Home Metamath Proof ExplorerTheorem List (p. 102 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 - 10101-10200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem9t5e45 10101 9 times 5 equals 45. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9t6e54 10102 9 times 6 equals 54. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9t7e63 10103 9 times 7 equals 63. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9t8e72 10104 9 times 8 equals 72. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem9t9e81 10105 9 times 9 equals 81. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theoremdecbin0 10106 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdecbin2 10107 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremdecbin3 10108 Decompose base 4 into base 2. (Contributed by Mario Carneiro, 18-Feb-2014.)

5.4.9  Upper partititions of integers

Syntaxcuz 10109 Extend class notation with the upper integer function. Read " " as "the set of integers greater than or equal to ."

Definitiondf-uz 10110* Define a function whose value at is the semi-infinite set of contiguous integers starting at , which we will also call the upper integers starting at . Read " " as "the set of integers greater than or equal to ." See uzval 10111 for its value, uzssz 10126 for its relationship to , nnuz 10142 and nn0uz 10141 for its relationships to and , and eluz1 10113 and eluz2 10115 for its membership relations. (Contributed by NM, 5-Sep-2005.)

Theoremuzval 10111* The value of the upper integers function. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremuzf 10112 The domain and range of the upper integers function. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremeluz1 10113 Membership in the set of upper integers starting at . (Contributed by NM, 5-Sep-2005.)

Theoremeluzel2 10114 Implication of membership in a set of upper integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremeluz2 10115 Membership in a set of upper integers. We use the fact that a function's value (under our function value definition) is empty outside of its domain to show . (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremeluz1i 10116 Membership in a set of upper integers. (Contributed by NM, 5-Sep-2005.)

Theoremeluzelz 10117 A member of a set of upper integers is an integer. (Contributed by NM, 6-Sep-2005.)

Theoremeluzelre 10118 A member of a set of upper integers is a real. (Contributed by Mario Carneiro, 31-Aug-2013.)

Theoremeluzle 10119 Implication of membership in a set of upper integers. (Contributed by NM, 6-Sep-2005.)

Theoremeluz 10120 Membership in a set of upper integers. (Contributed by NM, 2-Oct-2005.)

Theoremuzid 10121 Membership of the least member in a set of upper integers. (Contributed by NM, 2-Sep-2005.)

Theoremuzn0 10122 The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)

Theoremuztrn 10123 Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.)

Theoremuztrn2 10124 Transitive law for sets of upper integers. (Contributed by Mario Carneiro, 26-Dec-2013.)

Theoremuzneg 10125 Contraposition law for upper integers. (Contributed by NM, 28-Nov-2005.)

Theoremuzssz 10126 A set of upper integers is a subset of all integers. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)

Theoremuzss 10127 Subset relationship for two sets of upper integers. (Contributed by NM, 5-Sep-2005.)

Theoremuztric 10128 Totality of the ordering relation on integers, stated in terms of upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jun-2013.)

Theoremuz11 10129 The upper integers function is one-to-one. (Contributed by NM, 12-Dec-2005.)

Theoremeluzp1m1 10130 Membership in the next set of upper integers. (Contributed by NM, 12-Sep-2005.)

Theoremeluzp1l 10131 Strict ordering implied by membership in the next set of upper integers. (Contributed by NM, 12-Sep-2005.)

Theoremeluzp1p1 10132 Membership in the next set of upper integers. (Contributed by NM, 5-Oct-2005.)

Theoremeluzaddi 10133 Membership in a later set of upper integers. (Contributed by Paul Chapman, 22-Nov-2007.)

Theoremeluzsubi 10134 Membership in an earlier set of upper integers. (Contributed by Paul Chapman, 22-Nov-2007.)

Theoremeluzadd 10135 Membership in a later set of upper integers. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremeluzsub 10136 Membership in an earlier set of upper integers. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremuzm1 10137 Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremuznn0sub 10138 The nonnegative difference of integers is a nonnegative integer. (Contributed by NM, 4-Sep-2005.)

Theoremuzin 10139 Intersection of two upper intervals of integers. (Contributed by Mario Carneiro, 24-Dec-2013.)

Theoremuzp1 10140 Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremnn0uz 10141 Nonnegative integers expressed as a set of upper integers. (Contributed by NM, 2-Sep-2005.)

Theoremnnuz 10142 Natural numbers expressed as a set of upper integers. (Contributed by NM, 2-Sep-2005.)

Theoremelnnuz 10143 A natural number expressed as a member of a set of upper integers. (Contributed by NM, 6-Jun-2006.)

