Home Metamath Proof ExplorerTheorem List (p. 114 of 328) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-22421) Hilbert Space Explorer (22422-23944) Users' Mathboxes (23945-32762)

Theorem List for Metamath Proof Explorer - 11301-11400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremltweuz 11301 is a well-founded relation on any sequence of upper integers. (Contributed by Andrew Salmon, 13-Nov-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)

Theoremltwenn 11302 Less than well-orders the naturals. (Contributed by Scott Fenton, 6-Aug-2013.)

Theoremltwefz 11303 Less than well-orders a set of finite integers. (Contributed by Scott Fenton, 8-Aug-2013.)

Theoremuzenom 11304 An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.)

Theoremuzinf 11305 An upper integer set is infinite. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)

Theoremuzrdgxfr 11306* Transfer the value of the recursive sequence builder from one base to another. (Contributed by Mario Carneiro, 1-Apr-2014.)

Theoremfzennn 11307 The cardinality of a finite set of sequential integers. (See om2uz0i 11287 for a description of the hypothesis.) (Contributed by Mario Carneiro, 12-Feb-2013.) (Revised by Mario Carneiro, 7-Mar-2014.)

Theoremfzen2 11308 The cardinality of a finite set of sequential integers with arbitrary endpoints. (Contributed by Mario Carneiro, 13-Feb-2014.)

Theoremcardfz 11309 The cardinality of a finite set of sequential integers. (See om2uz0i 11287 for a description of the hypothesis.) (Contributed by NM, 7-Nov-2008.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremhashgf1o 11310 maps one-to-one onto . (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 13-Sep-2013.)

Theoremfzfi 11311 A finite interval of integers is finite. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.)

Theoremfzfid 11312 Commonly used special case of fzfi 11311. (Contributed by Mario Carneiro, 25-May-2014.)

Theoremfzofi 11313 Half-open integer sets are finite. (Contributed by Stefan O'Rear, 15-Aug-2015.)
..^

Theoremfsequb 11314* The values of a finite real sequence have an upper bound. (Contributed by NM, 19-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)

Theoremfsequb2 11315* The values of a finite real sequence have an upper bound. (Contributed by NM, 20-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)

Theoremfseqsupcl 11316 The values of a finite real sequence have a supremum. (Contributed by NM, 20-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremfseqsupubi 11317 The values of a finite real sequence are bounded by their supremum. (Contributed by NM, 20-Sep-2005.)

Theoremnn0ennn 11318 The nonnegative integers are equinumerous to the natural numbers. (Contributed by NM, 19-Jul-2004.)

Theoremnnenom 11319 The set of natural numbers (as a subset of complex numbers) is equinumerous to omega (the set of finite ordinal numbers). (Contributed by NM, 31-Jul-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremuzindi 11320* Indirect strong induction on the upper integers. (Contributed by Stefan O'Rear, 25-Aug-2015.)
..^

Theoremaxdc4uzlem 11321* Lemma for axdc4uz 11322. (Contributed by Mario Carneiro, 8-Jan-2014.) (Revised by Mario Carneiro, 26-Dec-2014.)

Theoremaxdc4uz 11322* A version of axdc4 8336 that works on a set of upper integers instead of . (Contributed by Mario Carneiro, 8-Jan-2014.)

Syntaxcseq 11323 Extend class notation with recursive sequence builder.

Definitiondf-seq 11324* Define a general-purpose operation that builds a recursive sequence (i.e. a function on the natural numbers or some other upper integer set) whose value at an index is a function of its previous value and the value of an input sequence at that index. This definition is complicated, but fortunately it is not intended to be used directly. Instead, the only purpose of this definition is to provide us with an object that has the properties expressed by seq1 11336 and seqp1 11338. Typically, those are the main theorems that would be used in practice.

The first operand in the parentheses is the operation that is applied to the previous value and the value of the input sequence (second operand). The operand to the left of the parenthesis is the integer to start from. For example, for the operation , an input sequence with values 1, 1/2, 1/4, 1/8,... would be transformed into the output sequence with values 1, 3/2, 7/4, 15/8,.., so that , 3/2, etc. In other words, transforms a sequence into an infinite series. means "the sum of F(n) from n = M to infinity is 2." Since limits are unique (climuni 12346), by climdm 12348 the "sum of F(n) from n = 1 to infinity" can be expressed as (provided the sequence converges) and evaluates to 2 in this example.

