Home Metamath Proof ExplorerTheorem List (p. 64 of 314) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-21444) Hilbert Space Explorer (21445-22967) Users' Mathboxes (22968-31305)

Theorem List for Metamath Proof Explorer - 6301-6400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremriotasvd 6301* Deduction version of riotasv 6306. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

TheoremriotasvdOLD 6302* Deduction version of riotasv 6306. (Contributed by NM, 1-Feb-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremriotasv2d 6303* Value of description binder for a single-valued class expression (as in e.g. reusv2 4498). Special case of riota2f 6280. (Contributed by NM, 2-Mar-2013.)

Theoremriotasv2dOLD 6304* Value of description binder for a single-valued class expression (as in e.g. reusv2 4498). Special case of riota2f 6280. (Contributed by NM, 1-Feb-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremriotasv2s 6305* The value of description binder for a single-valued class expression (as in e.g. reusv2 4498) in the form of a substitution instance. Special case of riota2f 6280. (Contributed by NM, 3-Mar-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)

Theoremriotasv 6306* Value of description binder for a single-valued class expression (as in e.g. reusv2 4498). Special case of riota2f 6280. (Contributed by NM, 26-Jan-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)

Theoremriotasv3d 6307* A property holding for a representative of a single-valued class expression (see e.g. reusv2 4498) also holds for its description binder (in the form of property ). (Contributed by NM, 5-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremriotasv3dOLD 6308* A property holding for a representative of a single-valued class expression (see e.g. reusv2 4498) also holds for its description binder (in the form of property ). (Contributed by NM, 1-Feb-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

2.4.18  Functions on ordinals; strictly monotone ordinal functions

Theoremiunon 6309* The indexed union of a set of ordinal numbers is an ordinal number. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 5-Dec-2016.)

TheoremiunonOLD 6310* The indexed union of a set of ordinal numbers is an ordinal number. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 5-Dec-2016.)

Theoremiinon 6311* The nonempty indexed intersection of a class of ordinal numbers is an ordinal number. (Contributed by NM, 13-Oct-2003.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)

Theoremonfununi 6312* A property of functions on ordinal numbers. Generalization of Theorem Schema 8E of [Enderton] p. 218. (Contributed by Eric Schmidt, 26-May-2009.)

Theoremonovuni 6313* A variant of onfununi 6312 for operations. (Contributed by Eric Schmidt, 26-May-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)

Theoremonoviun 6314* A variant of onovuni 6313 with indexed unions. (Contributed by Eric Schmidt, 26-May-2009.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)

Theoremonnseq 6315* There are no length decreasing sequences in the ordinals. See also noinfep 7314 for a stronger version assuming Regularity. (Contributed by Mario Carneiro, 19-May-2015.)

Syntaxwsmo 6316 Introduce the strictly monotone ordinal function. A strictly monotone function is one that is constantly increasing across the ordinals.

Definitiondf-smo 6317* Definition of a strictly monotone ordinal function. Definition 7.46 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 15-Nov-2011.)

Theoremdfsmo2 6318* Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 4-Mar-2013.)

Theoremissmo 6319* Conditions for which is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.)

Theoremissmo2 6320* Alternative definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 12-Mar-2013.)

Theoremsmoeq 6321 Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.)

Theoremsmodm 6322 The domain of a strictly monotone function is an ordinal. (Contributed by Andrew Salmon, 16-Nov-2011.)

Theoremsmores 6323 A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 16-Nov-2011.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)

Theoremsmores3 6324 A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 19-Nov-2011.)

Theoremsmores2 6325 A strictly monotone ordinal function restricted to an ordinal is still monotone. (Contributed by Mario Carneiro, 15-Mar-2013.)

Theoremsmodm2 6326 The domain of a strictly monotone ordinal function is an ordinal. (Contributed by Mario Carneiro, 12-Mar-2013.)

Theoremsmofvon2 6327 The function values of a strictly monotone ordinal function are ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.)

Theoremiordsmo 6328 The identity relation restricted to the ordinals is a strictly monotone function. (Contributed by Andrew Salmon, 16-Nov-2011.)

Theoremsmo0 6329 The null set is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 20-Nov-2011.)

Theoremsmofvon 6330 If is a strictly monotone ordinal function, and is in the domain of , then the value of the function at is an ordinal. (Contributed by Andrew Salmon, 20-Nov-2011.)

Theoremsmoel 6331 If is less than then a strictly monotone function's value will be strictly less at than at . (Contributed by Andrew Salmon, 22-Nov-2011.)

Theoremsmoiun 6332* The value of a strictly monotone ordinal function contains its indexed union. (Contributed by Andrew Salmon, 22-Nov-2011.)

Theoremsmoiso 6333 If is an isomorphism from an ordinal onto , which is a subset of the ordinals, then is a strictly monotonic function. Exercise 3 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 24-Nov-2011.)

