq0 20559 Complex exponentiation is zero iff the mantissa is zero and the exponent is nonzero. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremcxpadd 20560 Sum of exponents law for complex exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxpp1 20561 Value of a nonzero complex number raised to a complex power plus one. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxpneg 20562 Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxpsub 20563 Exponent subtraction law for complex exponentiation. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremcxpge0 20564 Nonnegative exponentiation with a real exponent is nonnegative. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremmulcxplem 20565 Lemma for mulcxp 20566. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremmulcxp 20566 Complex exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxprec 20567 Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremdivcxp 20568 Complex exponentiation of a quotient. (Contributed by Mario Carneiro, 8-Sep-2014.)

Theoremcxpmul 20569 Product of exponents law for complex exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxpmul2 20570 Product of exponents law for complex exponentiation. Variation on cxpmul 20569 with more general conditions on and when is an integer. (Contributed by Mario Carneiro, 9-Aug-2014.)

Theoremcxproot 20571 The complex power function allows us to write n-th roots via the idiom . (Contributed by Mario Carneiro, 6-May-2015.)

Theoremcxpmul2z 20572 Generalize cxpmul2 20570 to negative integers. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremabscxp 20573 Absolute value of a power, when the base is real. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremabscxp2 20574 Absolute value of a power, when the exponent is real. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremcxplt 20575 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxple 20576 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxplea 20577 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 10-Sep-2014.)

Theoremcxple2 20578 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 8-Sep-2014.)

Theoremcxplt2 20579 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremcxple2a 20580 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremcxplt3 20581 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016.)

Theoremcxple3 20582 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016.)

Theoremcxpsqrlem 20583 Lemma for cxpsqr 20584. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxpsqr 20584 The complex exponential function with exponent exactly matches the complex square root function (the branch cut is in the same place for both functions), and thus serves as a suitable generalization to other -th roots and irrational roots. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremlogsqr 20585 Logarithm of a square root. (Contributed by Mario Carneiro, 5-May-2016.)

Theoremcxp0d 20586 Value of the complex power function when the second argument is zero. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxp1d 20587 Value of the complex power function at one. (Contributed by Mario Carneiro, 30-May-2016.)

Theorem1cxpd 20588 Value of the complex power function at one. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpcld 20589 Closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpmul2d 20590 Product of exponents law for complex exponentiation. Variation on cxpmul 20569 with more general conditions on and when is an integer. (Contributed by Mario Carneiro, 30-May-2016.)

Theorem0cxpd 20591 Value of the complex power function when the first argument is zero. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpexpzd 20592 Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpefd 20593 Value of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpne0d 20594 Complex exponentiation is nonzero if its mantissa is nonzero. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpp1d 20595 Value of a nonzero complex number raised to a complex power plus one. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpnegd 20596 Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpmul2zd 20597 Generalize cxpmul2 20570 to negative integers. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpaddd 20598 Sum of exponents law for complex exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpsubd 20599 Exponent subtraction law for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpltd 20600 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)

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 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32663
 Copyright terms: Public domain < Previous  Next >