Home Metamath Proof ExplorerTheorem List (p. 192 of 330) < 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 (1-22459) Hilbert Space Explorer (22460-23982) Users' Mathboxes (23983-32936)

Theorem List for Metamath Proof Explorer - 19101-19200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempi1bas 19101 The base set of the fundamental group of a topological space at a given base point. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1blem 19102 Lemma for pi1buni 19103. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1buni 19103 Another way to write the loop space base in terms of the base of the fundamental group. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1bas2 19104 The base set of the fundamental group, written self-referentially. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1eluni 19105 Elementhood in the base set of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1bas3 19106 The base set of the fundamental group. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1cpbl 19107 The group operation, loop concatenation, is compatible with homotopy equivalence. (Contributed by Mario Carneiro, 10-Jul-2015.)
TopOn

Theoremelpi1 19108* The elements of the fundamental group. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
TopOn

Theoremelpi1i 19109 The elements of the fundamental group. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1addf 19110 The group operation of is a binary operation. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1addval 19111 The concatenation of two path-homotopy classes in the fundamental group. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
TopOn

Theorempi1grplem 19112 Lemma for pi1grp 19113. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 10-Aug-2015.)
TopOn

Theorempi1grp 19113 The fundamental group is a group. Proposition 1.3 of [Hatcher] p. 26. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof shortened by Mario Carneiro, 8-Jun-2014.) (Revised by Mario Carneiro, 10-Aug-2015.)
TopOn

Theorempi1id 19114 The identity element of the fundamental group. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 10-Aug-2015.)
TopOn

Theorempi1inv 19115* An inverse in the fundamental group. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 10-Aug-2015.)
TopOn

Theorempi1xfrf 19116* Functionality of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 23-Dec-2016.)
TopOn

Theorempi1xfrval 19117* The value of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 23-Dec-2016.)
TopOn

Theorempi1xfr 19118* Given a path and its inverse between two basepoints, there is an induced group homomorphism on the fundamental groups. (Contributed by Mario Carneiro, 12-Feb-2015.)
TopOn

Theorempi1xfrcnvlem 19119* Given a path between two basepoints, there is an induced group homomorphism on the fundamental groups. (Contributed by Mario Carneiro, 12-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
TopOn

Theorempi1xfrcnv 19120* Given a path between two basepoints, there is an induced group homomorphism on the fundamental groups. (Contributed by Mario Carneiro, 12-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
TopOn

Theorempi1xfrgim 19121* The mapping between fundamental groups is an isomorphism. (Contributed by Mario Carneiro, 12-Feb-2015.)
TopOn                     GrpIso

Theorempi1cof 19122* Functionality of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 23-Dec-2016.)
TopOn

Theorempi1coval 19123* The value of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 10-Aug-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
TopOn

Theorempi1coghm 19124* The mapping between fundamental groups is a group homomorphism. (Contributed by Mario Carneiro, 10-Aug-2015.) (Revised by Mario Carneiro, 23-Dec-2016.)
TopOn

11.5  Complex metric vector spaces

11.5.1  Complex left modules

Syntaxcclm 19125 Complex module.
CMod

Definitiondf-clm 19126* Define a complex module, which is just a left module over a subring of , which allows us to use conventional addition, multiplication, etc. in the left module theorems. (Contributed by Mario Carneiro, 16-Oct-2015.)
CMod Scalar flds SubRingfld

Theoremisclm 19127 A complex module is a left module over a subring of the complexes. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod flds SubRingfld

Theoremclmsca 19128 A complex module is a left module over a subring of the complexes. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod flds

Theoremclmsubrg 19129 A complex module is a left module over a subring of the complexes. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod SubRingfld

Theoremclmlmod 19130 A complex module is a left module. (Contributed by Mario Carneiro, 16-Oct-2015.)
CMod

Theoremclmgrp 19131 A complex module is an additive group. (Contributed by Mario Carneiro, 16-Oct-2015.)
CMod

Theoremclmabl 19132 A complex module is an abelian group. (Contributed by Mario Carneiro, 16-Oct-2015.)
CMod

Theoremclmrng 19133 The scalar ring of a complex module is a ring. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar       CMod

Theoremclmfgrp 19134 The scalar ring of a complex module is a group. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar       CMod

Theoremclm0 19135 The zero of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar       CMod

Theoremclm1 19136 The identity of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar       CMod

Theoremclmadd 19137 The addition of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar       CMod

Theoremclmmul 19138 The multiplication of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar       CMod

