Home Metamath Proof ExplorerTheorem List (p. 121 of 328) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-22421) Hilbert Space Explorer (22422-23944) Users' Mathboxes (23945-32762)

Theorem List for Metamath Proof Explorer - 12001-12100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcjcld 12001 Closure law for complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremreplimd 12002 Construct a complex number from its real and imaginary parts. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremremimd 12003 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 Mario Carneiro, 29-May-2016.)

Theoremcjcjd 12004 The conjugate of the conjugate is the original complex number. Proposition 10-3.4(e) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremreim0bd 12005 A number is real iff its imaginary part is 0. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremrerebd 12006 A real number equals its real part. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjrebd 12007 A number is real iff it equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjne0d 12008 A number is nonzero iff its complex conjugate is nonzero. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremrecjd 12009 Real part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimcjd 12010 Imaginary part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjmulrcld 12011 A complex number times its conjugate is real. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjmulvald 12012 A complex number times its conjugate. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjmulge0d 12013 A complex number times its conjugate is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremrenegd 12014 Real part of negative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimnegd 12015 Imaginary part of negative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjnegd 12016 Complex conjugate of negative. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremaddcjd 12017 A number plus its conjugate is twice its real part. Compare Proposition 10-3.4(h) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjexpd 12018 Complex conjugate of natural number exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremreaddd 12019 Real part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimaddd 12020 Imaginary part distributes over addition. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremresubd 12021 Real part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimsubd 12022 Imaginary part distributes over subtraction. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremremuld 12023 Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimmuld 12024 Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjaddd 12025 Complex conjugate distributes over addition. Proposition 10-3.4(a) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjmuld 12026 Complex conjugate distributes over multiplication. Proposition 10-3.4(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremipcnd 12027 Standard inner product on complex numbers. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjdivd 12028 Complex conjugate distributes over division. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremrered 12029 A real number equals its real part. One direction of Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremreim0d 12030 The imaginary part of a real number is 0. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcjred 12031 A real number equals its complex conjugate. Proposition 10-3.4(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremremul2d 12032 Real part of a product. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimmul2d 12033 Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremredivd 12034 Real part of a division. Related to remul2 11935. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremimdivd 12035 Imaginary part of a division. Related to remul2 11935. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcrred 12036 The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)

Theoremcrimd 12037 The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.)

5.7.3  Square root; absolute value

Syntaxcsqr 12038 Extend class notation to include square root of a complex number.

Syntaxcabs 12039 Extend class notation to include a function for the absolute value (modulus) of a complex number.

Definitiondf-sqr 12040* Define a function whose value is the square root of a complex number. Since iff , we ensure uniqueness by restricting the range to numbers with positive real part, or numbers with 0 real part and nonnegative imaginary part. A description can be found under "Principal square root of a complex number" at http://en.wikipedia.org/wiki/Square_root. The square root symbol was introduced in 1525 by Christoff Rudolff.

See sqrcl 12165 for its closure, sqrval 12042 for its value, sqrth 12168 and sqsqri 12179 for its relationship to squares, and sqr11i 12188 for uniqueness. (Contributed by NM, 27-Jul-1999.) (Revised by Mario Carneiro, 8-Jul-2013.)

Definitiondf-abs 12041 Define the function for the absolute value (modulus) of a complex number. See abscli 12198 for its closure and absval 12043 or absval2i 12200 for its value. (Contributed by NM, 27-Jul-1999.)

Theoremsqrval 12042* Value of square root function. (Contributed by Mario Carneiro, 8-Jul-2013.)

Theoremabsval 12043 The absolute value (modulus) of a complex number. Proposition 10-3.7(a) of [Gleason] p. 133. (Contributed by NM, 27-Jul-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)

Theoremrennim 12044 A real number does not lie on the negative imaginary axis. (Contributed by Mario Carneiro, 8-Jul-2013.)

Theoremcnpart 12045 The specification of restriction to the right half-plane partitions the complex plane without 0 into two disjoint pieces, which are related by a reflection about the origin (under the map ). (Contributed by Mario Carneiro, 8-Jul-2013.)

Theoremsqr0lem 12046 Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremsqr0 12047 Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremsqrlem1 12048* Lemma for 01sqrex 12055. (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremsqrlem2 12049* Lemma for 01sqrex 12055. (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremsqrlem3 12050* Lemma for 01sqrex 12055. (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremsqrlem4 12051* Lemma for 01sqrex 12055. (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremsqrlem5 12052* Lemma for 01sqrex 12055. (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremsqrlem6 12053* Lemma for 01sqrex 12055. (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremsqrlem7 12054* Lemma for 01sqrex 12055. (Contributed by Mario Carneiro, 10-Jul-2013.)

Theorem01sqrex 12055* Existence of a square root for reals in the interval . (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremresqrex 12056* Existence of a square root for positive reals. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremsqrmo 12057* Uniqueness for the square root function. (Contributed by Mario Carneiro, 9-Jul-2013.) (Revised by NM, 17-Jun-2017.)

