Home Metamath Proof ExplorerTheorem List (p. 303 of 329) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-22423) Hilbert Space Explorer (22424-23946) Users' Mathboxes (23947-32824)

Theorem List for Metamath Proof Explorer - 30201-30300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcvlatcvr1 30201 An atom is covered by its join with a different atom. (Contributed by NM, 5-Nov-2012.)

Theoremcvlatcvr2 30202 An atom is covered by its join with a different atom. (Contributed by NM, 5-Nov-2012.)

Theoremcvlsupr2 30203 Two equivalent ways of expressing that is a superposition of and . (Contributed by NM, 5-Nov-2012.)

Theoremcvlsupr3 30204 Two equivalent ways of expressing that is a superposition of and , which can replace the superposition part of ishlat1 30212, , with the simpler as shown in ishlat3N 30214. (Contributed by NM, 5-Nov-2012.)

Theoremcvlsupr4 30205 Consequence of superposition condition . (Contributed by NM, 9-Nov-2012.)

Theoremcvlsupr5 30206 Consequence of superposition condition . (Contributed by NM, 9-Nov-2012.)

Theoremcvlsupr6 30207 Consequence of superposition condition . (Contributed by NM, 9-Nov-2012.)

Theoremcvlsupr7 30208 Consequence of superposition condition . (Contributed by NM, 24-Nov-2012.)

Theoremcvlsupr8 30209 Consequence of superposition condition . (Contributed by NM, 24-Nov-2012.)

19.26.8  Hilbert lattices

Syntaxchlt 30210 Extend class notation with Hilbert lattices.

Definitiondf-hlat 30211* Define the class of Hilbert lattices, which are complete, atomic lattices satisfying the superposition principle and minimum height. (Contributed by NM, 5-Nov-2012.)

Theoremishlat1 30212* The predicate "is a Hilbert lattice," which is orthomodular ( ), complete ( ), atomic and satisfying the exchange (or covering) property ( ), satisfies the superposition principle, and has a minimum height of 4. (Contributed by NM, 5-Nov-2012.)

Theoremishlat2 30213* The predicate "is a Hilbert lattice". Here we replace with the weaker and show the exchange property explicitly. (Contributed by NM, 5-Nov-2012.)

Theoremishlat3N 30214* The predicate "is a Hilbert lattice". Note that the superposition principle is expressed in the compact form . The exchange property and atomicity are provided by , and "minimum height 4" is shown explicitly. (Contributed by NM, 8-Nov-2012.) (New usage is discouraged.)

TheoremishlatiN 30215* Properties that determine a Hilbert lattice. (Contributed by NM, 13-Nov-2011.) (New usage is discouraged.)

Theoremhlomcmcv 30216 A Hilbert lattice is orthomodular, complete, and has the covering (exchange) property. (Contributed by NM, 5-Nov-2012.)

Theoremhloml 30217 A Hilbert lattice is orthomodular. (Contributed by NM, 20-Oct-2011.)

Theoremhlclat 30218 A Hilbert lattice is complete. (Contributed by NM, 20-Oct-2011.)

Theoremhlcvl 30219 A Hilbert lattice is an atomic lattice with the covering property. (Contributed by NM, 5-Nov-2012.)

Theoremhlatl 30220 A Hilbert lattice is atomic. (Contributed by NM, 20-Oct-2011.)

Theoremhlol 30221 A Hilbert lattice is an ortholattice. (Contributed by NM, 20-Oct-2011.)

Theoremhlop 30222 A Hilbert lattice is an orthoposet. (Contributed by NM, 20-Oct-2011.)

Theoremhllat 30223 A Hilbert lattice is a lattice. (Contributed by NM, 20-Oct-2011.)

Theoremhlomcmat 30224 A Hilbert lattice is orthomodular, complete, and atomic. (Contributed by NM, 5-Nov-2012.)

Theoremhlpos 30225 A Hilbert lattice is a poset. (Contributed by NM, 20-Oct-2011.)

Theoremhlatjcl 30226 Closure of join operation. Frequently-used special case of latjcl 14481 for atoms. (Contributed by NM, 15-Jun-2012.)

Theoremhlatjcom 30227 Commutatitivity of join operation. Frequently-used special case of latjcom 14490 for atoms. (Contributed by NM, 15-Jun-2012.)

Theoremhlatjidm 30228 Idempotence of join operation. Frequently-used special case of latjcom 14490 for atoms. (Contributed by NM, 15-Jul-2012.)

Theoremhlatjass 30229 Lattice join is associative. Frequently-used special case of latjass 14526 for atoms. (Contributed by NM, 27-Jul-2012.)

Theoremhlatj12 30230 Swap 1st and 2nd members of lattice join. Frequently-used special case of latj32 14528 for atoms. (Contributed by NM, 4-Jun-2012.)

Theoremhlatj32 30231 Swap 2nd and 3rd members of lattice join. Frequently-used special case of latj32 14528 for atoms. (Contributed by NM, 21-Jul-2012.)

