Home Metamath Proof ExplorerTheorem List (p. 292 of 314) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-21444) Hilbert Space Explorer (21445-22967) Users' Mathboxes (22968-31305)

Theorem List for Metamath Proof Explorer - 29101-29200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempmaple 29101 The projective map of a Hilbert lattice preserves ordering. Part of Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 22-Oct-2011.)

Theorempmap11 29102 The projective map of a Hilbert lattice is one-to-one. Part of Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 22-Oct-2011.)

Theorempmapat 29103 The projective map of an atom. (Contributed by NM, 25-Jan-2012.)

Theoremelpmapat 29104 Member of the projective map of an atom. (Contributed by NM, 27-Jan-2012.)

Theorempmap0 29105 Value of the projective map of a Hilbert lattice at lattice zero. Part of Theorem 15.5.1 of [MaedaMaeda] p. 62. (Contributed by NM, 17-Oct-2011.)

Theorempmapeq0 29106 A projective map value is zero iff its argument is lattice zero. (Contributed by NM, 27-Jan-2012.)

Theorempmap1N 29107 Value of the projective map of a Hilbert lattice at lattice unit. Part of Theorem 15.5.1 of [MaedaMaeda] p. 62. (Contributed by NM, 22-Oct-2011.) (New usage is discouraged.)

Theorempmapsub 29108 The projective map of a Hilbert lattice maps to projective subspaces. Part of Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 17-Oct-2011.)

Theorempmapglbx 29109* The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb 29110, where we read as . Theorem 15.5.2 of [MaedaMaeda] p. 62. (Contributed by NM, 5-Dec-2011.)

Theorempmapglb 29110* The projective map of the GLB of a set of lattice elements . Variant of Theorem 15.5.2 of [MaedaMaeda] p. 62. (Contributed by NM, 5-Dec-2011.)

Theorempmapglb2N 29111* The projective map of the GLB of a set of lattice elements . Variant of Theorem 15.5.2 of [MaedaMaeda] p. 62. Allows . (Contributed by NM, 21-Jan-2012.) (New usage is discouraged.)

Theorempmapglb2xN 29112* The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb2N 29111, where we read as . Extension of Theorem 15.5.2 of [MaedaMaeda] p. 62 that allows . (Contributed by NM, 21-Jan-2012.) (New usage is discouraged.)

Theorempmapmeet 29113 The projective map of a meet. (Contributed by NM, 25-Jan-2012.)

Theoremisline2 29114* Definition of line in terms of projective map. (Contributed by NM, 25-Jan-2012.)

Theoremlinepmap 29115 A line described with a projective map. (Contributed by NM, 3-Feb-2012.)

Theoremisline3 29116* Definition of line in terms of original lattice elements. (Contributed by NM, 29-Apr-2012.)

Theoremisline4N 29117* Definition of line in terms of original lattice elements. (Contributed by NM, 16-Jun-2012.) (New usage is discouraged.)

Theoremlneq2at 29118 A line equals the join of any two of its distinct points (atoms). (Contributed by NM, 29-Apr-2012.)

TheoremlnatexN 29119* There is an atom in a line different from any other. (Contributed by NM, 30-Apr-2012.) (New usage is discouraged.)

TheoremlnjatN 29120* Given an atom in a line, there is another atom which when joined equals the line. (Contributed by NM, 30-Apr-2012.) (New usage is discouraged.)

TheoremlncvrelatN 29121 A lattice element covered by a line is an atom. (Contributed by NM, 28-Apr-2012.) (New usage is discouraged.)

Theoremlncvrat 29122 A line covers the atoms it contains. (Contributed by NM, 30-Apr-2012.)

Theoremlncmp 29123 If two lines are comparable, they are equal. (Contributed by NM, 30-Apr-2012.)

Theorem2lnat 29124 Two intersecting lines intersect at an atom. (Contributed by NM, 30-Apr-2012.)

Theorem2atm2atN 29125 Two joins with a common atom have a nonzero meet. (Contributed by NM, 4-Jul-2012.) (New usage is discouraged.)

Theorem2llnma1b 29126 Generalization of 2llnma1 29127. (Contributed by NM, 26-Apr-2013.)

Theorem2llnma1 29127 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 11-Oct-2012.)

