Home Metamath Proof ExplorerTheorem List (p. 319 of 324) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-22370) Hilbert Space Explorer (22371-23893) Users' Mathboxes (23894-32378)

Theorem List for Metamath Proof Explorer - 31801-31900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdochkrshp2 31801 Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LSHyp       LFnl       LKer

Theoremdochkrshp3 31802 Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LFnl       LKer

Theoremdochkrshp4 31803 Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
LFnl       LKer

Theoremdochdmj1 31804 De Morgan-like law for subspace orthocomplement. (Contributed by NM, 5-Aug-2014.)

Theoremdochnoncon 31805 Law of noncontradiction. The intersection of a subspace and its orthocomplement is the zero subspace. (Contributed by NM, 16-Apr-2014.)

Theoremdochnel2 31806 A nonzero member of a subspace doesn't belong to the orthocomplement of the subspace. (Contributed by NM, 28-Feb-2015.)

Theoremdochnel 31807 A nonzero vector doesn't belong to the orthocomplement of its singleton. (Contributed by NM, 27-Oct-2014.)

Syntaxcdjh 31808 Extend class notation with subspace join for vector space.
joinH

Definitiondf-djh 31809* Define (closed) subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theoremdjhffval 31810* Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theoremdjhfval 31811* Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theoremdjhval 31812 Subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theoremdjhval2 31813 Value of subspace join for vector space. (Contributed by NM, 6-Aug-2014.)
joinH

Theoremdjhcl 31814 Closure of subspace join for vector space. (Contributed by NM, 19-Jul-2014.)
joinH

Theoremdjhlj 31815 Transfer lattice join to vector space closed subspace join. (Contributed by NM, 19-Jul-2014.)
joinH

TheoremdjhljjN 31816 Lattice join in terms of vector space closed subspace join. (Contributed by NM, 17-Aug-2014.) (New usage is discouraged.)
joinH

Theoremdjhjlj 31817 vector space closed subspace join in terms of lattice join. (Contributed by NM, 9-Aug-2014.)
joinH

Theoremdjhj 31818 vector space closed subspace join in terms of lattice join. (Contributed by NM, 17-Aug-2014.)
joinH

Theoremdjhcom 31819 Subspace join commutes. (Contributed by NM, 8-Aug-2014.)
joinH

Theoremdjhspss 31820 Subspace span of union is a subset of subspace join. (Contributed by NM, 6-Aug-2014.)
joinH

Theoremdjhsumss 31821 Subspace sum is a subset of subspace join. (Contributed by NM, 6-Aug-2014.)
joinH

Theoremdihsumssj 31822 The subspace sum of two isomorphisms of lattice elements is less than the isomorphism of their lattice join. (Contributed by NM, 23-Sep-2014.)

TheoremdjhunssN 31823 Subspace union is a subset of subspace join. (Contributed by NM, 6-Aug-2014.) (New usage is discouraged.)
joinH

Theoremdochdmm1 31824 De Morgan-like law for closed subspace orthocomplement. (Contributed by NM, 13-Jan-2015.)
joinH

Theoremdjhexmid 31825 Excluded middle property of vector space closed subspace join. (Contributed by NM, 22-Jul-2014.)
joinH

Theoremdjh01 31826 Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.)
joinH

Theoremdjh02 31827 Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.)
joinH

Theoremdjhlsmcl 31828 A closed subspace sum equals subspace join. (shjshseli 22944 analog.) (Contributed by NM, 13-Aug-2014.)
joinH

Theoremdjhcvat42 31829* A covering property. (cvrat42 29857 analog.) (Contributed by NM, 17-Aug-2014.)
joinH

Theoremdihjatb 31830 Isomorphism H of lattice join of two atoms under the fiducial hyperplane. (Contributed by NM, 23-Sep-2014.)

Theoremdihjatc 31831 Isomorphism H of lattice join of an element under the fiducial hyperplane with atom not under it. (Contributed by NM, 26-Aug-2014.)

