Home Metamath Proof ExplorerTheorem List (p. 119 of 325) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-22374) Hilbert Space Explorer (22375-23897) Users' Mathboxes (23898-32447)

Theorem List for Metamath Proof Explorer - 11801-11900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorems6cli 11801 A length 6 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theorems7cli 11802 A length 7 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theorems8cli 11803 A length 8 string is a word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Word

Theorems2fv0 11804 Extract the first symbol from a doubleton. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

Theorems2fv1 11805 Extract the second symbol from a doubleton. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

Theorems2len 11806 The length of a doubleton. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

Theorems3fv0 11807 Extract the first symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)

Theorems3fv1 11808 Extract the second symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)

Theorems3fv2 11809 Extract the third symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017.)

Theorems3len 11810 The length of a length 3 string. (Contributed by Mario Carneiro, 26-Feb-2016.)

Theorems4len 11811 The length of a length 4 string. (Contributed by Mario Carneiro, 26-Feb-2016.)

Theorems5len 11812 The length of a length 5 string. (Contributed by Mario Carneiro, 26-Feb-2016.)

Theorems6len 11813 The length of a length 6 string. (Contributed by Mario Carneiro, 26-Feb-2016.)

Theorems7len 11814 The length of a length 7 string. (Contributed by Mario Carneiro, 26-Feb-2016.)

Theorems8len 11815 The length of a length 8 string. (Contributed by Mario Carneiro, 26-Feb-2016.)

Theorems2prop 11816 A length 2 word is an unordered pair of ordered pairs. (Contributed by Alexander van der Vekens, 14-Aug-2017.)

Theorems4prop 11817 A length 4 word is a union of two unordered pairs of ordered pairs. (Contributed by Alexander van der Vekens, 14-Aug-2017.)

Theorems2f1o 11818 A length 2 word with mutually different symbols is a one-to-one function onto the set of the symbols. (Contributed by Alexander van der Vekens, 14-Aug-2017.)

Theoremf1oun2prg 11819 A union of unordered pairs of ordered pairs with different elements is a one-to-one onto function. (Contributed by Alexander van der Vekens, 14-Aug-2017.)

Theorems4f1o 11820 A length 4 word with mutually different symbols is a one-to-one function onto the set of the symbols. (Contributed by Alexander van der Vekens, 14-Aug-2017.)

Theorems4dom 11821 The domain of a length 4 word is the union of two (disjunct) pairs. (Contributed by Alexander van der Vekens, 15-Aug-2017.)

Theorems2co 11822 Mapping a doubleton by a function. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theorems3co 11823 Mapping a length 3 string by a function. (Contributed by Mario Carneiro, 27-Feb-2016.)

Theorems0s1 11824 Concatenation of fixed length strings. (This special case of ccatlid 11703 is provided to complete the pattern s0s1 11824, df-s2 11767, df-s3 11768, ...) (Contributed by Mario Carneiro, 28-Feb-2016.)
concat

Theorems1s2 11825 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theorems1s3 11826 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theorems1s4 11827 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theorems1s5 11828 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theorems1s6 11829 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theorems1s7 11830 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theorems2s2 11831 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theorems4s2 11832 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theorems4s3 11833 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theorems4s4 11834 Concatenation of fixed length strings. (Contributed by Mario Carneiro, 26-Feb-2016.)
concat

Theoremswrds2 11835 Extract two adjacent symbols from a word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Word ..^ substr

5.7  Elementary real and complex functions

5.7.1  The "shift" operation

Syntaxcshi 11836 Extend class notation with function shifter.

Definitiondf-shft 11837* Define a function shifter. This operation offsets the value argument of a function (ordinarily on a subset of ) and produces a new function on . See shftval 11844 for its value. (Contributed by NM, 20-Jul-2005.)

Theoremshftlem 11838* Two ways to write a shifted set . (Contributed by Mario Carneiro, 3-Nov-2013.)

Theoremshftuz 11839* A shift of the upper integers. (Contributed by Mario Carneiro, 5-Nov-2013.)

Theoremshftfval 11840* The value of the sequence shifter operation is a function on . is ordinarily an integer. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremshftdm 11841* Domain of a relation shifted by . The set on the right is more commonly notated as (meaning add to every element of ). (Contributed by Mario Carneiro, 3-Nov-2013.)

Theoremshftfib 11842 Value of a fiber of the relation . (Contributed by Mario Carneiro, 4-Nov-2013.)

Theoremshftfn 11843* Functionality and domain of a sequence shifted by . (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremshftval 11844 Value of a sequence shifted by . (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 4-Nov-2013.)