Theoremhlatjrot 30232 Rotate lattice join of 3 classes. Frequently-used special case of latjrot 14531 for atoms. (Contributed by NM, 2-Aug-2012.)

Theoremhlatj4 30233 Rearrangement of lattice join of 4 classes. Frequently-used special case of latj4 14532 for atoms. (Contributed by NM, 9-Aug-2012.)

Theoremhlatlej1 30234 A join's first argument is less than or equal to the join. Special case of latlej1 14491 to show an atom is on a line. (Contributed by NM, 15-May-2013.)

Theoremhlatlej2 30235 A join's second argument is less than or equal to the join. Special case of latlej2 14492 to show an atom is on a line. (Contributed by NM, 15-May-2013.)

TheoremglbconN 30236* De Morgan's law for GLB and LUB. This holds in any complete ortholattice, although we assume for convenience. (Contributed by NM, 17-Jan-2012.) (New usage is discouraged.)

TheoremglbconxN 30237* De Morgan's law for GLB and LUB. Index-set version of glbconN 30236, where we read as . (Contributed by NM, 17-Jan-2012.) (New usage is discouraged.)

Theorematnlej1 30238 If an atom is not less than or equal to the join of two others, it is not equal to either. (This also holds for non-atoms, but in this form it is convenient.) (Contributed by NM, 8-Jan-2012.)

Theorematnlej2 30239 If an atom is not less than or equal to the join of two others, it is not equal to either. (This also holds for non-atoms, but in this form it is convenient.) (Contributed by NM, 8-Jan-2012.)

Theoremhlsuprexch 30240* A Hilbert lattice has the superposition and exchange properties. (Contributed by NM, 13-Nov-2011.)

Theoremhlexch1 30241 A Hilbert lattice has the exchange property. (Contributed by NM, 13-Nov-2011.)

Theoremhlexch2 30242 A Hilbert lattice has the exchange property. (Contributed by NM, 6-May-2012.)

Theoremhlexchb1 30243 A Hilbert lattice has the exchange property. (Contributed by NM, 16-Nov-2011.)

Theoremhlexchb2 30244 A Hilbert lattice has the exchange property. (Contributed by NM, 22-Jun-2012.)

Theoremhlsupr 30245* A Hilbert lattice has the superposition property. Theorem 13.2 in [Crawley] p. 107. (Contributed by NM, 30-Jan-2012.)

Theoremhlsupr2 30246* A Hilbert lattice has the superposition property. (Contributed by NM, 25-Nov-2012.)

Theoremhlhgt4 30247* A Hilbert lattice has a height of at least 4. (Contributed by NM, 4-Dec-2011.)

Theoremhlhgt2 30248* A Hilbert lattice has a height of at least 2. (Contributed by NM, 4-Dec-2011.)

Theoremhl0lt1N 30249 Lattice 0 is less than lattice 1 in a Hilbert lattice. (Contributed by NM, 4-Dec-2011.) (New usage is discouraged.)

Theoremhlexch3 30250 A Hilbert lattice has the exchange property. (atexch 23886 analog.) (Contributed by NM, 15-Nov-2011.)

Theoremhlexch4N 30251 A Hilbert lattice has the exchange property. Part of Definition 7.8 of [MaedaMaeda] p. 32. (Contributed by NM, 15-Nov-2011.) (New usage is discouraged.)

Theoremhlatexchb1 30252 A version of hlexchb1 30243 for atoms. (Contributed by NM, 15-Nov-2011.)

Theoremhlatexchb2 30253 A version of hlexchb2 30244 for atoms. (Contributed by NM, 7-Feb-2012.)

Theoremhlatexch1 30254 Atom exchange property. (Contributed by NM, 7-Jan-2012.)

Theoremhlatexch2 30255 Atom exchange property. (Contributed by NM, 8-Jan-2012.)

TheoremhlatmstcOLDN 30256* An atomic, complete, orthomodular lattice is atomistic i.e. every element is the join of the atoms under it. See remark before Proposition 1 in [Kalmbach] p. 140; also remark in [BeltramettiCassinelli] p. 98. (hatomistici 23867 analog.) (Contributed by NM, 21-Oct-2011.) (New usage is discouraged.)

Theoremhlatle 30257* The ordering of two Hilbert lattice elements is determined by the atoms under them. (chrelat3 23876 analog.) (Contributed by NM, 4-Nov-2011.)

Theoremhlateq 30258* The equality of two Hilbert lattice elements is determined by the atoms under them. (chrelat4i 23878 analog.) (Contributed by NM, 24-May-2012.)

Theoremhlrelat1 30259* An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 23868, with swapped, analog.) (Contributed by NM, 4-Dec-2011.)

Theoremhlrelat5N 30260* An atomistic lattice with 0 is relatively atomic, using the definition in Remark 2 of [Kalmbach] p. 149. (Contributed by NM, 21-Oct-2011.) (New usage is discouraged.)

Theoremhlrelat 30261* A Hilbert lattice is relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 23869 analog.) (Contributed by NM, 4-Feb-2012.)

