Home Metamath Proof ExplorerTheorem List (p. 301 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 - 30001-30100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremopoc0 30001 Orthocomplement of orthoposet zero. (Contributed by NM, 24-Jan-2012.)

Theoremopltcon3b 30002 Contraposition law for strict ordering in orthoposets. (chpsscon3 23005 analog.) (Contributed by NM, 4-Nov-2011.)

Theoremopltcon1b 30003 Contraposition law for strict ordering in orthoposets. (chpsscon1 23006 analog.) (Contributed by NM, 5-Nov-2011.)

Theoremopltcon2b 30004 Contraposition law for strict ordering in orthoposets. (chsscon2 23004 analog.) (Contributed by NM, 5-Nov-2011.)

Theoremopexmid 30005 Law of excluded middle for orthoposets. (chjo 23017 analog.) (Contributed by NM, 13-Sep-2011.)

Theoremopnoncon 30006 Law of contradiction for orthoposets. (chocin 22997 analog.) (Contributed by NM, 13-Sep-2011.)

TheoremriotaocN 30007* The orthocomplement of the unique poset element such that . (riotaneg 9983 analog.) (Contributed by NM, 16-Jan-2012.) (New usage is discouraged.)

TheoremcmtfvalN 30008* Value of commutes relation. (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)

TheoremcmtvalN 30009 Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 23086 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)

Theoremisolat 30010 The predicate "is an ortholattice." (Contributed by NM, 18-Sep-2011.)

Theoremollat 30011 An ortholattice is a lattice. (Contributed by NM, 18-Sep-2011.)

Theoremolop 30012 An ortholattice is an orthoposet. (Contributed by NM, 18-Sep-2011.)

TheoremolposN 30013 An ortholattice is a poset. (Contributed by NM, 16-Oct-2011.) (New usage is discouraged.)

TheoremisolatiN 30014 Properties that determine an ortholattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)

Theoremoldmm1 30015 De Morgan's law for meet in an ortholattice. (chdmm1 23027 analog.) (Contributed by NM, 6-Nov-2011.)

Theoremoldmm2 30016 De Morgan's law for meet in an ortholattice. (chdmm2 23028 analog.) (Contributed by NM, 6-Nov-2011.)

Theoremoldmm3N 30017 De Morgan's law for meet in an ortholattice. (chdmm3 23029 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

Theoremoldmm4 30018 De Morgan's law for meet in an ortholattice. (chdmm4 23030 analog.) (Contributed by NM, 6-Nov-2011.)

Theoremoldmj1 30019 De Morgan's law for join in an ortholattice. (chdmj1 23031 analog.) (Contributed by NM, 6-Nov-2011.)

Theoremoldmj2 30020 De Morgan's law for join in an ortholattice. (chdmj2 23032 analog.) (Contributed by NM, 7-Nov-2011.)

Theoremoldmj3 30021 De Morgan's law for join in an ortholattice. (chdmj3 23033 analog.) (Contributed by NM, 7-Nov-2011.)

Theoremoldmj4 30022 De Morgan's law for join in an ortholattice. (chdmj4 23034 analog.) (Contributed by NM, 7-Nov-2011.)

Theoremolj01 30023 An ortholattice element joined with zero equals itself. (chj0 22999 analog.) (Contributed by NM, 19-Oct-2011.)

Theoremolj02 30024 An ortholattice element joined with zero equals itself. (Contributed by NM, 28-Jan-2012.)

Theoremolm11 30025 The meet of an ortholattice element with one equals itself. (chm1i 22958 analog.) (Contributed by NM, 22-May-2012.)

Theoremolm12 30026 The meet of an ortholattice element with one equals itself. (Contributed by NM, 22-May-2012.)

TheoremlatmassOLD 30027 Ortholattice meet is associative. (This can also be proved for lattices with a longer proof.) (inass 3551 analog.) (Contributed by NM, 7-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremlatm12 30028 A rearrangement of lattice meet. (in12 3552 analog.) (Contributed by NM, 8-Nov-2011.)

Theoremlatm32 30029 A rearrangement of lattice meet. (in12 3552 analog.) (Contributed by NM, 13-Nov-2012.)

Theoremlatmrot 30030 Rotate lattice meet of 3 classes. (Contributed by NM, 9-Oct-2012.)

Theoremlatm4 30031 Rearrangement of lattice meet of 4 classes. (in4 3557 analog.) (Contributed by NM, 8-Nov-2011.)