Theoremelnn0uz 10144 A nonnegative integer expressed as a member a set of upper integers. (Contributed by NM, 6-Jun-2006.)

Theoremuznnssnn 10145 The upper integers starting from a natural are a subset of the naturals. (Contributed by Scott Fenton, 29-Jun-2013.)

Theoremraluz 10146* Restricted universal quantification in a set of upper integers. (Contributed by NM, 9-Sep-2005.)

Theoremraluz2 10147* Restricted universal quantification in a set of upper integers. (Contributed by NM, 9-Sep-2005.)

Theoremrexuz 10148* Restricted existential quantification in a set of upper integers. (Contributed by NM, 9-Sep-2005.)

Theoremrexuz2 10149* Restricted existential quantification in a set of upper integers. (Contributed by NM, 9-Sep-2005.)

Theorem2rexuz 10150* Double existential quantification in a set of upper integers. (Contributed by NM, 3-Nov-2005.)

Theorempeano2uz 10151 Second Peano postulate for a set of upper integers. (Contributed by NM, 7-Sep-2005.)

Theorempeano2uzs 10152 Second Peano postulate for a set of upper integers. (Contributed by Mario Carneiro, 26-Dec-2013.)

Theorempeano2uzr 10153 Reversed second Peano axiom for upper integers. (Contributed by NM, 2-Jan-2006.)

Theoremuzaddcl 10154 Addition closure law for a set of upper integers. (Contributed by NM, 4-Jun-2006.)

Theoremuzind4 10155* Induction on the set of upper integers that starts at an integer . The first four hypotheses give us the substitution instances we need, and the last two are the basis and the induction hypothesis. (Contributed by NM, 7-Sep-2005.)

Theoremuzind4ALT 10156* Alternate version of uzind4 10155 with different hypothesis order for easier use with the Metamath Proof Assistant, since "assign last" will assign the substitutions first. (This may or may not be kept permanenently, or it may replace uzind4 10155- I haven't decided yet. --NM) (Contributed by NM, 7-Sep-2005.)

Theoremuzind4s 10157* Induction on the set of upper integers that starts at an integer , using explicit substitution. The hypotheses are the basis and the induction hypothesis. (Contributed by NM, 4-Nov-2005.)

Theoremuzind4s2 10158* Induction on the set of upper integers that starts at an integer , using explicit substitution. The hypotheses are the basis and the induction hypothesis. Use this instead of uzind4s 10157 when and must be distinct in . (Contributed by NM, 16-Nov-2005.)

Theoremuzind4i 10159* Induction on the upper integers that start at . The first hypothesis specifies the lower bound, the next four give us the substitution instances we need, and the last two are the basis and the induction hypothesis. (Contributed by NM, 4-Sep-2005.)

Theoremuzwo 10160* Well-ordering principle: any non-empty subset of a set of upper integers has a least element. (Contributed by NM, 8-Oct-2005.)

TheoremuzwoOLD 10161* Well-ordering principle: any non-empty subset of the upper integers has a least element. (Contributed by NM, 8-Oct-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremuzwo2 10162* Well-ordering principle: any non-empty subset of upper integers has a unique least element. (Contributed by NM, 8-Oct-2005.)

Theoremnnwo 10163* Well-ordering principle: any non-empty set of natural numbers has a least element. Theorem I.37 (well-ordering principle) of [Apostol] p. 34. (Contributed by NM, 17-Aug-2001.)

Theoremnnwof 10164* Well-ordering principle: any non-empty set of natural numbers has a least element. This version allows and to be present in as long as they are effectively not free. (Contributed by NM, 17-Aug-2001.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremnnwos 10165* Well-ordering principle: any non-empty set of natural numbers has a least element (schema form). (Contributed by NM, 17-Aug-2001.)

Theoremindstr 10166* Strong Mathematical Induction for positive integers (inference schema). (Contributed by NM, 17-Aug-2001.)

Theoremeluznn0 10167 Membership in a nonegative set of upper integers implies membership in . (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremeluz2b1 10168 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)

Theoremeluz2b2 10169 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)

Theoremeluz2b3 10170 Two ways to say "an integer greater than or equal to 2." (Contributed by Paul Chapman, 23-Nov-2012.)

Theoremuz2m1nn 10171 One less than an integer greater than or equal to 2 is a positive integer. (Contributed by Paul Chapman, 17-Nov-2012.)

Theorem1nuz2 10172 1 is not in . (Contributed by Paul Chapman, 21-Nov-2012.)

