Home Metamath Proof ExplorerTheorem List (p. 312 of 321) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-22270) Hilbert Space Explorer (22271-23793) Users' Mathboxes (23794-32080)

Theorem List for Metamath Proof Explorer - 31101-31200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdvhb1dimN 31101* Two expressions for the 1-dimensional subspaces of vector space H, in the isomorphism B case where the 2nd vector component is zero. (Contributed by NM, 23-Feb-2014.) (New usage is discouraged.)

Theoremerng1lem 31102 Value of the endomorphism division ring unit. (Contributed by NM, 12-Oct-2013.)

Theoremerngdvlem1 31103* Lemma for erngrng 31107. (Contributed by NM, 4-Aug-2013.)

Theoremerngdvlem2N 31104* Lemma for erngrng 31107. (Contributed by NM, 6-Aug-2013.) (New usage is discouraged.)

Theoremerngdvlem3 31105* Lemma for erngrng 31107. (Contributed by NM, 6-Aug-2013.)

Theoremerngdvlem4 31106* Lemma for erngdv 31108. (Contributed by NM, 11-Aug-2013.)

Theoremerngrng 31107 An endomorphism ring is a ring. Todo: fix comment. (Contributed by NM, 4-Aug-2013.)

Theoremerngdv 31108 An endomorphism ring is a division ring. Todo: fix comment. (Contributed by NM, 11-Aug-2013.)

Theoremerng0g 31109* The division ring zero of an endomorphism ring. (Contributed by NM, 5-Nov-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)

Theoremerng1r 31110 The division ring unit of an endomorphism ring. (Contributed by NM, 5-Nov-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)

Theoremerngdvlem1-rN 31111* Lemma for erngrng 31107. (Contributed by NM, 4-Aug-2013.) (New usage is discouraged.)

Theoremerngdvlem2-rN 31112* Lemma for erngrng 31107. (Contributed by NM, 6-Aug-2013.) (New usage is discouraged.)

Theoremerngdvlem3-rN 31113* Lemma for erngrng 31107. (Contributed by NM, 6-Aug-2013.) (New usage is discouraged.)

Theoremerngdvlem4-rN 31114* Lemma for erngdv 31108. (Contributed by NM, 11-Aug-2013.) (New usage is discouraged.)

Theoremerngrng-rN 31115 An endomorphism ring is a ring. Todo: fix comment. (Contributed by NM, 4-Aug-2013.) (New usage is discouraged.)

Theoremerngdv-rN 31116 An endomorphism ring is a division ring. Todo: fix comment. (Contributed by NM, 11-Aug-2013.) (New usage is discouraged.)

Syntaxcdveca 31117 Extend class notation with constructed vector space A.

Definitiondf-dveca 31118* Define constructed partial vector space A. (Contributed by NM, 8-Oct-2013.)
Scalar

Theoremdvafset 31119* The constructed partial vector space A for a lattice . (Contributed by NM, 8-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremdvaset 31120* The constructed partial vector space A for a lattice . (Contributed by NM, 8-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremdvasca 31121 The ring base set of the constructed partial vector space A are all translation group endomorphisms (for a fiducial co-atom ). (Contributed by NM, 22-Jun-2014.)
Scalar

Theoremdvabase 31122 The ring base set of the constructed partial vector space A are all translation group endomorphisms (for a fiducial co-atom ). (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremdvafplusg 31123* Ring addition operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremdvaplusg 31124* Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
Scalar

Theoremdvaplusgv 31125 Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
Scalar

Theoremdvafmulr 31126* Ring multiplication operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremdvamulr 31127 Ring multiplication operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
Scalar

Theoremdvavbase 31128 The vectors (vector base set) of the constructed partial vector space A are all translations (for a fiducial co-atom ). (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)

Theoremdvafvadd 31129* The vector sum operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)

Theoremdvavadd 31130 Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)

Theoremdvafvsca 31131* Ring addition operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)

Theoremdvavsca 31132 Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)

Theoremtendospid 31133 Identity property of endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)

Theoremtendospcl 31134 Closure of endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)

Theoremtendospass 31135 Associative law for endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)

Theoremtendospdi1 31136 Forward distributive law for endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)

Theoremtendocnv 31137 Converse of a trace-preserving endomorphism value. (Contributed by NM, 7-Apr-2014.)