Theoremdihjatcclem1 31832 Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014.)

Theoremdihjatcclem2 31833 Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014.)

Theoremdihjatcclem3 31834* Lemma for dihjatcc 31836. (Contributed by NM, 28-Sep-2014.)

Theoremdihjatcclem4 31835* Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014.)

Theoremdihjatcc 31836 Isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014.)

Theoremdihjat 31837 Isomorphism H of lattice join of two atoms. (Contributed by NM, 29-Sep-2014.)

Theoremdihprrnlem1N 31838 Lemma for dihprrn 31840, showing one of 4 cases. (Contributed by NM, 30-Aug-2014.) (New usage is discouraged.)

Theoremdihprrnlem2 31839 Lemma for dihprrn 31840. (Contributed by NM, 29-Sep-2014.)

Theoremdihprrn 31840 The span of a vector pair belongs to the range of isomorphism H i.e. is a closed subspace. (Contributed by NM, 29-Sep-2014.)

Theoremdjhlsmat 31841 The sum of two subspace atoms equals their join. TODO: seems convoluted to go via dihprrn 31840; should we directly use dihjat 31837? (Contributed by NM, 13-Aug-2014.)
joinH

Theoremdihjat1lem 31842 Subspace sum of a closed subspace and an atom. (pmapjat1 30266 analog.) TODO: merge into dihjat1 31843? (Contributed by NM, 18-Aug-2014.)
joinH

Theoremdihjat1 31843 Subspace sum of a closed subspace and an atom. (pmapjat1 30266 analog.) (Contributed by NM, 1-Oct-2014.)
joinH

Theoremdihsmsprn 31844 Subspace sum of a closed subspace and the span of a singleton. (Contributed by NM, 17-Jan-2015.)

Theoremdihjat2 31845 The subspace sum of a closed subspace and an atom is the same as their subspace join. (Contributed by NM, 1-Oct-2014.)
joinH                     LSAtoms

Theoremdihjat3 31846 Isomorphism H of lattice join with an atom. (Contributed by NM, 25-Apr-2015.)

Theoremdihjat4 31847 Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015.)
LSAtoms

Theoremdihjat6 31848 Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015.)
LSAtoms

Theoremdihsmsnrn 31849 The subspace sum of two singleton spans is closed. (Contributed by NM, 27-Feb-2015.)

Theoremdihsmatrn 31850 The subspace sum of a closed subspace and an atom is closed. TODO: see if proof at http://math.stackexchange.com/a/1233211/50776 and Mon, 13 Apr 2015 20:44:07 -0400 email could be used instead of this and dihjat2 31845. (Contributed by NM, 15-Jan-2015.)
LSAtoms

Theoremdihjat5N 31851 Transfer lattice join with atom to subspace sum. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)

Theoremdvh4dimat 31852* There is an atom that is outside the subspace sum of 3 others. (Contributed by NM, 25-Apr-2015.)
LSAtoms

Theoremdvh3dimatN 31853* There is an atom that is outside the subspace sum of 2 others. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)
LSAtoms

Theoremdvh2dimatN 31854* Given an atom, there exists another. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)
LSAtoms

Theoremdvh1dimat 31855* There exists an atom. (Contributed by NM, 25-Apr-2015.)
LSAtoms

Theoremdvh1dim 31856* There exists a nonzero vector. (Contributed by NM, 26-Apr-2015.)

Theoremdvh4dimlem 31857* Lemma for dvh4dimN 31861. (Contributed by NM, 22-May-2015.)

Theoremdvhdimlem 31858* Lemma for dvh2dim 31859 and dvh3dim 31860. TODO: make this obsolete and use dvh4dimlem 31857 directly? (Contributed by NM, 24-Apr-2015.)

Theoremdvh2dim 31859* There is a vector that is outside the span of another. (Contributed by NM, 25-Apr-2015.)

Theoremdvh3dim 31860* There is a vector that is outside the span of 2 others. (Contributed by NM, 24-Apr-2015.)