Theorem2llnma3r 29128 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 30-Apr-2013.)

Theorem2llnma2 29129 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 28-May-2012.)

Theorem2llnma2rN 29130 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 2-May-2013.) (New usage is discouraged.)

16.24.14  Construction of a vector space from a Hilbert lattice

Theoremcdlema1N 29131 A condition for required for proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 29-Apr-2012.) (New usage is discouraged.)

Theoremcdlema2N 29132 A condition for required for proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 9-May-2012.) (New usage is discouraged.)

Theoremcdlemblem 29133 Lemma for cdlemb 29134. (Contributed by NM, 8-May-2012.)

Theoremcdlemb 29134* Given two atoms not less than or equal to an element covered by 1, there is a third. Lemma B in [Crawley] p. 112. (Contributed by NM, 8-May-2012.)

Syntaxcpadd 29135 Extend class notation with projective subspace sum.

Definitiondf-padd 29136* Define projective sum of two subspaces (or more generally two sets of atoms), which is the union of all lines generated by pairs of atoms from each subspace. Lemma 16.2 of [MaedaMaeda] p. 68. For convenience, our definition is generalized to apply to empty sets. (Contributed by NM, 29-Dec-2011.)

Theorempaddfval 29137* Projective subspace sum operation. (Contributed by NM, 29-Dec-2011.)

Theorempaddval 29138* Projective subspace sum operation value. (Contributed by NM, 29-Dec-2011.)

Theoremelpadd 29139* Member of a projective subspace sum. (Contributed by NM, 29-Dec-2011.)

Theoremelpaddn0 29140* Member of projective subspace sum of non-empty sets. (Contributed by NM, 3-Jan-2012.)

Theorempaddvaln0N 29141* Projective subspace sum operation value for non-empty sets. (Contributed by NM, 27-Jan-2012.) (New usage is discouraged.)

Theoremelpaddri 29142 Condition implying membership in a projective subspace sum. (Contributed by NM, 8-Jan-2012.)

TheoremelpaddatriN 29143 Condition implying membership in a projective subspace sum with a point. (Contributed by NM, 1-Feb-2012.) (New usage is discouraged.)

Theoremelpaddat 29144* Membership in a projective subspace sum with a point. (Contributed by NM, 29-Jan-2012.)

TheoremelpaddatiN 29145* Consequence of membership in a projective subspace sum with a point. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)

Theoremelpadd2at 29146 Membership in a projective subspace sum of two points. (Contributed by NM, 29-Jan-2012.)

Theoremelpadd2at2 29147 Membership in a projective subspace sum of two points. (Contributed by NM, 8-Mar-2012.)

TheorempaddunssN 29148 Projective subspace sum includes the set union of its arguments. (Contributed by NM, 12-Jan-2012.) (New usage is discouraged.)

Theoremelpadd0 29149 Member of projective subspace sum with at least one empty set.. (Contributed by NM, 29-Dec-2011.)

Theorempaddval0 29150 Projective subspace sum with at least one empty set. (Contributed by NM, 11-Jan-2012.)

Theorempadd01 29151 Projective subspace sum with an empty set. (Contributed by NM, 11-Jan-2012.)

Theorempadd02 29152 Projective subspace sum with an empty set. (Contributed by NM, 11-Jan-2012.)

Theorempaddcom 29153 Projective subspace sum commutes. (Contributed by NM, 3-Jan-2012.)

Theorempaddssat 29154 A projective subspace sum is a set of atoms. (Contributed by NM, 3-Jan-2012.)

Theoremsspadd1 29155 A projective subspace sum is a superset of its first summand. (ssun1 3299 analog.) (Contributed by NM, 3-Jan-2012.)

Theoremsspadd2 29156 A projective subspace sum is a superset of its second summand. (ssun2 3300 analog.) (Contributed by NM, 3-Jan-2012.)

Theorempaddss1 29157 Subset law for projective subspace sum. (unss1 3305 analog.) (Contributed by NM, 7-Mar-2012.)

Theorempaddss2 29158 Subset law for projective subspace sum. (unss2 3307 analog.) (Contributed by NM, 7-Mar-2012.)

Theorempaddss12 29159 Subset law for projective subspace sum. (unss12 3308 analog.) (Contributed by NM, 7-Mar-2012.)