TheoremlatmmdiN 30032 Lattice meet distributes over itself. (inindi 3558 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

Theoremlatmmdir 30033 Lattice meet distributes over itself. (inindir 3559 analog.) (Contributed by NM, 6-Jun-2012.)

Theoremolm01 30034 Meet with lattice zero is zero. (chm0 22993 analog.) (Contributed by NM, 8-Nov-2011.)

Theoremolm02 30035 Meet with lattice zero is zero. (Contributed by NM, 9-Oct-2012.)

Theoremisoml 30036* The predicate "is an orthomodular lattice." (Contributed by NM, 18-Sep-2011.)

TheoremisomliN 30037* Properties that determine an orthomodular lattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)

Theoremomlol 30038 An orthomodular lattice is an ortholattice. (Contributed by NM, 18-Sep-2011.)

Theoremomlop 30039 An orthomodular lattice is an orthoposet. (Contributed by NM, 6-Nov-2011.)

Theoremomllat 30040 An orthomodular lattice is a lattice. (Contributed by NM, 6-Nov-2011.)

Theoremomllaw 30041 The orthomodular law. (Contributed by NM, 18-Sep-2011.)

Theoremomllaw2N 30042 Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 23087 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)

Theoremomllaw3 30043 Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 22938 analog.) (Contributed by NM, 19-Oct-2011.)

Theoremomllaw4 30044 Orthomodular law equivalent. Remark in [Holland95] p. 223. (Contributed by NM, 19-Oct-2011.)

Theoremomllaw5N 30045 The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 23115 analog.) (Contributed by NM, 14-Nov-2011.) (New usage is discouraged.)

TheoremcmtcomlemN 30046 Lemma for cmtcomN 30047. (cmcmlem 23093 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)

TheoremcmtcomN 30047 Commutation is symmetric. Theorem 2(v) in [Kalmbach] p. 22. (cmcmi 23094 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)

Theoremcmt2N 30048 Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 23095 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

Theoremcmt3N 30049 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 23097 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

Theoremcmt4N 30050 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 23097 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

Theoremcmtbr2N 30051 Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (cmbr2i 23098 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

Theoremcmtbr3N 30052 Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 23110 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

Theoremcmtbr4N 30053 Alternate definition for the commutes relation. (cmbr4i 23103 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.)

TheoremlecmtN 30054 Ordered elements commute. (lecmi 23104 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.)

TheoremcmtidN 30055 Any element commutes with itself. (cmidi 23112 analog.) (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

Theoremomlfh1N 30056 Foulis-Holland Theorem, part 1. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. Part of Theorem 5 in [Kalmbach] p. 25. (fh1 23120 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

Theoremomlfh3N 30057 Foulis-Holland Theorem, part 3. Dual of omlfh1N 30056. (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)

Theoremomlmod1i2N 30058 Analog of modular law atmod1i2 30656 that holds in any OML. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremomlspjN 30059 Contraction of a Sasaki projection. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

19.26.7  Atomic lattices with covering property

Syntaxccvr 30060 Extend class notation with covers relation.

Syntaxcatm 30061 Extend class notation with atoms.

Syntaxcal 30062 Extend class notation with atomic lattices.

Syntaxclc 30063 Extend class notation with lattices with the covering property.

Definitiondf-covers 30064* Define the covers relation ("is covered by") for posets. " is covered by " means that is strictly less than and there is nothing in between. See cvrval 30067 for the relation form. (Contributed by NM, 18-Sep-2011.)

Definitiondf-ats 30065* Define the class of poset atoms. (Contributed by NM, 18-Sep-2011.)

Theoremcvrfval 30066* Value of covers relation "is covered by". (Contributed by NM, 18-Sep-2011.)

Theoremcvrval 30067* Binary relation expressing covers , which means that is larger than and there is nothing in between. Definition 3.2.18 of [PtakPulmannova] p. 68. (cvbr 23785 analog.) (Contributed by NM, 18-Sep-2011.)

Theoremcvrlt 30068 The covers relation implies the less-than relation. (cvpss 23788 analog.) (Contributed by NM, 8-Oct-2011.)

Theoremcvrnbtwn 30069 There is no element between the two arguments of the covers relation. (cvnbtwn 23789 analog.) (Contributed by NM, 18-Oct-2011.)

Theoremncvr1 30070 No element covers the lattice unit. (Contributed by NM, 8-Jul-2013.)