Theoremhlrelat2 30262* A consequence of relative atomicity. (chrelat2i 23870 analog.) (Contributed by NM, 5-Feb-2012.)

TheoremexatleN 30263 A condition for an atom to be less than or equal to a lattice element. Part of proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 28-Apr-2012.) (New usage is discouraged.)

Theoremhl2at 30264* A Hilbert lattice has at least 2 atoms. (Contributed by NM, 5-Dec-2011.)

Theorematex 30265 At least one atom exists. (Contributed by NM, 15-Jul-2012.)

TheoremintnatN 30266 If the intersection with a non-majorizing element is an atom, the intersecting element is not an atom. (Contributed by NM, 26-Jun-2012.) (New usage is discouraged.)

Theorem2llnne2N 30267 Condition implying that two intersecting lines are different. (Contributed by NM, 13-Jun-2012.) (New usage is discouraged.)

Theorem2llnneN 30268 Condition implying that two intersecting lines are different. (Contributed by NM, 29-May-2012.) (New usage is discouraged.)

Theoremcvr1 30269 A Hilbert lattice has the covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 23860 analog.) (Contributed by NM, 17-Nov-2011.)

Theoremcvr2N 30270 Less-than and covers equivalence in a Hilbert lattice. (chcv2 23861 analog.) (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)

Theoremhlrelat3 30271* The Hilbert lattice is relatively atomic. Stronger version of hlrelat 30261. (Contributed by NM, 2-May-2012.)

Theoremcvrval3 30272* Binary relation expressing covers . (Contributed by NM, 16-Jun-2012.)

Theoremcvrval4N 30273* Binary relation expressing covers . (Contributed by NM, 16-Jun-2012.) (New usage is discouraged.)

Theoremcvrval5 30274* Binary relation expressing covers . (Contributed by NM, 7-Dec-2012.)

Theoremcvrp 30275 A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 23880 analog.) (Contributed by NM, 18-Nov-2011.)

Theorematcvr1 30276 An atom is covered by its join with a different atom. (Contributed by NM, 7-Feb-2012.)

Theorematcvr2 30277 An atom is covered by its join with a different atom. (Contributed by NM, 7-Feb-2012.)

Theoremcvrexchlem 30278 Lemma for cvrexch 30279. (cvexchlem 23873 analog.) (Contributed by NM, 18-Nov-2011.)

Theoremcvrexch 30279 A Hilbert lattice satisfies the exchange axiom. Proposition 1(iii) of [Kalmbach] p. 140 and its converse. Originally proved by Garrett Birkhoff in 1933. (cvexchi 23874 analog.) (Contributed by NM, 18-Nov-2011.)

Theoremcvratlem 30280 Lemma for cvrat 30281. (atcvatlem 23890 analog.) (Contributed by NM, 22-Nov-2011.)

Theoremcvrat 30281 A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 23891 analog.) (Contributed by NM, 22-Nov-2011.)

Theoremltltncvr 30282 A chained strong ordering is not a covers relation. (Contributed by NM, 18-Jun-2012.)

Theoremltcvrntr 30283 Non-transitive condition for the covers relation. (Contributed by NM, 18-Jun-2012.)

Theoremcvrntr 30284 The covers relation is not transitive. (cvntr 23797 analog.) (Contributed by NM, 18-Jun-2012.)

Theorematcvr0eq 30285 The covers relation is not transitive. (atcv0eq 23884 analog.) (Contributed by NM, 29-Nov-2011.)

Theoremlnnat 30286 A line (the join of two distinct atoms) is not an atom. (Contributed by NM, 14-Jun-2012.)

Theorematcvrj0 30287 Two atoms covering the zero subspace are equal. (atcv1 23885 analog.) (Contributed by NM, 29-Nov-2011.)

Theoremcvrat2 30288 A Hilbert lattice element covered by the join of two distinct atoms is an atom. (atcvat2i 23892 analog.) (Contributed by NM, 30-Nov-2011.)

TheorematcvrneN 30289 Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)

Theorematcvrj1 30290 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)

Theorematcvrj2b 30291 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)

Theorematcvrj2 30292 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)

TheorematleneN 30293 Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)

Theorematltcvr 30294 An equivalence of less-than ordering and covers relation. (Contributed by NM, 7-Feb-2012.)

Theorematle 30295* Any non-zero element has an atom under it. (Contributed by NM, 28-Jun-2012.)

Theorematlt 30296 Two atoms are unequal iff their join is greater than one of them. (Contributed by NM, 6-May-2012.)

Theorematlelt 30297 Transfer less-than relation from one atom to another. (Contributed by NM, 7-May-2012.)

Theorem2atlt 30298* Given an atom less than an element, there is another atom less than the element. (Contributed by NM, 6-May-2012.)

TheorematexchcvrN 30299 Atom exchange property. Version of hlatexch2 30255 with covers relation. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)

TheorematexchltN 30300 Atom exchange property. Version of hlatexch2 30255 with less-than ordering. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)

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 32801-32824
 Copyright terms: Public domain < Previous  Next >