Theoremresqreu 12058* Existence and uniqueness for the real square root function. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremresqrcl 12059 Closure of the square root function. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremresqrthlem 12060 Lemma for resqrth 12061. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremresqrth 12061 Square root theorem over the reals. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremremsqsqr 12062 Square of square root. (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremsqrge0 12063 The square root function is nonnegative for nonnegative input. (Contributed by NM, 26-May-1999.) (Revised by Mario Carneiro, 9-Jul-2013.)

Theoremsqrgt0 12064 The square root function is positive for positive input. (Contributed by Mario Carneiro, 10-Jul-2013.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremsqrmul 12065 Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremsqrle 12066 Square root is monotonic. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremsqrlt 12067 Square root is strictly monotonic. Closed form of sqrlti 12193. (Contributed by Scott Fenton, 17-Apr-2014.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremsqr11 12068 The square root function is one-to-one. (Contributed by Scott Fenton, 11-Jun-2013.)

Theoremsqr00 12069 A square root is zero iff its argument is 0. (Contributed by NM, 27-Jul-1999.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremrpsqrcl 12070 The square root of a positive real is a postive real. (Contributed by NM, 22-Feb-2008.)

Theoremsqrdiv 12071 Square root distributes over division. (Contributed by Mario Carneiro, 5-May-2016.)

Theoremsqrneglem 12072 The square root of a negative number. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremsqrneg 12073 The square root of a negative number. (Contributed by Mario Carneiro, 9-Jul-2013.)

Theoremsqrsq2 12074 Relationship between square root and squares. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremsqrsq 12075 Square root of square. (Contributed by NM, 14-Jan-2006.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremsqrmsq 12076 Square root of square. (Contributed by NM, 2-Aug-1999.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremsqr1 12077 The square root of 1 is 1. (Contributed by NM, 31-Jul-1999.)

Theoremsqr4 12078 The square root of 4 is 2. (Contributed by NM, 3-Aug-1999.)

Theoremsqr9 12079 The square root of 9 is 3. (Contributed by NM, 11-May-2004.)

Theoremsqr2gt1lt2 12080 The square root of 2 is bounded by 1 and 2. (Contributed by Roy F. Longton, 8-Aug-2005.) (Revised by Mario Carneiro, 6-Sep-2013.)

Theoremsqrm1 12081 The imaginary unit is the square root of negative 1. A lot of people like to call this the "definition" of , but the definition of df-sqr 12040 has already been crafted with being mentioned explicitly, and in any case it doesn't make too much sense to define a value based on a function evaluated outside its domain. A more appropriate view is to take ax-i2m1 9058 or i2 11481 as the "definition", and simply postulate the existence of a number satisfying this property. This is the approach we take here. (Contributed by Mario Carneiro, 10-Jul-2013.)

Theoremabsneg 12082 Absolute value of negative. (Contributed by NM, 27-Feb-2005.)

Theoremabscl 12083 Real closure of absolute value. (Contributed by NM, 3-Oct-1999.)

Theoremabscj 12084 The absolute value of a number and its conjugate are the same. Proposition 10-3.7(b) of [Gleason] p. 133. (Contributed by NM, 28-Apr-2005.)

Theoremabsvalsq 12085 Square of value of absolute value function. (Contributed by NM, 16-Jan-2006.)

Theoremabsvalsq2 12086 Square of value of absolute value function. (Contributed by NM, 1-Feb-2007.)

Theoremsqabsadd 12087 Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason] p. 133. (Contributed by NM, 21-Jan-2007.)

Theoremsqabssub 12088 Square of absolute value of difference. (Contributed by NM, 21-Jan-2007.)

Theoremabsval2 12089 Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by NM, 17-Mar-2005.)

Theoremabs0 12090 The absolute value of 0. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabsi 12091 The absolute value of the imaginary unit. (Contributed by NM, 26-Mar-2005.)

Theoremabsge0 12092 Absolute value is nonnegative. (Contributed by NM, 20-Nov-2004.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabsrpcl 12093 The absolute value of a nonzero number is a positive real. (Contributed by FL, 27-Dec-2007.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremabs00 12094 The absolute value of a number is zero iff the number is zero. Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by NM, 26-Sep-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.)

Theoremabs00ad 12095 A complex number is zero iff its absolute value is zero. Deduction form of abs00 12094. (Contributed by David Moews, 28-Feb-2017.)

Theoremabs00bd 12096 If a complex number is zero, its absolute value is zero. Converse of abs00d 12248. One-way deduction form of abs00 12094. (Contributed by David Moews, 28-Feb-2017.)

Theoremabsreimsq 12097 Square of the absolute value of a number that has been decomposed into real and imaginary parts. (Contributed by NM, 1-Feb-2007.)

Theoremabsreim 12098 Absolute value of a number that has been decomposed into real and imaginary parts. (Contributed by NM, 14-Jan-2006.)

Theoremabsmul 12099 Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.)

Theoremabsdiv 12100 Absolute value distributes over division. (Contributed by NM, 27-Apr-2005.)

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 32601-32700 328 32701-32762
 Copyright terms: Public domain < Previous  Next >