Theoremdvh4dimN 31861* There is a vector that is outside the span of 3 others. (Contributed by NM, 22-May-2015.) (New usage is discouraged.)

Theoremdvh3dim2 31862* There is a vector that is outside of 2 spans with a common vector. (Contributed by NM, 13-May-2015.)

Theoremdvh3dim3N 31863* There is a vector that is outside of 2 spans. TODO: decide to use either this or dvh3dim2 31862 everywhere. If this one is needed, make dvh3dim2 31862 into a lemma. (Contributed by NM, 21-May-2015.) (New usage is discouraged.)

Theoremdochsnnz 31864 The orthocomplement of a singleton is nonzero. (Contributed by NM, 13-Jun-2015.)

Theoremdochsatshp 31865 The orthocomplement of a subspace atom is a hyperplane. (Contributed by NM, 27-Jul-2014.) (Revised by Mario Carneiro, 1-Oct-2014.)
LSAtoms       LSHyp

Theoremdochsatshpb 31866 The orthocomplement of a subspace atom is a hyperplane. (Contributed by NM, 29-Oct-2014.)
LSAtoms       LSHyp

Theoremdochsnshp 31867 The orthocomplement of a nonzero singleton is a hyperplane. (Contributed by NM, 3-Jan-2015.)
LSHyp

Theoremdochshpsat 31868 A hyperplane is closed iff its orthocomplement is an atom. (Contributed by NM, 29-Oct-2014.)
LSAtoms       LSHyp

Theoremdochkrsat 31869 The orthocomplement of a kernel is an atom iff it is nonzero. (Contributed by NM, 1-Nov-2014.)
LSAtoms       LFnl       LKer

Theoremdochkrsat2 31870 The orthocomplement of a kernel is an atom iff the double orthocomplement is not the vector space. (Contributed by NM, 1-Jan-2015.)
LSAtoms       LFnl       LKer

Theoremdochsat0 31871 The orthocomplement of a kernel is either an atom or zero. (Contributed by NM, 29-Jan-2015.)
LSAtoms       LFnl       LKer

Theoremdochkrsm 31872 The subspace sum of a closed subspace and a kernel orthocomplement is closed. (djhlsmcl 31828 can be used to convert sum to join.) (Contributed by NM, 29-Jan-2015.)
LFnl       LKer

Theoremdochexmidat 31873 Special case of excluded middle for the singleton of a vector. (Contributed by NM, 27-Oct-2014.)

Theoremdochexmidlem1 31874 Lemma for dochexmid 31882. Holland's proof implicitly requires , which we prove here. (Contributed by NM, 14-Jan-2015.)
LSAtoms

Theoremdochexmidlem2 31875 Lemma for dochexmid 31882. (Contributed by NM, 14-Jan-2015.)
LSAtoms

Theoremdochexmidlem3 31876 Lemma for dochexmid 31882. Use atom exchange lsatexch1 29460 to swap and . (Contributed by NM, 14-Jan-2015.)
LSAtoms

Theoremdochexmidlem4 31877 Lemma for dochexmid 31882. (Contributed by NM, 15-Jan-2015.)
LSAtoms

Theoremdochexmidlem5 31878 Lemma for dochexmid 31882. (Contributed by NM, 15-Jan-2015.)
LSAtoms

Theoremdochexmidlem6 31879 Lemma for dochexmid 31882. (Contributed by NM, 15-Jan-2015.)
LSAtoms

Theoremdochexmidlem7 31880 Lemma for dochexmid 31882. Contradict dochexmidlem6 31879. (Contributed by NM, 15-Jan-2015.)
LSAtoms

Theoremdochexmidlem8 31881 Lemma for dochexmid 31882. The contradiction of dochexmidlem6 31879 and dochexmidlem7 31880 shows that there can be no atom that is not in , which is therefore the whole atom space. (Contributed by NM, 15-Jan-2015.)
LSAtoms

Theoremdochexmid 31882 Excluded middle law for closed subspaces, which is equivalent to (and derived from) the orthomodular law dihoml4 31791. Lemma 3.3(2) in [Holland95] p. 215. In our proof, we use the variables , , , , in place of Hollands' l, m, P, Q, L respectively. (pexmidALTN 30391 analog.) (Contributed by NM, 15-Jan-2015.)

