Home Metamath Proof ExplorerTheorem List (p. 321 of 328) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-21514) Hilbert Space Explorer (21515-23037) Users' Mathboxes (23038-32776)

Theorem List for Metamath Proof Explorer - 32001-32100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdicelval2nd 32001 Membership in value of the partial isomorphism C for a lattice . (Contributed by NM, 16-Feb-2014.)

Theoremdicvaddcl 32002 Membership in value of the partial isomorphism C is closed under vector sum. (Contributed by NM, 16-Feb-2014.)

Theoremdicvscacl 32003 Membership in value of the partial isomorphism C is closed under scalar product. (Contributed by NM, 16-Feb-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)

Theoremdicn0 32004 The value of the partial isomorphism C is not empty. (Contributed by NM, 15-Feb-2014.)

Theoremdiclss 32005 The value of partial isomorphism C is a subspace of partial vector space H. (Contributed by NM, 16-Feb-2014.)

Theoremdiclspsn 32006* The value of isomorphism C is spanned by vector . Part of proof of Lemma N of [Crawley] p. 121 line 29. (Contributed by NM, 21-Feb-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)

Theoremcdlemn2 32007* Part of proof of Lemma N of [Crawley] p. 121 line 30. (Contributed by NM, 21-Feb-2014.)

Theoremcdlemn2a 32008* Part of proof of Lemma N of [Crawley] p. 121. (Contributed by NM, 24-Feb-2014.)

Theoremcdlemn3 32009* Part of proof of Lemma N of [Crawley] p. 121 line 31. (Contributed by NM, 21-Feb-2014.)

Theoremcdlemn4 32010* Part of proof of Lemma N of [Crawley] p. 121 line 31. (Contributed by NM, 21-Feb-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)

Theoremcdlemn4a 32011* Part of proof of Lemma N of [Crawley] p. 121 line 32. (Contributed by NM, 24-Feb-2014.)

Theoremcdlemn5pre 32012* Part of proof of Lemma N of [Crawley] p. 121 line 32. (Contributed by NM, 25-Feb-2014.)

Theoremcdlemn5 32013 Part of proof of Lemma N of [Crawley] p. 121 line 32. (Contributed by NM, 25-Feb-2014.)

Theoremcdlemn6 32014* Part of proof of Lemma N of [Crawley] p. 121 line 35. (Contributed by NM, 26-Feb-2014.)

Theoremcdlemn7 32015* Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 26-Feb-2014.)

Theoremcdlemn8 32016* Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 26-Feb-2014.)

Theoremcdlemn9 32017* Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn10 32018 Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn11a 32019* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn11b 32020* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn11c 32021* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn11pre 32022* Part of proof of Lemma N of [Crawley] p. 121 line 37. TODO: combine cdlemn11a 32019, cdlemn11b 32020, cdlemn11c 32021, cdlemn11pre into one? (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn11 32023 Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)

Theoremcdlemn 32024 Lemma N of [Crawley] p. 121 line 27. (Contributed by NM, 27-Feb-2014.)

Theoremdihordlem6 32025* Part of proof of Lemma N of [Crawley] p. 122 line 35. (Contributed by NM, 3-Mar-2014.)

Theoremdihordlem7 32026* Part of proof of Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihordlem7b 32027* Part of proof of Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihjustlem 32028 Part of proof after Lemma N of [Crawley] p. 122 line 4, "the definition of phi(x) is independent of the atom q." (Contributed by NM, 2-Mar-2014.)

Theoremdihjust 32029 Part of proof after Lemma N of [Crawley] p. 122 line 4, "the definition of phi(x) is independent of the atom q." (Contributed by NM, 2-Mar-2014.)

Theoremdihord1 32030 Part of proof after Lemma N of [Crawley] p. 122. Forward ordering property. TODO: change to using lhpmcvr3 30836, here and all theorems below. (Contributed by NM, 2-Mar-2014.)

Theoremdihord2a 32031 Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord2b 32032 Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord2cN 32033* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. TODO: needed? shorten other proof with it? (Contributed by NM, 3-Mar-2014.) (New usage is discouraged.)

Theoremdihord11b 32034* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord10 32035* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord11c 32036* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord2pre 32037* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Theoremdihord2pre2 32038* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 4-Mar-2014.)