Theoremelnn1uz2 10173 A positive integer is either 1 or greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.)

Theoremuz2mulcl 10174 Closure of multiplication of integers greater than or equal to 2. (Contributed by Paul Chapman, 26-Oct-2012.)

Theoremindstr2 10175* Strong Mathematical Induction for positive integers (inference schema). The first two hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis. (Contributed by Paul Chapman, 21-Nov-2012.)

Theoremuzinfmi 10176 Extract the lower bound of a set of upper integers as its infimum. Note that the " " argument turns supremum into infimum (for which we do not currently have a separate notation). (Contributed by NM, 7-Oct-2005.)

Theoremnninfm 10177 The infimum of the set of natural numbers is one. Note that " " turns sup into inf. (Contributed by NM, 16-Jun-2005.)

Theoremnn0infm 10178 The infimum of the set of nonnegative integers is zero. Note that " " turns sup into inf. (Contributed by NM, 16-Jun-2005.)

Theoreminfmssuzle 10179 The infimum of a subset of a set of upper integers is less than or equal to all members of the subset. Note that the " " argument turns supremum into infimum (for which we do not currently have a separate notation). (Contributed by NM, 11-Oct-2005.)

Theoreminfmssuzcl 10180 The infimum of a subset of a set of upper integers belongs to the subset. (Contributed by NM, 11-Oct-2005.)

Theoremublbneg 10181* The image under negation of a bounded-above set of reals is bounded below. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremeqreznegel 10182* Two ways to express the image under negation of a set of integers. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremnegn0 10183* The image under negation of a nonempty set of reals is nonempty. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremsupminf 10184* The supremum of a bounded-above set of reals is the negation of the supremum of that set's image under negation. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremlbzbi 10185* If a set of reals is bounded below, it is bounded below by an integer. (Contributed by Paul Chapman, 21-Mar-2011.)

Theoremzsupss 10186* Any nonempty bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-sup 8695.) (Contributed by Mario Carneiro, 21-Apr-2015.)

Theoremsuprzcl2 10187* The supremum of a bounded-above set of integers is a member of the set. (This version of suprzcl 9970 avoids ax-pre-sup 8695.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Mario Carneiro, 24-Dec-2016.)

Theoremsuprzub 10188* The supremum of a bounded-above set of integers is greater than any member of the set. (Contributed by Mario Carneiro, 21-Apr-2015.)

Theoremuzsupss 10189* Any bounded subset of upper integers has a supremum. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 21-Apr-2015.)

5.4.10  Well-ordering principle for bounded-below sets of integers

Theoremuzwo3 10190* Well-ordering principle: any non-empty subset of upper integers has a unique least element. This generalization of uzwo2 10162 allows the lower bound to be any real number. See also nnwo 10163 and nnwos 10165. (Contributed by NM, 12-Nov-2004.) (Proof shortened by Mario Carneiro, 2-Oct-2015.)

Theoremzmin 10191* There is a unique smallest integer greater than or equal to a given real number. (Contributed by NM, 12-Nov-2004.) (Revised by Mario Carneiro, 13-Jun-2014.)

Theoremzmax 10192* There is a unique largest integer less than or equal to a given real number. (Contributed by NM, 15-Nov-2004.)

Theoremzbtwnre 10193* There is a unique integer between a real number and the number plus one. Exercise 5 of [Apostol] p. 28. (Contributed by NM, 13-Nov-2004.)

Theoremrebtwnz 10194* There is a unique greatest integer less than or equal to a real number. Exercise 4 of [Apostol] p. 28. (Contributed by NM, 15-Nov-2004.)

5.4.11  Rational numbers (as a subset of complex numbers)

Syntaxcq 10195 Extend class notation to include the class of rationals.

Definitiondf-q 10196 Define the set of rational numbers. Based on definition of rationals in [Apostol] p. 22. See elq 10197 for the relation "is rational." (Contributed by NM, 8-Jan-2002.)

Theoremelq 10197* Membership in the set of rationals. (Contributed by NM, 8-Jan-2002.) (Revised by Mario Carneiro, 28-Jan-2014.)

Theoremqmulz 10198* If is rational, then some integer multiple of it is an integer. (Contributed by NM, 7-Nov-2008.) (Revised by Mario Carneiro, 22-Jul-2014.)

Theoremznq 10199 The ratio of an integer and a natural number is a rational number. (Contributed by NM, 12-Jan-2002.)

Theoremqre 10200 A rational number is a real number. (Contributed by NM, 14-Nov-2002.)

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 >