Theoremtendospdi2 31138* Reverse distributive law for endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)

TheoremtendospcanN 31139* Cancellation law for trace-perserving endomorphism values (used as scalar product). (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)

Theoremdvaabl 31140 The constructed partial vector space A for a lattice is an abelian group. (Contributed by NM, 11-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)

Theoremdvalveclem 31141 Lemma for dvalvec 31142. (Contributed by NM, 11-Oct-2013.) (Proof shortened by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremdvalvec 31142 The constructed partial vector space A for a lattice is a left vector space. (Contributed by NM, 11-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)

Theoremdva0g 31143 The zero vector of partial vector space A. (Contributed by NM, 9-Sep-2014.)

Syntaxcdia 31144 Extend class notation with partial isomorphism A.

Definitiondf-disoa 31145* Define partial isomorphism A. (Contributed by NM, 15-Oct-2013.)

Theoremdiaffval 31146* The partial isomorphism A for a lattice . (Contributed by NM, 15-Oct-2013.)

Theoremdiafval 31147* The partial isomorphism A for a lattice . (Contributed by NM, 15-Oct-2013.)

Theoremdiaval 31148* The partial isomorphism A for a lattice . Definition of isomorphism map in [Crawley] p. 120 line 24. (Contributed by NM, 15-Oct-2013.)

Theoremdiaelval 31149 Member of the partial isomorphism A for a lattice . (Contributed by NM, 3-Dec-2013.)

Theoremdiafn 31150* Functionality and domain of the partial isomorphism A. (Contributed by NM, 26-Nov-2013.)

Theoremdiadm 31151* Domain of the partial isomorphism A. (Contributed by NM, 3-Dec-2013.)

Theoremdiaeldm 31152 Member of domain of the partial isomorphism A. (Contributed by NM, 4-Dec-2013.)

TheoremdiadmclN 31153 A member of domain of the partial isomorphism A is a lattice element. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

TheoremdiadmleN 31154 A member of domain of the partial isomorphism A is under the fiducial hyperplane. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

Theoremdian0 31155 The value of the partial isomorphism A is not empty. (Contributed by NM, 17-Jan-2014.)

Theoremdia0eldmN 31156 The lattice zero belongs to the domain of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

Theoremdia1eldmN 31157 The fiducial hyperplane (the largest allowed lattice element) belongs to the domain of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

Theoremdiass 31158 The value of the partial isomorphism A is a set of translations i.e. a set of vectors. (Contributed by NM, 26-Nov-2013.)

Theoremdiael 31159 A member of the value of the partial isomorphism A is a translation i.e. a vector. (Contributed by NM, 17-Jan-2014.)

Theoremdiatrl 31160 Trace of a member of the partial isomorphism A. (Contributed by NM, 17-Jan-2014.)

TheoremdiaelrnN 31161 Any value of the partial isomorphism A is a set of translations i.e. a set of vectors. (Contributed by NM, 26-Nov-2013.) (New usage is discouraged.)

Theoremdialss 31162 The value of partial isomorphism A is a subspace of partial vector space A. Part of Lemma M of [Crawley] p. 120 line 26. (Contributed by NM, 17-Jan-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)

Theoremdiaord 31163 The partial isomorphism A for a lattice is order-preserving in the region under co-atom . Part of Lemma M of [Crawley] p. 120 line 28. (Contributed by NM, 26-Nov-2013.)

Theoremdia11N 31164 The partial isomorphism A for a lattice is one-to-one in the region under co-atom . Part of Lemma M of [Crawley] p. 120 line 28. (Contributed by NM, 25-Nov-2013.) (New usage is discouraged.)

Theoremdiaf11N 31165 The partial isomorphism A for a lattice is a one-to-one function. . Part of Lemma M of [Crawley] p. 120 line 27. (Contributed by NM, 4-Dec-2013.) (New usage is discouraged.)

TheoremdiaclN 31166 Closure of partial isomorphism A for a lattice . (Contributed by NM, 4-Dec-2013.) (New usage is discouraged.)

TheoremdiacnvclN 31167 Closure of partial isomorphism A converse. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

Theoremdia0 31168 The value of the partial isomorphism A at the lattice zero is the singleton of the identity translation i.e. the zero subspace. (Contributed by NM, 26-Nov-2013.)