TheoremcvrletrN 30071 Property of an element above a covering. (Contributed by NM, 7-Dec-2012.) (New usage is discouraged.)

Theoremcvrval2 30072* Binary relation expressing covers . Definition of covers in [Kalmbach] p. 15. (cvbr2 23786 analog.) (Contributed by NM, 16-Nov-2011.)

Theoremcvrnbtwn2 30073 The covers relation implies no in-betweenness. (cvnbtwn2 23790 analog.) (Contributed by NM, 17-Nov-2011.)

Theoremcvrnbtwn3 30074 The covers relation implies no in-betweenness. (cvnbtwn3 23791 analog.) (Contributed by NM, 4-Nov-2011.)

Theoremcvrcon3b 30075 Contraposition law for the covers relation. (cvcon3 23787 analog.) (Contributed by NM, 4-Nov-2011.)

Theoremcvrle 30076 The covers relation implies the less-than-or-equal relation. (Contributed by NM, 12-Oct-2011.)

Theoremcvrnbtwn4 30077 The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 23792 analog.) (Contributed by NM, 18-Oct-2011.)

Theoremcvrnle 30078 The covers relation implies the negation of the reverse less-than-or-equal relation. (Contributed by NM, 18-Oct-2011.)

Theoremcvrne 30079 The covers relation implies inequality. (Contributed by NM, 13-Oct-2011.)

TheoremcvrnrefN 30080 The covers relation is not reflexive. (cvnref 23794 analog.) (Contributed by NM, 1-Nov-2012.) (New usage is discouraged.)

Theoremcvrcmp 30081 If two lattice elements that cover a third are comparable, then they are equal. (Contributed by NM, 6-Feb-2012.)

Theoremcvrcmp2 30082 If two lattice elements covered by a third are comparable, then they are equal. (Contributed by NM, 20-Jun-2012.)

Theorempats 30083* The set of atoms in a poset. (Contributed by NM, 18-Sep-2011.)

Theoremisat 30084 The predicate "is an atom". (ela 23842 analog.) (Contributed by NM, 18-Sep-2011.)

Theoremisat2 30085 The predicate "is an atom". (elatcv0 23844 analog.) (Contributed by NM, 18-Jun-2012.)

Theorematcvr0 30086 An atom covers zero. (atcv0 23845 analog.) (Contributed by NM, 4-Nov-2011.)

Theorematbase 30087 An atom is a member of the lattice base set (i.e. a lattice element). (atelch 23847 analog.) (Contributed by NM, 10-Oct-2011.)

Theorematssbase 30088 The set of atoms is a subset of the base set. (atssch 23846 analog.) (Contributed by NM, 21-Oct-2011.)

Theorem0ltat 30089 An atom is greater than zero. (Contributed by NM, 4-Jul-2012.)

Theoremleatb 30090 A poset element less than or equal to an atom equals either zero or the atom. (atss 23849 analog.) (Contributed by NM, 17-Nov-2011.)

Theoremleat 30091 A poset element less than or equal to an atom equals either zero or the atom. (Contributed by NM, 15-Oct-2013.)

Theoremleat2 30092 A nonzero poset element less than or equal to an atom equals the atom. (Contributed by NM, 6-Mar-2013.)

Theoremleat3 30093 A poset element less than or equal to an atom is either an atom or zero. (Contributed by NM, 2-Dec-2012.)

Theoremmeetat 30094 The meet of any element with an atom is either the atom or zero. (Contributed by NM, 28-Aug-2012.)

Theoremmeetat2 30095 The meet of any element with an atom is either the atom or zero. (Contributed by NM, 30-Aug-2012.)

Definitiondf-atl 30096* Define the class of atomic lattices, in which every nonzero element is greater than or equal to an atom. . We also ensure the existence of a lattice zero, since a lattice by itself may not have a zero. (Contributed by NM, 18-Sep-2011.)

Theoremisatl 30097* The predicate "is an atomic lattice." Every nonzero element is less than or equal to an atom. (Contributed by NM, 18-Sep-2011.)

Theorematllat 30098 An atomic lattice is a lattice. (Contributed by NM, 21-Oct-2011.)

Theorematlpos 30099 An atomic lattice is a poset. (Contributed by NM, 5-Nov-2012.)

TheoremisatliN 30100* Properties that determine an atomic lattice. (Contributed by NM, 18-Sep-2011.) (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 32701-32762
 Copyright terms: Public domain < Previous  Next >