Internally, the function generates as its values a set of ordered pairs starting at , with the first member of each pair incremented by one in each successive value. So, the range of is exactly the sequence we want, and we just extract the range (restricted to omega) and throw away the domain.

This definition has its roots in a series of theorems from om2uz0i 11287 through om2uzf1oi 11293, originally proved by Raph Levien for use with df-exp 11383 and later generalized for arbitrary recursive sequences. Definition df-sum 12480 extracts the summation values from partial (finite) and complete (infinite) series. (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 4-Sep-2013.)

Theoremseqex 11325 Existence of the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)

Theoremseqeq1 11326 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)

Theoremseqeq2 11327 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)

Theoremseqeq3 11328 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)

Theoremseqeq1d 11329 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)

Theoremseqeq2d 11330 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)

Theoremseqeq3d 11331 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)

Theoremseqeq123d 11332 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)

Theoremnfseq 11333 Hypothesis builder for the sequence builder operation. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremseqval 11334* Value of the sequence builder function. (Contributed by Mario Carneiro, 24-Jun-2013.)

Theoremseqfn 11335 The sequence builder function is a function. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremseq1 11336 Value of the sequence builder function at its initial value. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremseq1i 11337 Value of the sequence builder function at its initial value. (Contributed by Mario Carneiro, 30-Apr-2014.)

Theoremseqp1 11338 Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremseqp1i 11339 Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 30-Apr-2014.)

Theoremseqm1 11340 Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.)

Theoremseqcl2 11341* Closure properties of the recursive sequence builder. (Contributed by Mario Carneiro, 2-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqf2 11342* Range of the recursive sequence builder. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqcl 11343* Closure properties of the recursive sequence builder. (Contributed by Mario Carneiro, 2-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqf 11344* Range of the recursive sequence builder (special case of seqf2 11342). (Contributed by Mario Carneiro, 24-Jun-2013.)

Theoremseqfveq2 11345* Equality of sequences. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqfeq2 11346* Equality of sequences. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqfveq 11347* Equality of sequences. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqfeq 11348* Equality of sequences. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqshft2 11349* Shifting the index set of a sequence. (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqres 11350 Restricting its characteristic function to does not affect the function. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremserf 11351* An infinite series of complex terms is a function from to . (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremserfre 11352* An infinite series of real numbers is a function from to . (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremmonoord 11353* Ordering relation for a monotonic sequence, increasing case. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)

Theoremmonoord2 11354* Ordering relation for a monotonic sequence, decreasing case. (Contributed by Mario Carneiro, 18-Jul-2014.)

Theoremsermono 11355* The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 30-Jun-2013.)

Theoremseqsplit 11356* Split a sequence into two sequences. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseq1p 11357* Removing the first term from a sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqcaopr3 11358* Lemma for seqcaopr2 11359. (Contributed by Mario Carneiro, 25-Apr-2016.)
..^

Theoremseqcaopr2 11359* The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014.)

Theoremseqcaopr 11360* The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 30-May-2014.)

Theoremseqf1olem2a 11361* Lemma for seqf1o 11364. (Contributed by Mario Carneiro, 24-Apr-2016.)

Theoremseqf1olem1 11362* Lemma for seqf1o 11364. (Contributed by Mario Carneiro, 26-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqf1olem2 11363* Lemma for seqf1o 11364. (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)

Theoremseqf1o 11364* Rearrange a sum via an arbitrary bijection on . (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)

Theoremseradd 11365* The sum of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 26-May-2014.)

Theoremsersub 11366* The difference of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqid3 11367* A sequence that consists entirely of zeroes (or whatever the identity is for operation ) sums to zero. (Contributed by Mario Carneiro, 15-Dec-2014.)

Theoremseqid 11368* Discard the first few terms of a sequence that starts with all zeroes (or whatever the identity is for operation ). (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqid2 11369* The last few terms of a sequence that ends with all zeroes (or whatever the identity is for operation ) are all the same. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqhomo 11370* Apply a homomorphism to a sequence. (Contributed by Mario Carneiro, 28-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremseqz 11371* If the operation has an absorbing element (a.k.a. zero element), then any sequence containing a evaluates to . (Contributed by Mario Carneiro, 27-May-2014.)

