Home Metamath Proof ExplorerTheorem List (p. 100 of 329) < Previous  Next > Browser slow? Try the Unicode version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-22426) Hilbert Space Explorer (22427-23949) Users' Mathboxes (23950-32835)

Theorem List for Metamath Proof Explorer - 9901-10000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremltdiv2OLD 9901 Division of a positive number by both sides of 'less than'. (Contributed by NM, 27-Apr-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremltrec1 9902 Reciprocal swap in a 'less than' relation. (Contributed by NM, 24-Feb-2005.)

Theoremlerec2 9903 Reciprocal swap in a 'less than or equal to' relation. (Contributed by NM, 24-Feb-2005.)

Theoremledivdiv 9904 Invert ratios of positive numbers and swap their ordering. (Contributed by NM, 9-Jan-2006.)

Theoremlediv2 9905 Division of a positive number by both sides of 'less than or equal to'. (Contributed by NM, 10-Jan-2006.)

Theoremltdiv23 9906 Swap denominator with other side of 'less than'. (Contributed by NM, 3-Oct-1999.)

Theoremlediv23 9907 Swap denominator with other side of 'less than or equal to'. (Contributed by NM, 30-May-2005.)

Theoremlediv12a 9908 Comparison of ratio of two nonnegative numbers. (Contributed by NM, 31-Dec-2005.)

Theoremlediv2a 9909 Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)

Theoremreclt1 9910 The reciprocal of a positive number less than 1 is greater than 1. (Contributed by NM, 23-Feb-2005.)

Theoremrecgt1 9911 The reciprocal of a positive number greater than 1 is less than 1. (Contributed by NM, 28-Dec-2005.)

Theoremrecgt1i 9912 The reciprocal of a number greater than 1 is positive and less than 1. (Contributed by NM, 23-Feb-2005.)

Theoremrecp1lt1 9913 Construct a number less than 1 from any nonnegative number. (Contributed by NM, 30-Dec-2005.)

Theoremrecreclt 9914 Given a positive number , construct a new positive number less than both and 1. (Contributed by NM, 28-Dec-2005.)

Theoremle2msq 9915 The square function on nonnegative reals is monotonic. (Contributed by NM, 3-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremmsq11 9916 The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremledivp1 9917 Less-than-or-equal-to and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 28-Sep-2005.)

Theoremsqueeze0 9918* If a nonnegative number is less than any positive number, it is zero. (Contributed by NM, 11-Feb-2006.)

Theoremltp1i 9919 A number is less than itself plus 1. (Contributed by NM, 20-Aug-2001.)

Theoremrecgt0i 9920 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)

Theoremrecgt0ii 9921 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 15-May-1999.)

Theoremprodgt0i 9922 Infer that a multiplicand is positive from a nonnegative muliplier and positive product. (Contributed by NM, 15-May-1999.)

Theoremprodge0i 9923 Infer that a multiplicand is nonnegative from a positive muliplier and nonnegative product. (Contributed by NM, 2-Jul-2005.)

Theoremdivgt0i 9924 The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)

Theoremdivge0i 9925 The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 12-Aug-1999.)

Theoremltreci 9926 The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)

Theoremlereci 9927 The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 16-Sep-1999.)

Theoremlt2msqi 9928 The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 3-Aug-1999.)

Theoremle2msqi 9929 The square function on nonnegative reals is monotonic. (Contributed by NM, 2-Aug-1999.)

Theoremmsq11i 9930 The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.)

Theoremdivgt0i2i 9931 The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)

Theoremltrecii 9932 The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)

Theoremdivgt0ii 9933 The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999.)

Theoremltmul1i 9934 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.)

Theoremltdiv1i 9935 Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)

Theoremltmuldivi 9936 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.)

Theoremltmul2i 9937 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.)

Theoremlemul1i 9938 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 2-Aug-1999.)

Theoremlemul2i 9939 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 1-Aug-1999.)