Theoremdochsnkrlem1 31883 Lemma for dochsnkr 31886. (Contributed by NM, 1-Jan-2015.)
LFnl       LKer

Theoremdochsnkrlem2 31884 Lemma for dochsnkr 31886. (Contributed by NM, 1-Jan-2015.)
LFnl       LKer                            LSAtoms

Theoremdochsnkrlem3 31885 Lemma for dochsnkr 31886. (Contributed by NM, 2-Jan-2015.)
LFnl       LKer

Theoremdochsnkr 31886 A (closed) kernel expressed in terms of a nonzero vector in its orthocomplement. TODO: consolidate lemmas unless they're needed for something else (in which case break out as theorems) (Contributed by NM, 2-Jan-2015.)
LFnl       LKer

Theoremdochsnkr2 31887* Kernel of the explicit functional determined by a nonzero vector . Compare the more general lshpkr 29531. (Contributed by NM, 27-Oct-2014.)
LKer       Scalar

Theoremdochsnkr2cl 31888* The determining functional belongs to the atom formed by the orthocomplement of the kernel. (Contributed by NM, 4-Jan-2015.)
LKer       Scalar

Theoremdochflcl 31889* Closure of the explicit functional determined by a nonzero vector . Compare the more general lshpkrcl 29530. (Contributed by NM, 27-Oct-2014.)
LFnl       Scalar

Theoremdochfl1 31890* The value of the explicit functional is 1 at the that determines it. (Contributed by NM, 27-Oct-2014.)
Scalar

Theoremdochfln0 31891 The value of a functional is nonzero at a nonzero vector in the orthocomplement of its kernel. (Contributed by NM, 2-Jan-2015.)
Scalar                     LFnl       LKer

Theoremdochkr1 31892* A non-zero functional has a value of 1 at some argument belonging to the orthocomplement of its kernel (when its kernel is a closed hyperplane). Tighter version of lfl1 29484. (Contributed by NM, 2-Jan-2015.)
Scalar                     LFnl       LKer

Theoremdochkr1OLDN 31893* A non-zero functional has a value of 1 at some argument belonging to the orthocomplement of its kernel (when its kernel is a closed hyperplane). Tighter version of lfl1 29484. (Contributed by NM, 2-Jan-2015.) (New usage is discouraged.)
Scalar                     LFnl       LKer

19.26.11  Construction of involution and inner product from a Hilbert lattice

SyntaxclpoN 31894 Extend class notation with all polarities of a left module or left vector space.
LPol

Definitiondf-lpolN 31895* Define the set of all polarities of a left module or left vector space. A polarity is a kind of complementation operation on a subspace. The double polarity of a subspace is a closure operation. Based on Definition 3.2 of [Holland95] p. 214 for projective geometry polarities. For convenience, we open up the domain to include all vector subsets and not just subspaces, but any more restricted polarity can be converted to this one by taking the span of its argument. (Contributed by NM, 24-Nov-2014.)
LPol LSAtoms LSHyp

TheoremlpolsetN 31896* The set of polarities of a left module or left vector space. (Contributed by NM, 24-Nov-2014.) (New usage is discouraged.)
LSAtoms       LSHyp       LPol

TheoremislpolN 31897* The predicate "is a polarity". (Contributed by NM, 24-Nov-2014.) (New usage is discouraged.)
LSAtoms       LSHyp       LPol

TheoremislpoldN 31898* Properties that determine a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
LSAtoms       LSHyp       LPol

TheoremlpolfN 31899 Functionality of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
LPol

TheoremlpolvN 31900 The polarity of the whole space is the zero subspace. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
LPol

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 32301-32378
 Copyright terms: Public domain < Previous  Next >