Theoremseqfeq4 11372* Equality of series under different addition operations which agree on an additively closed subset. (Contributed by Mario Carneiro, 25-Apr-2016.)

Theoremseqfeq3 11373* Equality of series under different addition operations which agree on an additively closed subset. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 25-Apr-2016.)

Theoremseqdistr 11374* The distributive property for series. (Contributed by Mario Carneiro, 28-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremser0 11375 The value of the partial sums in a zero-valued infinite series. (Contributed by Mario Carneiro, 31-Aug-2013.) (Revised by Mario Carneiro, 15-Dec-2014.)

Theoremser0f 11376 A zero-valued infinite series is equal to the constant zero function. (Contributed by Mario Carneiro, 8-Feb-2014.)

Theoremserge0 11377* A finite sum of nonnegative terms is nonnegative. (Contributed by Mario Carneiro, 8-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremserle 11378* Comparison of partial sums of two infinite series of reals. (Contributed by NM, 27-Dec-2005.) (Revised by Mario Carneiro, 27-May-2014.)

Theoremser1const 11379 Value of the partial series sum of a constant function. (Contributed by NM, 8-Aug-2005.) (Revised by Mario Carneiro, 16-Feb-2014.)

Theoremseqof 11380* Distribute function operation through a sequence. Note that is an implicit function on . (Contributed by Mario Carneiro, 3-Mar-2015.)

Theoremseqof2 11381* Distribute function operation through a sequence. Maps-to notation version of seqof 11380. (Contributed by Mario Carneiro, 7-Jul-2017.)

5.6.4  Integer powers

Syntaxcexp 11382 Extend class notation to include exponentiation of a complex number to an integer power.

Definitiondf-exp 11383* Define exponentiation to nonnegative integer powers. This definition is not meant to be used directly; instead, exp0 11386 and expp1 11388 provide the standard recursive definition. The up-arrow notation is used by Donald Knuth for iterated exponentiation (Science 194, 1235-1242, 1976) and is convenient for us since we don't have superscripts. 10-Jun-2005: The definition was extended to include zero exponents, so that per the convention of Definition 10-4.1 of [Gleason] p. 134. 4-Jun-2014: The definition was extended to include negative integer exponents. The case gives the value , so we will avoid this case in our theorems. (Contributed by Raph Levien, 20-May-2004.) (Revised by NM, 15-Oct-2004.)

Theoremexpval 11384 Value of exponentiation to integer powers. (Contributed by NM, 20-May-2004.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremexpnnval 11385 Value of exponentiation to positive integer powers. (Contributed by Mario Carneiro, 4-Jun-2014.)

Theoremexp0 11386 Value of a complex number raised to the 0th power. Note that under our definition, , following the convention used by Gleason. Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by NM, 20-May-2004.) (Revised by Mario Carneiro, 4-Jun-2014.)

Theoremexp1 11387 Value of a complex number raised to the first power. (Contributed by NM, 20-Oct-2004.) (Revised by Mario Carneiro, 2-Jul-2013.)

Theoremexpp1 11388 Value of a complex number raised to a nonnegative integer power plus one. Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by NM, 20-May-2005.) (Revised by Mario Carneiro, 2-Jul-2013.)

Theoremexpneg 11389 Value of a complex number raised to a negative integer power. (Contributed by Mario Carneiro, 4-Jun-2014.)

Theoremexpneg2 11390 Value of a complex number raised to a negative integer power. (Contributed by Mario Carneiro, 4-Jun-2014.)

Theoremexpn1 11391 A number to the negative one power is the reciprocal. (Contributed by Mario Carneiro, 4-Jun-2014.)

Theoremexpcllem 11392* Lemma for proving nonnegative integer exponentiation closure laws. (Contributed by NM, 14-Dec-2005.)

Theoremexpcl2lem 11393* Lemma for proving integer exponentiation closure laws. (Contributed by Mario Carneiro, 4-Jun-2014.) (Revised by Mario Carneiro, 9-Sep-2014.)

Theoremnnexpcl 11394 Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.)

Theoremnn0expcl 11395 Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)

Theoremzexpcl 11396 Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)

Theoremqexpcl 11397 Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.)

Theoremreexpcl 11398 Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.)

Theoremexpcl 11399 Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.)

Theoremrpexpcl 11400 Closure law for exponentiation of positive reals. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 9-Sep-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 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 32701-32762
 Copyright terms: Public domain < Previous  Next >