Theoremclmcj 19139 The conjugation of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar       CMod

Theoremisclmi 19140 Reverse direction of isclm 19127. (Contributed by Mario Carneiro, 30-Oct-2015.)
Scalar       flds SubRingfld CMod

Theoremclmzss 19141 The scalar ring of a complex module contains the integers. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmsscn 19142 The scalar ring of a complex module is a subset of the complexes. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmsub 19143 Subtraction in the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmneg 19144 Negation in the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmabs 19145 Norm in the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmacl 19146 Closure of ring addition for a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmmcl 19147 Closure of ring multiplication for a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmsubcl 19148 Closure of ring subtraction for a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremlmhmclm 19149 The domain of a linear operator is a complex module iff the range is. (Contributed by Mario Carneiro, 21-Oct-2015.)
LMHom CMod CMod

Theoremclmvsass 19150 Associative law for scalar product. (lmodvsass 16013 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar                     CMod

Theoremclmvsdir 19151 Distributive law for scalar product. (lmodvsdir 16012 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar                            CMod

Theoremclmvs1 19152 Scalar product with ring unit. (lmodvs1 16016 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
CMod

Theoremclm0vs 19153 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (lmod0vs 16021 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar                     CMod

Theoremclmvneg1 19154 Minus 1 times a vector is the negative of the vector. Equation 2 of [Kreyszig] p. 51. (lmodvneg1 16025 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmvsneg 19155 Multiplication of a vector by a negated scalar. (lmodvsneg 16026 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar                            CMod

Theoremclmmulg 19156 The group multiple function matches the scalar multiplication function. (Contributed by Mario Carneiro, 15-Oct-2015.)
.g              CMod

Theoremclmsubdir 19157 Scalar multiplication distributive law for subtraction. (lmodsubdir 16040 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar                     CMod

Theoremzlmclm 19158 The -module operation turns an arbitrary abelian group into a complex module. (Contributed by Mario Carneiro, 30-Oct-2015.)
Mod       CMod

Theoremclmzlmvsca 19159 The scalar product of a complex module matches the scalar product of the derived -module, which implies, together with zlmbas 16837 and zlmplusg 16838, that any module over is structure-equivalent to the canonical -module Mod. (Contributed by Mario Carneiro, 30-Oct-2015.)
Mod              CMod

Theoremnmoleub2lem 19160* Lemma for nmoleub2a 19163 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)
Scalar              NrmMod CMod       NrmMod CMod       LMHom

Theoremnmoleub2lem3 19161* Lemma for nmoleub2a 19163 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)
Scalar              NrmMod CMod       NrmMod CMod       LMHom

Theoremnmoleub2lem2 19162* Lemma for nmoleub2a 19163 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)
Scalar              NrmMod CMod       NrmMod CMod       LMHom

Theoremnmoleub2a 19163* The operator norm is the supremum of the value of a linear operator in the closed unit ball. (Contributed by Mario Carneiro, 19-Oct-2015.)
Scalar              NrmMod CMod       NrmMod CMod       LMHom

Theoremnmoleub2b 19164* The operator norm is the supremum of the value of a linear operator in the open unit ball. (Contributed by Mario Carneiro, 19-Oct-2015.)
Scalar              NrmMod CMod       NrmMod CMod       LMHom

Theoremnmoleub3 19165* The operator norm is the supremum of the value of a linear operator on the closed unit sphere. (Contributed by Mario Carneiro, 19-Oct-2015.)
Scalar              NrmMod CMod       NrmMod CMod       LMHom

Theoremnmhmcn 19166 A linear operator over a normed complex module is bounded iff it is continuous. (Contributed by Mario Carneiro, 22-Oct-2015.)
Scalar              NrmMod CMod NrmMod CMod NMHom LMHom

11.5.2  Complex pre-Hilbert space

Syntaxccph 19167 Extend class notation with a complex pre-Hilbert space.

