Home Metamath Proof ExplorerTheorem List (p. 305 of 329) < Previous  Next > Browser slow? Try the Unicode version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-22426) Hilbert Space Explorer (22427-23949) Users' Mathboxes (23950-32836)

Theorem List for Metamath Proof Explorer - 30401-30500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremislpln 30401* The predicate "is a lattice plane". (Contributed by NM, 16-Jun-2012.)

Theoremislpln4 30402* The predicate "is a lattice plane". (Contributed by NM, 17-Jun-2012.)

Theoremlplni 30403 Condition implying a lattice plane. (Contributed by NM, 20-Jun-2012.)

Theoremislpln3 30404* The predicate "is a lattice plane". (Contributed by NM, 17-Jun-2012.)

Theoremlplnbase 30405 A lattice plane is a lattice element. (Contributed by NM, 17-Jun-2012.)

Theoremislpln5 30406* The predicate "is a lattice plane" in terms of atoms. (Contributed by NM, 24-Jun-2012.)

Theoremislpln2 30407* The predicate "is a lattice plane" in terms of atoms. (Contributed by NM, 25-Jun-2012.)

Theoremlplni2 30408 The join of 3 different atoms is a lattice plane. (Contributed by NM, 4-Jul-2012.)

Theoremlvolex3N 30409* There is an atom outside of a lattice plane i.e. a 3-dimensional lattice volume exists. (Contributed by NM, 28-Jul-2012.) (New usage is discouraged.)

TheoremllnmlplnN 30410 The intersection of a line with a plane not containing it is an atom. (Contributed by NM, 29-Jun-2012.) (New usage is discouraged.)

Theoremlplnle 30411* Any element greater than 0 and not an atom and not a lattice line majorizes a lattice plane. (Contributed by NM, 28-Jun-2012.)

Theoremlplnnle2at 30412 A lattice line (or atom) cannot majorize a lattice plane. (Contributed by NM, 8-Jul-2012.)

Theoremlplnnleat 30413 A lattice plane cannot majorize an atom. (Contributed by NM, 14-Jul-2012.)

Theoremlplnnlelln 30414 A lattice plane is not less than or equal to a lattice line. (Contributed by NM, 14-Jul-2012.)

Theorem2atnelpln 30415 The join of two atoms is not a lattice plane. (Contributed by NM, 16-Jul-2012.)

Theoremlplnneat 30416 No lattice plane is an atom. (Contributed by NM, 15-Jul-2012.)

Theoremlplnnelln 30417 No lattice plane is a lattice line. (Contributed by NM, 19-Jun-2012.)

Theoremlplnn0N 30418 A lattice plane is non-zero. (Contributed by NM, 15-Jul-2012.) (New usage is discouraged.)

Theoremislpln2a 30419 The predicate "is a lattice plane" for join of atoms. (Contributed by NM, 16-Jul-2012.)

Theoremislpln2ah 30420 The predicate "is a lattice plane" for join of atoms. Version of islpln2a 30419 expressed with an abbreviation hypothesis. (Contributed by NM, 30-Jul-2012.)

TheoremlplnriaN 30421 Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)

TheoremlplnribN 30422 Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)

Theoremlplnric 30423 Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.)

Theoremlplnri1 30424 Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.)

Theoremlplnri2N 30425 Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)

Theoremlplnri3N 30426 Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)

TheoremlplnllnneN 30427 Two lattice lines defined by atoms defining a lattice plane are not equal. (Contributed by NM, 9-Oct-2012.) (New usage is discouraged.)

Theoremllncvrlpln2 30428 A lattice line under a lattice plane is covered by it. (Contributed by NM, 24-Jun-2012.)

Theoremllncvrlpln 30429 An element covering a lattice line is a lattice plane and vice-versa. (Contributed by NM, 26-Jun-2012.)

Theorem2lplnmN 30430 If the join of two lattice planes covers one of them, their meet is a lattice line. (Contributed by NM, 30-Jun-2012.) (New usage is discouraged.)

Theorem2llnmj 30431 The meet of two lattice lines is an atom iff their join is a lattice plane. (Contributed by NM, 27-Jun-2012.)

Theorem2atmat 30432 The meet of two intersecting lines (expressed as joins of atoms) is an atom. (Contributed by NM, 21-Nov-2012.)

Theoremlplncmp 30433 If two lattice planes are comparable, they are equal. (Contributed by NM, 24-Jun-2012.)