Theoremdihord2 32039 Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. Todo: do we need and ? (Contributed by NM, 4-Mar-2014.)

Syntaxcdih 32040 Extend class notation with isomorphism H.

Definitiondf-dih 32041* Define isomorphism H. (Contributed by NM, 28-Jan-2014.)

Theoremdihffval 32042* The isomorphism H for a lattice . Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 28-Jan-2014.)

Theoremdihfval 32043* Isomorphism H for a lattice . Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 28-Jan-2014.)

Theoremdihval 32044* Value of isomorphism H for a lattice . Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 3-Feb-2014.)

Theoremdihvalc 32045* Value of isomorphism H for a lattice when . (Contributed by NM, 4-Mar-2014.)

Theoremdihlsscpre 32046 Closure of isomorphism H for a lattice when . (Contributed by NM, 6-Mar-2014.)

Theoremdihvalcqpre 32047 Value of isomorphism H for a lattice when , given auxiliary atom . (Contributed by NM, 6-Mar-2014.)

Theoremdihvalcq 32048 Value of isomorphism H for a lattice when , given auxiliary atom . TODO: Use dihvalcq2 32059 (with lhpmcvr3 30836 for simplification) that changes and to and make this obsolete. Do to other theorems as well. (Contributed by NM, 6-Mar-2014.)

Theoremdihvalb 32049 Value of isomorphism H for a lattice when . (Contributed by NM, 4-Mar-2014.)

TheoremdihopelvalbN 32050* Ordered pair member of the partial isomorphism H for argument under . (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)

Theoremdihvalcqat 32051 Value of isomorphism H for a lattice at an atom not under . (Contributed by NM, 27-Mar-2014.)

Theoremdih1dimb 32052* Two expressions for a 1-dimensional subspace of vector space H (when is a nonzero vector i.e. non-identity translation). (Contributed by NM, 27-Apr-2014.)

Theoremdih1dimb2 32053* Isomorphism H at an atom under . (Contributed by NM, 27-Apr-2014.)

Theoremdih1dimc 32054* Isomorphism H at an atom not under . (Contributed by NM, 27-Apr-2014.)

Theoremdib2dim 32055 Extend dia2dim 31889 to partial isomorphism B. (Contributed by NM, 22-Sep-2014.)

Theoremdih2dimb 32056 Extend dib2dim 32055 to isomorphism H. (Contributed by NM, 22-Sep-2014.)

Theoremdih2dimbALTN 32057 Extend dia2dim 31889 to isomorphism H. (This version combines dib2dim 32055 and dih2dimb 32056 for shorter overall proof, but may be less easy to understand. TODO: decide which to use.) (Contributed by NM, 22-Sep-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdihopelvalcqat 32058* Ordered pair member of the partial isomorphism H for atom argument not under . TODO: remove .t hypothesis. (Contributed by NM, 30-Mar-2014.)

Theoremdihvalcq2 32059 Value of isomorphism H for a lattice when , given auxiliary atom . (Contributed by NM, 26-Sep-2014.)

Theoremdihopelvalcpre 32060* Member of value of isomorphism H for a lattice when , given auxiliary atom . TODO: refactor to be shorter and more understandable; add lemmas? (Contributed by NM, 13-Mar-2014.)

Theoremdihopelvalc 32061* Member of value of isomorphism H for a lattice when , given auxiliary atom . (Contributed by NM, 13-Mar-2014.)

Theoremdihlss 32062 The value of isomorphism H is a subspace. (Contributed by NM, 6-Mar-2014.)

Theoremdihss 32063 The value of isomorphism H is a set of vectors. (Contributed by NM, 14-Mar-2014.)

Theoremdihssxp 32064 An isomorphism H value is included in the vector space (expressed as ). (Contributed by NM, 26-Sep-2014.)

Theoremdihopcl 32065 Closure of an ordered pair (vector) member of a value of isomorphism H. (Contributed by NM, 26-Sep-2014.)

TheoremxihopellsmN 32066* Ordered pair membership in a subspace sum of isomorphism H values. (Contributed by NM, 26-Sep-2014.) (New usage is discouraged.)

Theoremdihopellsm 32067* Ordered pair membership in a subspace sum of isomorphism H values. (Contributed by NM, 26-Sep-2014.)