Theoremdia1N 31169 The value of the partial isomorphism A at the fiducial co-atom is the set of all translations i.e. the entire vector space. (Contributed by NM, 26-Nov-2013.) (New usage is discouraged.)

Theoremdia1elN 31170 The largest subspace in the range of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

TheoremdiaglbN 31171* Partial isomorphism A of a lattice glb. (Contributed by NM, 3-Dec-2013.) (New usage is discouraged.)

TheoremdiameetN 31172 Partial isomorphism A of a lattice meet. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

TheoremdiainN 31173 Inverse partial isomorphism A of an intersection. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)

TheoremdiaintclN 31174 The intersection of partial isomorphism A closed subspaces is a closed subspace. (Contributed by NM, 3-Dec-2013.) (New usage is discouraged.)

TheoremdiasslssN 31175 The partial isomorphism A maps to subspaces of partial vector space A. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)

TheoremdiassdvaN 31176 The partial isomorphism A maps to a set of vectors in partial vector space A. (Contributed by NM, 1-Jan-2014.) (New usage is discouraged.)

Theoremdia1dim 31177* Two expressions for the 1-dimensional subspaces of partial vector space A (when is a nonzero vector i.e. non-identity translation). Remark after Lemma L in [Crawley] p. 120 line 21. (Contributed by NM, 15-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)

Theoremdia1dim2 31178 Two expressions for a 1-dimensional subspace of partial vector space A (when is a nonzero vector i.e. non-identity translation). (Contributed by NM, 15-Jan-2014.) (Revised by Mario Carneiro, 22-Jun-2014.)

Theoremdia1dimid 31179 A vector (translation) belongs to the 1-dim subspace it generates. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem1 31180 Lemma for dia2dim 31193. Show properties of the auxiliary atom . Part of proof of Lemma M in [Crawley] p. 121 line 3. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem2 31181 Lemma for dia2dim 31193. Define a translation whose trace is atom . Part of proof of Lemma M in [Crawley] p. 121 line 4. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem3 31182 Lemma for dia2dim 31193. Define a translation whose trace is atom . Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem4 31183 Lemma for dia2dim 31193. Show that the composition (sum) of translations (vectors) and equals . Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem5 31184 Lemma for dia2dim 31193. The sum of vectors and belongs to the sum of the subspaces generated by them. Thus, belongs to the subspace sum. Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem6 31185 Lemma for dia2dim 31193. Eliminate auxiliary translations and . (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem7 31186 Lemma for dia2dim 31193. Eliminate condition. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem8 31187 Lemma for dia2dim 31193. Eliminate no-longer used auxiliary atoms and . (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem9 31188 Lemma for dia2dim 31193. Eliminate , conditions. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem10 31189 Lemma for dia2dim 31193. Convert membership in closed subspace to a lattice ordering. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem11 31190 Lemma for dia2dim 31193. Convert ordering hypothesis on to subspace membership . (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem12 31191 Lemma for dia2dim 31193. Obtain subset relation. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dimlem13 31192 Lemma for dia2dim 31193. Eliminate condition. (Contributed by NM, 8-Sep-2014.)

Theoremdia2dim 31193 A two-dimensional subspace of partial vector space A is closed, or equivalently, the isomorphism of a join of two atoms is a subset of the subspace sum of the isomorphisms of each atom (and thus they are equal, as shown later for the full vector space H). (Contributed by NM, 9-Sep-2014.)

Syntaxcdvh 31194 Extend class notation with constructed full vector space H.

Definitiondf-dvech 31195* Define constructed full vector space H. (Contributed by NM, 17-Oct-2013.)
Scalar

Theoremdvhfset 31196* The constructed full vector space H for a lattice . (Contributed by NM, 17-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremdvhset 31197* The constructed full vector space H for a lattice . (Contributed by NM, 17-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremdvhsca 31198 The ring of scalars of the constructed full vector space H. (Contributed by NM, 22-Jun-2014.)
Scalar

Theoremdvhbase 31199 The ring base set of the constructed full vector space H. (Contributed by NM, 29-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar

Theoremdvhfplusr 31200* Ring addition operation for the constructed full vector space H. (Contributed by NM, 29-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar

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 32001-32080
 Copyright terms: Public domain < Previous  Next >