Theoremshftval2 11845 Value of a sequence shifted by . (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftval3 11846 Value of a sequence shifted by . (Contributed by NM, 20-Jul-2005.)

Theoremshftval4 11847 Value of a sequence shifted by . (Contributed by NM, 18-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftval5 11848 Value of a shifted sequence. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftf 11849* Functionality of a shifted sequence. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theorem2shfti 11850 Composite shift operations. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftidt2 11851 Identity law for the shift operation. (Contributed by Mario Carneiro, 5-Nov-2013.)

Theoremshftidt 11852 Identity law for the shift operation. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftcan1 11853 Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremshftcan2 11854 Cancellation law for the shift operation. (Contributed by NM, 4-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremseqshft 11855 Shifting the index set of a sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-Feb-2014.)

5.7.2  Real and imaginary parts; conjugate

Syntaxccj 11856 Extend class notation to include complex conjugate function.

Syntaxcre 11857 Extend class notation to include real part of a complex number.

Syntaxcim 11858 Extend class notation to include imaginary part of a complex number.

Definitiondf-cj 11859* Define the complex conjugate function. See cjcli 11929 for its closure and cjval 11862 for its value. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Definitiondf-re 11860 Define a function whose value is the real part of a complex number. See reval 11866 for its value, recli 11927 for its closure, and replim 11876 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)

Definitiondf-im 11861 Define a function whose value is the imaginary part of a complex number. See imval 11867 for its value, imcli 11928 for its closure, and replim 11876 for its use in decomposing a complex number. (Contributed by NM, 9-May-1999.)

Theoremcjval 11862* The value of the conjugate of a complex number. (Contributed by Mario Carneiro, 6-Nov-2013.)

Theoremcjth 11863 The defining property of the complex conjugate. (Contributed by Mario Carneiro, 6-Nov-2013.)

Theoremcjf 11864 Domain and codomain of the conjugate function. (Contributed by Mario Carneiro, 6-Nov-2013.)

Theoremcjcl 11865 The conjugate of a complex number is a complex number (closure law). (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremreval 11866 The value of the real part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremimval 11867 The value of the imaginary part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremimre 11868 The imaginary part of a complex number in terms of the real part function. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremreim 11869 The real part of a complex number in terms of the imaginary part function. (Contributed by Mario Carneiro, 31-Mar-2015.)

Theoremrecl 11870 The real part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremimcl 11871 The imaginary part of a complex number is real. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremref 11872 Domain and codomain of the real part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremimf 11873 Domain and codomain of the imaginary part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.)

Theoremcrre 11874 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theoremcrim 11875 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theoremreplim 11876 Reconstruct a complex number from its real and imaginary parts. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theoremremim 11877 Value of the conjugate of a complex number. The value is the real part minus times the imaginary part. Definition 10-3.2 of [Gleason] p. 132. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theoremreim0 11878 The imaginary part of a real number is 0. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theoremreim0b 11879 A number is real iff its imaginary part is 0. (Contributed by NM, 26-Sep-2005.)

Theoremrereb 11880 A number is real iff it equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 20-Aug-2008.)

Theoremmulre 11881 A product with a nonzero real multiplier is real iff the multiplicand is real. (Contributed by NM, 21-Aug-2008.)

Theoremrere 11882 A real number equals its real part. One direction of Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremcjreb 11883 A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 2-Jul-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremrecj 11884 Real part of a complex conjugate. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremreneg 11885 Real part of negative. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremreadd 11886 Real part distributes over addition. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremresub 11887 Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.)

Theoremremullem 11888 Lemma for remul 11889, immul 11896, and cjmul 11902. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremremul 11889 Real part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremremul2 11890 Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremrediv 11891 Real part of a division. Related to remul2 11890. (Contributed by David A. Wheeler, 10-Jun-2015.)

Theoremimcj 11892 Imaginary part of a complex conjugate. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremimneg 11893 The imaginary part of a negative number. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremimadd 11894 Imaginary part distributes over addition. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremimsub 11895 Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.)

Theoremimmul 11896 Imaginary part of a product. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)

Theoremimmul2 11897 Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremimdiv 11898 Imaginary part of a division. Related to immul2 11897. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremcjre 11899 A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by NM, 8-Oct-1999.)

Theoremcjcj 11900 The conjugate of the conjugate is the original complex number. Proposition 10-3.4(e) of [Gleason] p. 133. (Contributed by NM, 29-Jul-1999.) (Proof shortened by Mario Carneiro, 14-Jul-2014.)

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 32401-32447
 Copyright terms: Public domain < Previous  Next >