Home Metamath Proof ExplorerTheorem List (p. 97 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 - 9601-9700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsupmullem2 9601* Lemma for supmul 9602. (Contributed by Mario Carneiro, 5-Jul-2013.)

Theoremsupmul 9602* 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 8488). (Contributed by Mario Carneiro, 5-Jul-2013.) (Revised by Mario Carneiro, 6-Sep-2014.)

Theoremsup3ii 9603* A version of the completeness axiom for reals. (Contributed by NM, 23-Aug-1999.)

Theoremsuprclii 9604* Closure of supremum of a non-empty bounded set of reals. (Contributed by NM, 12-Sep-1999.)

Theoremsuprubii 9605* 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 9606* 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 9607* 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 9608* 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 9609* The negative of the unique real such that . (Contributed by NM, 13-Jun-2005.)

Theoremnegiso 9610 Negation is an order anti-isomorphism of the real numbers, which is its own inverse. (Contributed by Mario Carneiro, 24-Dec-2016.)

Theoremdfinfmr 9611* The infimum (expressed as supremum with converse 'less-than') of a set of reals . (Contributed by NM, 9-Oct-2005.)

Theoreminfmsup 9612* 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 9613* Closure of infimum of a non-empty bounded set of reals. (Contributed by NM, 8-Oct-2005.)

Theoreminfmrgelb 9614* 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 9615* 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 9616 The imaginary unit is not a real number. (Contributed by NM, 6-May-1999.)

Theoremrimul 9617 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 9618 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 9619 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 9620* 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 9621* 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.)

Theoremcju 9622* The complex conjugate of a complex number is unique. (Contributed by Mario Carneiro, 6-Nov-2013.)

5.3.10  Function operation analogue theorems

Theoremofsubeq0 9623 Function analog of subeq0 8953. (Contributed by Mario Carneiro, 24-Jul-2014.)

Theoremofnegsub 9624 Function analog of negsub 8975. (Contributed by Mario Carneiro, 24-Jul-2014.)

Theoremofsubge0 9625 Function analog of subge0 9167. (Contributed by Mario Carneiro, 24-Jul-2014.)

5.4  Integer sets

5.4.1  Natural numbers (as a subset of complex numbers)

Syntaxcn 9626 Extend class notation to include the class of positive integers.

Definitiondf-n 9627 The natural numbers of analysis start at one (unlike the ordinal natural numbers, i.e. the members of the set , df-om 4548, which start at zero). This is the convention used by most analysis books, and it is often convenient in proofs because we don't have to worry about division by zero. See nnind 9644 for the principle of mathematical induction. See dfnn2 9639 for a slight variant. See df-n0 9845 for the set of nonnegative integers starting at zero. See dfn2 9857 for defined in terms of .

This is a technical definition that helps us avoid the Axiom of Infinity in certain proofs. For a more conventional and intuitive definition ("the smallest set of reals containing as well as the successor of every member") see dfnn3 9640. (Contributed by NM, 10-Jan-1997.)

TheoremnnexALT 9628 The set of natural numbers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 3-May-2014.) (New usage is discouraged.)

Theorempeano5nni 9629* Peano's inductive postulate. Theorem I.36 (principle of mathematical induction) of [Apostol] p. 34. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremnnssre 9630 The natural numbers are a subset of the reals. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)

Theoremnnsscn 9631 The natural numbers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)

Theoremnnex 9632 The set of natural numbers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremnnre 9633 A natural number is a real number. (Contributed by NM, 18-Aug-1999.)

Theoremnncn 9634 A natural number is a complex number. (Contributed by NM, 18-Aug-1999.)

Theoremnnrei 9635 A natural number is a real number. (Contributed by NM, 18-Aug-1999.)

Theoremnncni 9636 A natural number is a complex number. (Contributed by NM, 18-Aug-1999.)

Theorem1nn 9637 Peano postulate: 1 is a natural number. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theorempeano2nn 9638 Peano postulate: a successor of a natural number is a natural number. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)

Theoremdfnn2 9639* Alternate definition of the set of natural numbers. This was our original definition, before the current df-n 9627 replaced it. This definition requires the axiom of infinity to ensure it has the properties we expect. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.)

Theoremdfnn3 9640* Alternate definition of the set of natural numbers. Definition of positive integers in [Apostol] p. 22. (Contributed by NM, 3-Jul-2005.)

Theoremnnred 9641 A natural number is a real number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnncnd 9642 A natural number is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)

Theorempeano2nnd 9643 Peano postulate: a successor of a natural number is a natural number. (Contributed by Mario Carneiro, 27-May-2016.)

5.4.2  Principle of mathematical induction

Theoremnnind 9644* Principle of Mathematical Induction (inference schema). The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis. See nnaddcl 9648 for an example of its use. See nn0ind 9987 for induction on nonnegative integers and uzind 9982, uzind4 10155 for induction on an arbitrary set of upper integers. See indstr 10166 for strong induction. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)

TheoremnnindALT 9645* Principle of Mathematical Induction (inference schema). The last four hypotheses give us the substitution instances we need; the first two are the induction hypothesis and the basis. (This ALT version of nnind 9644 is easier to use with the Proof Assistant since 'assign last' will be applied to the substitution instances first. We may switch to it as the official version.) (Contributed by NM, 7-Dec-2005.)

Theoremnn1m1nn 9646 Every natural number is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)

Theoremnn1suc 9647* If a statement holds for 1 and also holds for a successor, it holds for all natural numbers. The first three hypotheses give us the substitution instances we need; the last two show that it holds for 1 and for a successor. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 16-May-2014.)