TheoremlplnexatN 30434* Given a lattice line on a lattice plane, there is an atom whose join with the line equals the plane. (Contributed by NM, 29-Jun-2012.) (New usage is discouraged.)

TheoremlplnexllnN 30435* Given an atom on a lattice plane, there is a lattice line whose join with the atom equals the plane. (Contributed by NM, 26-Jun-2012.) (New usage is discouraged.)

Theoremlplnnlt 30436 Two lattice planes cannot satisfy the less than relation. (Contributed by NM, 7-Jul-2012.)

Theorem2llnjaN 30437 The join of two different lattice lines in a lattice plane equals the plane (version of 2llnjN 30438 in terms of atoms). (Contributed by NM, 5-Jul-2012.) (New usage is discouraged.)

Theorem2llnjN 30438 The join of two different lattice lines in a lattice plane equals the plane. (Contributed by NM, 4-Jul-2012.) (New usage is discouraged.)

Theorem2llnm2N 30439 The meet of two different lattice lines in a lattice plane is an atom. (Contributed by NM, 5-Jul-2012.) (New usage is discouraged.)

Theorem2llnm3N 30440 Two lattice lines in a lattice plane always meet. (Contributed by NM, 5-Jul-2012.) (New usage is discouraged.)

Theorem2llnm4 30441 Two lattice lines that majorize the same atom always meet. (Contributed by NM, 20-Jul-2012.)

Theorem2llnmeqat 30442 An atom equals the intersection of two majorizing lines. (Contributed by NM, 3-Apr-2013.)

Theoremlvolset 30443* The set of 3-dim lattice volumes in a Hilbert lattice. (Contributed by NM, 1-Jul-2012.)

Theoremislvol 30444* The predicate "is a 3-dim lattice volume". (Contributed by NM, 1-Jul-2012.)

Theoremislvol4 30445* The predicate "is a 3-dim lattice volume". (Contributed by NM, 1-Jul-2012.)

Theoremlvoli 30446 Condition implying a 3-dim lattice volume. (Contributed by NM, 1-Jul-2012.)

Theoremislvol3 30447* The predicate "is a 3-dim lattice volume". (Contributed by NM, 1-Jul-2012.)

Theoremlvoli3 30448 Condition implying a 3-dim lattice volume. (Contributed by NM, 2-Aug-2012.)

Theoremlvolbase 30449 A 3-dim lattice volume is a lattice element. (Contributed by NM, 1-Jul-2012.)

Theoremislvol5 30450* The predicate "is a 3-dim lattice volume" in terms of atoms. (Contributed by NM, 1-Jul-2012.)

Theoremislvol2 30451* The predicate "is a 3-dim lattice volume" in terms of atoms. (Contributed by NM, 1-Jul-2012.)

Theoremlvoli2 30452 The join of 4 different atoms is a lattice volume. (Contributed by NM, 8-Jul-2012.)

Theoremlvolnle3at 30453 A lattice plane (or lattice line or atom) cannot majorize a lattice volume. (Contributed by NM, 8-Jul-2012.)

Theoremlvolnleat 30454 An atom cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)

Theoremlvolnlelln 30455 A lattice line cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)

Theoremlvolnlelpln 30456 A lattice plane cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)

Theorem3atnelvolN 30457 The join of 3 atoms is not a lattice volume. (Contributed by NM, 17-Jul-2012.) (New usage is discouraged.)

Theorem2atnelvolN 30458 The join of two atoms is not a lattice volume. (Contributed by NM, 17-Jul-2012.) (New usage is discouraged.)

TheoremlvolneatN 30459 No lattice volume is an atom. (Contributed by NM, 15-Jul-2012.) (New usage is discouraged.)

Theoremlvolnelln 30460 No lattice volume is a lattice line. (Contributed by NM, 15-Jul-2012.)

Theoremlvolnelpln 30461 No lattice volume is a lattice plane. (Contributed by NM, 19-Jun-2012.)

Theoremlvoln0N 30462 A lattice volume is non-zero. (Contributed by NM, 17-Jul-2012.) (New usage is discouraged.)

Theoremislvol2aN 30463 The predicate "is a lattice volume". (Contributed by NM, 16-Jul-2012.) (New usage is discouraged.)

Theorem4atlem0a 30464 Lemma for 4at 30484. (Contributed by NM, 10-Jul-2012.)

Theorem4atlem0ae 30465 Lemma for 4at 30484. (Contributed by NM, 10-Jul-2012.)