Syntaxctch 19168 Function to put a norm on a Hilbert space.
toCHil

Definitiondf-cph 19169* Define a complex pre-Hilbert space. By restricting the scalar field to a quadratically closed subfield of , we have enough structure to define a norm, with the associated connection to a metric and topology. (Contributed by Mario Carneiro, 8-Oct-2015.)
NrmMod Scalar flds

Definitiondf-tch 19170* Define a function to augment a (pre-)Hilbert space with a norm. No extra parameters are needed, but some conditions must be satisfied to ensure that this in fact creates a normed pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
toCHil toNrmGrp

Theoremiscph 19171* A complex pre-Hilbert space is a pre-Hilbert space over a quadratically closed subfield of the complexes, with a norm defined (Contributed by Mario Carneiro, 8-Oct-2015.)
Scalar              NrmMod flds

Theoremcphphl 19172 A complex pre-Hilbert space is a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremcphnlm 19173 A complex pre-Hilbert space is a normed module. (Contributed by Mario Carneiro, 7-Oct-2015.)
NrmMod

Theoremcphngp 19174 A complex pre-Hilbert space is a normed group. (Contributed by Mario Carneiro, 13-Oct-2015.)
NrmGrp

Theoremcphlmod 19175 A complex pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremcphlvec 19176 A complex pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremcphnvc 19177 A complex pre-Hilbert space is a normed vector space. (Contributed by Mario Carneiro, 8-Oct-2015.)
NrmVec

Theoremcphsubrglem 19178 Lemma for cphsubrg 19181. (Contributed by Mario Carneiro, 9-Oct-2015.)
flds               flds SubRingfld

Theoremcphreccllem 19179 Lemma for cphreccl 19182. (Contributed by Mario Carneiro, 8-Oct-2015.)
flds

Theoremcphsca 19180 A complex pre-Hilbert space is a vector space over a subfield of . (Contributed by Mario Carneiro, 8-Oct-2015.)
Scalar              flds

Theoremcphsubrg 19181 The scalar field of a complex pre-Hilbert space is a subring of . (Contributed by Mario Carneiro, 8-Oct-2015.)
Scalar              SubRingfld

Theoremcphreccl 19182 The scalar field of a complex pre-Hilbert space is closed under reciprocal. (Contributed by Mario Carneiro, 8-Oct-2015.)
Scalar

Theoremcphdivcl 19183 The scalar field of a complex pre-Hilbert space is closed under reciprocal. (Contributed by Mario Carneiro, 11-Oct-2015.)
Scalar

Theoremcphcjcl 19184 The scalar field of a complex pre-Hilbert space is closed under conjugation. (Contributed by Mario Carneiro, 11-Oct-2015.)
Scalar

Theoremcphsqrcl 19185 The scalar field of a complex pre-Hilbert space is closed under square roots of positive reals (i.e. it is quadratically closed relative to ). (Contributed by Mario Carneiro, 8-Oct-2015.)
Scalar

Theoremcphabscl 19186 The scalar field of a complex pre-Hilbert space is closed under the absolute value operation. (Contributed by Mario Carneiro, 11-Oct-2015.)
Scalar

Theoremcphsqrcl2 19187 The scalar field of a complex pre-Hilbert space is closed under square roots of all numbers except possibly the negative reals. (Contributed by Mario Carneiro, 8-Oct-2015.)
Scalar

Theoremcphsqrcl3 19188 If the scalar field contains , it is completely closed under square roots (i.e. it is quadratically closed). (Contributed by Mario Carneiro, 11-Oct-2015.)
Scalar

Theoremcphqss 19189 The scalar field of a complex pre-Hilbert space contains all rational numbers. (Contributed by Mario Carneiro, 15-Oct-2015.)
Scalar

Theoremcphclm 19190 A complex pre-Hilbert space is a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
CMod

Theoremcphnmvs 19191 Norm of a scalar product. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar

Theoremcphipcl 19192 An inner product is a member of the complex numbers. (Contributed by Mario Carneiro, 13-Oct-2015.)

Theoremcphnmfval 19193* The value of the norm in a complex pre-Hilbert space is the square root of the inner product of a vector with itself. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremcphnm 19194 The square of the norm is the norm of an inner product in a normed pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremnmsq 19195 The square of the norm is the norm of an inner product in a normed pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremcphnmf 19196 The norm of a vector is a member of the scalar field in a complex pre-Hilbert space. (Contributed by Mario Carneiro, 9-Oct-2015.)
Scalar

Theoremcphnmcl 19197 The norm of a vector is a member of the scalar field in a complex pre-Hilbert space. (Contributed by Mario Carneiro, 9-Oct-2015.)
Scalar

Theoremreipcl 19198 An inner product of an element with itself is real. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremipge0 19199 The inner product in a complex pre-Hilbert space is positive definite. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremcphipcj 19200 Conjugate of an inner product in a complex pre-Hilbert space. Complex version of ipcj 16903. (Contributed by Mario Carneiro, 16-Oct-2015.)

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 32801-32900 330 32901-32936
 Copyright terms: Public domain < Previous  Next >