Theoremnnaddcl 9648 Closure of addition of natural numbers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.)

Theoremnnmulcl 9649 Closure of multiplication of natural numbers. (Contributed by NM, 12-Jan-1997.)

Theoremnnmulcli 9650 Closure of multiplication of natural numbers. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremnn2ge 9651* There exists a natural number greater than or equal to any two others. (Contributed by NM, 18-Aug-1999.)

Theoremnnge1 9652 A natural number is one or greater. (Contributed by NM, 25-Aug-1999.)

Theoremnngt1ne1 9653 A natural number is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.)

Theoremnnle1eq1 9654 A natural number is less than or equal to one iff it is equal to one. (Contributed by NM, 3-Apr-2005.)

Theoremnngt0 9655 A natural number is positive. (Contributed by NM, 26-Sep-1999.)

Theoremnnnlt1 9656 A natural number is not less than one. (Contributed by NM, 18-Jan-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theorem0nnn 9657 Zero is not a natural number. (Contributed by NM, 25-Aug-1999.)

Theoremnnne0 9658 A natural number is nonzero. (Contributed by NM, 27-Sep-1999.)

Theoremnngt0i 9659 A natural number is positive (inference version). (Contributed by NM, 17-Sep-1999.)

Theoremnnne0i 9660 A natural number is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)

Theoremnndivre 9661 The quotient of a real and a natural number is real. (Contributed by NM, 28-Nov-2008.)

Theoremnnrecre 9662 The reciprocal of a natural number is real. (Contributed by NM, 8-Feb-2008.)

Theoremnnrecgt0 9663 The reciprocal of a natural number is positive. (Contributed by NM, 25-Aug-1999.)

Theoremnnsub 9664 Subtraction of natural numbers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.)

Theoremnnsubi 9665 Subtraction of natural numbers. (Contributed by NM, 19-Aug-2001.)

Theoremnndiv 9666* Two ways to express " divides " for natural numbers. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)

Theoremnndivtr 9667 Transitive property of divisibility: if divides and divides , then divides . Typically would be an integer, although the theorem holds for complex . (Contributed by NM, 3-May-2005.)

Theoremnnge1d 9668 A natural number is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnngt0d 9669 A natural number is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnnne0d 9670 A natural number is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnnrecred 9671 The reciprocal of a natural number is real. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnnaddcld 9672 Closure of addition of natural numbers. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnnmulcld 9673 Closure of multiplication of natural numbers. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnndivred 9674 A natural number is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)

5.4.3  Decimal representation of numbers

Note that the numbers 0 and 1 are constants defined as primitives of the complex number axiom system (see df-0 8624 and df-1 8625).

Only the digits 0 through 9 (df-0 8624 through df-9 9691) and the number 10 (df-10 9692) are explicitly defined.

We will later define the decimal constructor df-dec 10004, which will allow us to easily express larger integers in base 10. See deccl 10017 and the theorems that follow it. See also 4001prm 13017 (4001 is prime) and the proof of bpos 20364. Note that the decimal constructor builds on the definitions in this section.

Integers can also be exhibited as sums of powers of 10 or as some other expression built from operations on the numbers 0 through 10. For example, the prime number 823541 can be expressed as . Decimals can be expressed as ratios of integers, as in cos2bnd 12342.

Most abstract math rarely requires numbers larger than 4. Even in Wiles' proof of Fermat's Last Theorem, the largest number used appears to be 12.

Syntaxc2 9675 Extend class notation to include the number 2.

Syntaxc3 9676 Extend class notation to include the number 3.

Syntaxc4 9677 Extend class notation to include the number 4.

Syntaxc5 9678 Extend class notation to include the number 5.

Syntaxc6 9679 Extend class notation to include the number 6.

Syntaxc7 9680 Extend class notation to include the number 7.

Syntaxc8 9681 Extend class notation to include the number 8.

Syntaxc9 9682 Extend class notation to include the number 9.

Syntaxc10 9683 Extend class notation to include the number 10.

Definitiondf-2 9684 Define the number 2. (Contributed by NM, 27-May-1999.)

Definitiondf-3 9685 Define the number 3. (Contributed by NM, 27-May-1999.)

Definitiondf-4 9686 Define the number 4. (Contributed by NM, 27-May-1999.)

Definitiondf-5 9687 Define the number 5. (Contributed by NM, 27-May-1999.)

Definitiondf-6 9688 Define the number 6. (Contributed by NM, 27-May-1999.)

Definitiondf-7 9689 Define the number 7. (Contributed by NM, 27-May-1999.)

Definitiondf-8 9690 Define the number 8. (Contributed by NM, 27-May-1999.)

Definitiondf-9 9691 Define the number 9. (Contributed by NM, 27-May-1999.)

Definitiondf-10 9692 Define the number 10. See remarks under df-2 9684. (Contributed by NM, 5-Feb-2007.)

Theoremneg1cn 9693 -1 is a complex number. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)

Theorem1m1e0 9694 . Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)

Theorem2re 9695 The number 2 is real. (Contributed by NM, 27-May-1999.)

Theorem2cn 9696 The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.)

Theorem3re 9697 The number 3 is real. (Contributed by NM, 27-May-1999.)

Theorem3cn 9698 The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.)

Theorem4re 9699 The number 4 is real. (Contributed by NM, 27-May-1999.)

Theorem4cn 9700 The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-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 30801-30843
 Copyright terms: Public domain < Previous  Next >