Theoremltdiv23i 9940 Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)

Theoremledivp1i 9941 Less-than-or-equal-to and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 17-Sep-2005.)

Theoremltdivp1i 9942 Less-than and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 17-Sep-2005.)

Theoremltdiv23ii 9943 Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)

Theoremltmul1ii 9944 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.) (Proof shortened by Paul Chapman, 25-Jan-2008.)

Theoremltdiv1ii 9945 Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)

Theoremltp1d 9946 A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlep1d 9947 A number is less than or equal to itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltm1d 9948 A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlem1d 9949 A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrecgt0d 9950 The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremdivgt0d 9951 The ratio of two positive numbers is positive. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremmulgt1d 9952 The product of two numbers greater than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemulge11d 9953 Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemulge12d 9954 Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemul1ad 9955 The square of a nonnegative number is a one-to-one function. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemul2ad 9956 The square of a nonnegative number is a one-to-one function. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremltmul12ad 9957 The square of a nonnegative number is a one-to-one function. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemul12ad 9958 The square of a nonnegative number is a one-to-one function. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremlemul12bd 9959 The square of a nonnegative number is a one-to-one function. (Contributed by Mario Carneiro, 28-May-2016.)

5.3.8  Completeness Axiom and Suprema

Theoremfimaxre 9960* A finite set of real numbers has a maximum. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremfimaxre2 9961* A nonempty finite set of real numbers has a maximum. (Contributed by Jeff Madsen, 27-May-2011.) (Revised by Mario Carneiro, 13-Feb-2014.)

Theoremfimaxre3 9962* A nonempty finite set of real numbers has a maximum (image set version). (Contributed by Mario Carneiro, 13-Feb-2014.)

Theoremlbreu 9963* If a set of reals contains a lower bound, it contains a unique lower bound. (Contributed by NM, 9-Oct-2005.)

Theoremlbcl 9964* If a set of reals contains a lower bound, it contains a unique lower bound that belongs to the set. (Contributed by NM, 9-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)

Theoremlble 9965* If a set of reals contains a lower bound, the lower bound is less than or equal to all members of the set. (Contributed by NM, 9-Oct-2005.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

Theoremlbinfm 9966* If a set of reals contains a lower bound, the lower bound is its infimum. (Contributed by NM, 9-Oct-2005.) (Revised by Mario Carneiro, 16-Jun-2013.)

Theoremlbinfmcl 9967* If a set of reals contains a lower bound, it contains its infimum. (Contributed by NM, 11-Oct-2005.)

Theoremlbinfmle 9968* If a set of reals contains a lower bound, its infmimum is less than or equal to all members of the set. (Contributed by NM, 11-Oct-2005.)

Theoremsup2 9969* A non-empty, bounded-above set of reals has a supremum. Stronger version of completeness axiom (it has a slightly weaker antecedent). (Contributed by NM, 19-Jan-1997.)

Theoremsup3 9970* A version of the completeness axiom for reals. (Contributed by NM, 12-Oct-2004.)

Theoreminfm3lem 9971* Lemma for infm3 9972. (Contributed by NM, 14-Jun-2005.)

Theoreminfm3 9972* The completeness axiom for reals in terms of infimum: a non-empty, bounded-below set of reals has an infimum. (This theorem is the dual of sup3 9970.) (Contributed by NM, 14-Jun-2005.)

Theoremsuprcl 9973* Closure of supremum of a non-empty bounded set of reals. (Contributed by NM, 12-Oct-2004.)

Theoremsuprub 9974* A member of a non-empty bounded set of reals is less than or equal to the set's upper bound. (Contributed by NM, 12-Oct-2004.)

Theoremsuprlub 9975* The supremum of a non-empty bounded set of reals is the least upper bound. (Contributed by NM, 15-Nov-2004.) (Revised by Mario Carneiro, 6-Sep-2014.)

Theoremsuprnub 9976* An upper bound is not less than the supremum of a non-empty bounded set of reals. (Contributed by NM, 15-Nov-2004.) (Revised by Mario Carneiro, 6-Sep-2014.)