Theorempaddasslem3 29162* Lemma for paddass 29178. Restate projective space axiom ps-2 28818. (Contributed by NM, 8-Jan-2012.)

Theorempaddasslem11 29170 Lemma for paddass 29178. The case when . (Contributed by NM, 11-Jan-2012.)

Theorempaddasslem12 29171 Lemma for paddass 29178. The case when . (Contributed by NM, 11-Jan-2012.)

Theorempaddasslem13 29172 Lemma for paddass 29178. The case when . (Unlike the proof in Maeda and Maeda, we don't need .) (Contributed by NM, 11-Jan-2012.)

Theorempaddasslem17 29176 Lemma for paddass 29178. The case when at least one sum argument is empty. (Contributed by NM, 12-Jan-2012.)

Theorempaddass 29178 Projective subspace sum is associative. Equation 16.2.1 of [MaedaMaeda] p. 68. In our version, the subspaces do not have to be non-empty. (Contributed by NM, 29-Dec-2011.)

Theorempadd12N 29179 Commutative/associative law for projective subspace sum. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)

Theorempadd4N 29180 Rearrangement of 4 terms in a projective subspace sum. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)

Theorempaddidm 29181 Projective subspace sum is idempotent. Part of Lemma 16.2 of [MaedaMaeda] p. 68. (Contributed by NM, 13-Jan-2012.)

TheorempaddclN 29182 The projective sum of two subspaces is a subspace. Part of Lemma 16.2 of [MaedaMaeda] p. 68. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)

Theorempaddssw1 29183 Subset law for projective subspace sum valid for all subsets of atoms. (Contributed by NM, 14-Mar-2012.)

Theorempaddssw2 29184 Subset law for projective subspace sum valid for all subsets of atoms. (Contributed by NM, 14-Mar-2012.)

Theorempaddss 29185 Subset law for projective subspace sum. (unss 3310 analog.) (Contributed by NM, 7-Mar-2012.)

Theorempmodlem1 29186* Lemma for pmod1i 29188. (Contributed by NM, 9-Mar-2012.)

Theorempmodlem2 29187 Lemma for pmod1i 29188. (Contributed by NM, 9-Mar-2012.)

Theorempmod1i 29188 The modular law holds in a projective subspace. (Contributed by NM, 10-Mar-2012.)

Theorempmod2iN 29189 Dual of the modular law. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)

TheorempmodN 29190 The modular law for projective subspaces. (Contributed by NM, 26-Mar-2012.) (New usage is discouraged.)

Theorempmodl42N 29191 Lemma derived from modular law. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)

Theorempmapjoin 29192 The projective map of the join of two lattice elements. Part of Equation 15.5.3 of [MaedaMaeda] p. 63. (Contributed by NM, 27-Jan-2012.)

Theorempmapjat1 29193 The projective map of the join of a lattice element and an atom. (Contributed by NM, 28-Jan-2012.)

Theorempmapjat2 29194 The projective map of the join of an atom with a lattice element. (Contributed by NM, 12-May-2012.)

Theorempmapjlln1 29195 The projective map of the join of a lattice element and a lattice line (expressed as the join of two atoms). (Contributed by NM, 16-Sep-2012.)

Theoremhlmod1i 29196 A version of the modular law pmod1i 29188 that holds in a Hilbert lattice. (Contributed by NM, 13-May-2012.)

Theorematmod1i1 29197 Version of modular law pmod1i 29188 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 11-May-2012.) (Revised by Mario Carneiro, 10-May-2013.)

Theorematmod1i1m 29198 Version of modular law pmod1i 29188 that holds in a Hilbert lattice, when an element meets an atom. (Contributed by NM, 2-Sep-2012.) (Revised by Mario Carneiro, 10-May-2013.)

Theorematmod1i2 29199 Version of modular law pmod1i 29188 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 14-May-2012.) (Revised by Mario Carneiro, 10-May-2013.)

Theoremllnmod1i2 29200 Version of modular law pmod1i 29188 that holds in a Hilbert lattice, when one element is a lattice line (expressed as the join ). (Contributed by NM, 16-Sep-2012.) (Revised by Mario Carneiro, 10-May-2013.)

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 31301-31305
 Copyright terms: Public domain < Previous  Next >