Theorem4atlem0be 30466 Lemma for 4at 30484. (Contributed by NM, 10-Jul-2012.)

Theorem4atlem3 30467 Lemma for 4at 30484. Break inequality into 4 cases. (Contributed by NM, 8-Jul-2012.)

Theorem4atlem3a 30468 Lemma for 4at 30484. Break inequality into 3 cases. (Contributed by NM, 9-Jul-2012.)

Theorem4atlem3b 30469 Lemma for 4at 30484. Break inequality into 2 cases. (Contributed by NM, 9-Jul-2012.)

Theorem4atlem4a 30470 Lemma for 4at 30484. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)

Theorem4atlem4b 30471 Lemma for 4at 30484. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)

Theorem4atlem4c 30472 Lemma for 4at 30484. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)

Theorem4atlem4d 30473 Lemma for 4at 30484. Frequently used associative law. (Contributed by NM, 9-Jul-2012.)

Theorem4atlem9 30474 Lemma for 4at 30484. Substitute for . (Contributed by NM, 9-Jul-2012.)

Theorem4atlem10a 30475 Lemma for 4at 30484. Substitute for . (Contributed by NM, 9-Jul-2012.)

Theorem4atlem10b 30476 Lemma for 4at 30484. Substitute for (cont.). (Contributed by NM, 10-Jul-2012.)

Theorem4atlem10 30477 Lemma for 4at 30484. Combine both possible cases. (Contributed by NM, 9-Jul-2012.)

Theorem4atlem11a 30478 Lemma for 4at 30484. Substitute for . (Contributed by NM, 9-Jul-2012.)

Theorem4atlem11b 30479 Lemma for 4at 30484. Substitute for (cont.). (Contributed by NM, 10-Jul-2012.)

Theorem4atlem11 30480 Lemma for 4at 30484. Combine all three possible cases. (Contributed by NM, 10-Jul-2012.)

Theorem4atlem12a 30481 Lemma for 4at 30484. Substitute for . (Contributed by NM, 9-Jul-2012.)

Theorem4atlem12b 30482 Lemma for 4at 30484. Substitute for (cont.). (Contributed by NM, 11-Jul-2012.)

Theorem4atlem12 30483 Lemma for 4at 30484. Combine all four possible cases. (Contributed by NM, 11-Jul-2012.)

Theorem4at 30484 Four atoms determine a lattice volume uniquely. Three-dimensional analog of ps-1 30348 and 3at 30361. (Contributed by NM, 11-Jul-2012.)

Theorem4at2 30485 Four atoms determine a lattice volume uniquely. (Contributed by NM, 11-Jul-2012.)

Theoremlplncvrlvol2 30486 A lattice line under a lattice plane is covered by it. (Contributed by NM, 12-Jul-2012.)

Theoremlplncvrlvol 30487 An element covering a lattice plane is a lattice volume and vice-versa. (Contributed by NM, 15-Jul-2012.)

Theoremlvolcmp 30488 If two lattice planes are comparable, they are equal. (Contributed by NM, 12-Jul-2012.)

TheoremlvolnltN 30489 Two lattice volumes cannot satisfy the less than relation. (Contributed by NM, 12-Jul-2012.) (New usage is discouraged.)

Theorem2lplnja 30490 The join of two different lattice planes in a lattice volume equals the volume (version of 2lplnj 30491 in terms of atoms). (Contributed by NM, 12-Jul-2012.)

Theorem2lplnj 30491 The join of two different lattice planes in a (3-dimensional) lattice volume equals the volume. (Contributed by NM, 12-Jul-2012.)

Theorem2lplnm2N 30492 The meet of two different lattice planes in a lattice volume is a lattice line. (Contributed by NM, 12-Jul-2012.) (New usage is discouraged.)

Theorem2lplnmj 30493 The meet of two lattice planes is a lattice line iff their join is a lattice volume. (Contributed by NM, 13-Jul-2012.)

Theoremdalemkehl 30494 Lemma for dath 30607. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemkelat 30495 Lemma for dath 30607. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemkeop 30496 Lemma for dath 30607. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalempea 30497 Lemma for dath 30607. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemqea 30498 Lemma for dath 30607. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemrea 30499 Lemma for dath 30607. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Theoremdalemsea 30500 Lemma for dath 30607. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

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 32801-32836
 Copyright terms: Public domain < Previous  Next >