Theoremsuprleub 9977* The supremum of a non-empty bounded set of reals is less than or equal to an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 6-Sep-2014.)

Theoremsupmul1 9978* The supremum function distributes over multiplication, in the sense that , where is shorthand for and is defined as below. This is the simple version, with only one set argument; see supmul 9981 for the more general case with two set arguments. (Contributed by Mario Carneiro, 5-Jul-2013.)

Theoremsupmullem1 9979* Lemma for supmul 9981. (Contributed by Mario Carneiro, 5-Jul-2013.)

Theoremsupmullem2 9980* Lemma for supmul 9981. (Contributed by Mario Carneiro, 5-Jul-2013.)

Theoremsupmul 9981* The supremum function distributes over multiplication, in the sense that , where is shorthand for and is defined as below. We made use of this in our definition of multiplication in the Dedekind cut construction of the reals (see df-mp 8866). (Contributed by Mario Carneiro, 5-Jul-2013.) (Revised by Mario Carneiro, 6-Sep-2014.)

Theoremsup3ii 9982* A version of the completeness axiom for reals. (Contributed by NM, 23-Aug-1999.)

Theoremsuprclii 9983* Closure of supremum of a non-empty bounded set of reals. (Contributed by NM, 12-Sep-1999.)

Theoremsuprubii 9984* A member of a non-empty bounded set of reals is less than or equal to the set's upper bound. (Contributed by NM, 12-Sep-1999.)

Theoremsuprlubii 9985* The supremum of a non-empty bounded set of reals is the least upper bound. (Contributed by NM, 15-Oct-2004.) (Revised by Mario Carneiro, 6-Sep-2014.)

Theoremsuprnubii 9986* An upper bound is not less than the supremum of a non-empty bounded set of reals. (Contributed by NM, 15-Oct-2004.) (Revised by Mario Carneiro, 6-Sep-2014.)

Theoremsuprleubii 9987* The supremum of a non-empty bounded set of reals is less than or equal to an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 6-Sep-2014.)

Theoremriotaneg 9988* The negative of the unique real such that . (Contributed by NM, 13-Jun-2005.)

Theoremnegiso 9989 Negation is an order anti-isomorphism of the real numbers, which is its own inverse. (Contributed by Mario Carneiro, 24-Dec-2016.)

Theoremdfinfmr 9990* The infimum (expressed as supremum with converse 'less-than') of a set of reals . (Contributed by NM, 9-Oct-2005.)

Theoreminfmsup 9991* The infimum (expressed as supremum with converse 'less-than') of a set of reals is the negative of the supremum of the negatives of its elements. The antecedent ensures that is nonempty and has a lower bound. (Contributed by NM, 14-Jun-2005.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

Theoreminfmrcl 9992* Closure of infimum of a non-empty bounded set of reals. (Contributed by NM, 8-Oct-2005.)

Theoreminfmrgelb 9993* Any lower bound of a nonempty set of real numbers is less than or equal to its infimum. (Contributed by Jeff Hankins, 1-Sep-2013.) (Revised by Mario Carneiro, 6-Sep-2014.)

Theoreminfmrlb 9994* If a nonempty set of real numbers has a lower bound, its infimum is less than or equal to any of its elements. (Contributed by Jeff Hankins, 15-Sep-2013.)

5.3.9  Imaginary and complex number properties

Theoreminelr 9995 The imaginary unit is not a real number. (Contributed by NM, 6-May-1999.)

Theoremrimul 9996 A real number times the imaginary unit is real only if the number is 0. (Contributed by NM, 28-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremcru 9997 The representation of complex numbers in terms of real and imaginary parts is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremcrne0 9998 The real representation of complex numbers is nonzero iff one of its terms is nonzero. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremcreur 9999* The real part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremcreui 10000* The imaginary part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-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 32601-32700 328 32701-32800 329 32801-32835
 Copyright terms: Public domain < Previous  Next >