Theoremdihord6apre 32068* Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)

Theoremdihord3 32069 The isomorphism H for a lattice is order-preserving in the region under co-atom . (Contributed by NM, 6-Mar-2014.)

Theoremdihord4 32070 The isomorphism H for a lattice is order-preserving in the region not under co-atom . TODO: reformat q e. A /\ -. q .<_ W to eliminate adant*. (Contributed by NM, 6-Mar-2014.)

Theoremdihord5b 32071 Part of proof that isomorphism H is order-preserving. TODO: eliminate 3ad2ant1; combine w/ other way to have one lhpmcvr2 (Contributed by NM, 7-Mar-2014.)

Theoremdihord6b 32072 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)

Theoremdihord6a 32073 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)

Theoremdihord5apre 32074 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)

Theoremdihord5a 32075 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)

Theoremdihord 32076 The isomorphism H is order-preserving. Part of proof after Lemma N of [Crawley] p. 122 line 6. (Contributed by NM, 7-Mar-2014.)

Theoremdih11 32077 The isomorphism H is one-to-one. Part of proof after Lemma N of [Crawley] p. 122 line 6. (Contributed by NM, 7-Mar-2014.)

Theoremdihf11lem 32078 Functionality of the isomorphism H. (Contributed by NM, 6-Mar-2014.)

Theoremdihf11 32079 The isomorphism H for a lattice is a one-to-one function. . Part of proof after Lemma N of [Crawley] p. 122 line 6. (Contributed by NM, 7-Mar-2014.)

Theoremdihfn 32080 Functionality and domain of isomorphism H. (Contributed by NM, 9-Mar-2014.)

Theoremdihdm 32081 Domain of isomorphism H. (Contributed by NM, 9-Mar-2014.)

Theoremdihcl 32082 Closure of isomorphism H. (Contributed by NM, 8-Mar-2014.)

Theoremdihcnvcl 32083 Closure of isomorphism H converse. (Contributed by NM, 8-Mar-2014.)

Theoremdihcnvid1 32084 The converse isomorphism of an isomorphism. (Contributed by NM, 5-Aug-2014.)

Theoremdihcnvid2 32085 The isomorphism of a converse isomorphism. (Contributed by NM, 5-Aug-2014.)

Theoremdihcnvord 32086 Ordering property for converse of isomorphism H. (Contributed by NM, 17-Aug-2014.)

Theoremdihcnv11 32087 The converse of isomorphism H is one-to-one. (Contributed by NM, 17-Aug-2014.)

Theoremdihsslss 32088 The isomorphism H maps to subspaces. (Contributed by NM, 14-Mar-2014.)

Theoremdihrnlss 32089 The isomorphism H maps to subspaces. (Contributed by NM, 14-Mar-2014.)

Theoremdihrnss 32090 The isomorphism H maps to a set of vectors. (Contributed by NM, 14-Mar-2014.)

Theoremdihvalrel 32091 The value of isomorphism H is a relation. (Contributed by NM, 9-Mar-2014.)

Theoremdih0 32092 The value of isomorphism H at the lattice zero is the singleton of the zero vector i.e. the zero subspace. (Contributed by NM, 9-Mar-2014.)

Theoremdih0bN 32093 A lattice element is zero iff its isomorphism is the zero subspace. (Contributed by NM, 16-Aug-2014.) (New usage is discouraged.)

Theoremdih0vbN 32094 A vector is zero iff its span is the isomorphism of lattice zero. (Contributed by NM, 16-Aug-2014.) (New usage is discouraged.)

Theoremdih0cnv 32095 The isomorphism H converse value of the zero subspace is the lattice zero. (Contributed by NM, 19-Jun-2014.)

Theoremdih0rn 32096 The zero subspace belongs to the range of isomorphism H. (Contributed by NM, 27-Apr-2014.)

Theoremdih0sb 32097 A subspace is zero iff the converse of its isomorphism is lattice zero. (Contributed by NM, 17-Aug-2014.)

Theoremdih1 32098 The value of isomorphism H at the lattice unit is the set of all vectors. (Contributed by NM, 13-Mar-2014.)

Theoremdih1rn 32099 The full vector space belongs to the range of isomorphism H. (Contributed by NM, 19-Jun-2014.)

Theoremdih1cnv 32100 The isomorphism H converse value of the full vector space is the lattice one. (Contributed by NM, 19-Jun-2014.)

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