Theoremsmoel2 6334 A strictly monotone ordinal function preserves the epsilon relation. (Contributed by Mario Carneiro, 12-Mar-2013.)

Theoremsmo11 6335 A strictly monotone ordinal function is one-to-one. (Contributed by Mario Carneiro, 28-Feb-2013.)

Theoremsmoord 6336 A strictly monotone ordinal function preserves strict ordering. (Contributed by Mario Carneiro, 4-Mar-2013.)

Theoremsmoword 6337 A strictly monotone ordinal function preserves weak ordering. (Contributed by Mario Carneiro, 4-Mar-2013.)

Theoremsmogt 6338 A strictly monotone ordinal function is greater than or equal to its argument. Exercise 1 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 23-Nov-2011.) (Revised by Mario Carneiro, 28-Feb-2013.)

Theoremsmorndom 6339 The range of a strictly monotone ordinal function dominates the domain. (Contributed by Mario Carneiro, 13-Mar-2013.)

Theoremsmoiso2 6340 The strictly monotone ordinal functions are also epsilon isomorphisms of subclasses of . (Contributed by Mario Carneiro, 20-Mar-2013.)

2.4.19  "Strong" transfinite recursion

Syntaxcrecs 6341 Notation for a function defined by strong transfinite recursion.
recs

Definitiondf-recs 6342* Define a function recs on , the class of ordinal numbers, by transfinite recursion given a rule which sets the next value given all values so far. See df-rdg 6377 for more details on why this definition is desirable. Unlike df-rdg 6377 which restricts the update rule to use only the previous value, this version allows the update rule to use all previous values, which is why it is described as "strong", although it is actually more primitive. See recsfnon 6370 and recsval 6371 for the primary contract of this definition.

EDITORIAL: there are several existing versions of this construction without the definition, notably in ordtype 7201, zorn2 8087, and dfac8alem 7610. (Contributed by Stefan O'Rear, 18-Jan-2015.) (New usage is discouraged.)

recs

Theoremrecseq 6343 Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs recs

Theoremnfrecs 6344 Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs

Theoremtfrlem1 6345* A technical lemma for transfinite recursion. Compare Lemma 1 of [TakeutiZaring] p. 47. (Contributed by NM, 23-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremtfrlem2 6346* Lemma for transfinite recursion. This provides some messy details needed to link tfrlem1 6345 into the main proof. (Contributed by NM, 23-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremtfrlem3 6347* Lemma for transfinite recursion. Let be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in for later use. (Contributed by NM, 9-Apr-1995.)

Theoremtfrlem3a 6348* Lemma for transfinite recursion. Let be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in for later use. (Contributed by NM, 22-Jul-2012.)

Theoremtfrlem4 6349* Lemma for transfinite recursion. is the class of all "acceptable" functions, and is their union. First we show that an acceptable function is in fact a function. (Contributed by NM, 9-Apr-1995.)

Theoremtfrlem5 6350* Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains. (Contributed by NM, 9-Apr-1995.)

Theoremrecsfval 6351* Lemma for transfinite recursion. The definition recs is the union of all acceptable functions. (Contributed by Mario Carneiro, 9-May-2015.)
recs

Theoremtfrlem6 6352* Lemma for transfinite recursion. The union of all acceptable functions is a relation. (Contributed by NM, 8-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
recs

Theoremtfrlem7 6353* Lemma for transfinite recursion. The union of all acceptable functions is a function. (Contributed by NM, 9-Aug-1994.)
recs

Theoremtfrlem8 6354* Lemma for transfinite recursion. The domain of recs is ordinal. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Alan Sare, 11-Mar-2008.)
recs

Theoremtfrlem9 6355* Lemma for transfinite recursion. Here we compute the value of recs (the union of all acceptable functions). (Contributed by NM, 17-Aug-1994.)
recs recs recs

Theoremtfrlem9a 6356* Lemma for transfinite recursion. Without using ax-rep 4091, show that all the restrictions of recs are sets. (Contributed by Mario Carneiro, 16-Nov-2014.)
recs recs

Theoremtfrlem10 6357* Lemma for transfinite recursion. We define class by extending recs with one ordered pair. We will assume, falsely, that domain of recs is a member of, and thus not equal to, . Using this assumption we will prove facts about that will lead to a contradiction in tfrlem14 6361, thus showing the domain of recs does in fact equal . Here we show (under the false assumption) that is a function extending the domain of recs by one. (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
recs recs recs       recs recs

Theoremtfrlem11 6358* Lemma for transfinite recursion. Compute the value of . (Contributed by NM, 18-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
recs recs recs       recs recs

Theoremtfrlem12 6359* Lemma for transfinite recursion. Show is an acceptable function. (Contributed by NM, 15-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
recs recs recs       recs

Theoremtfrlem13 6360* Lemma for transfinite recursion. If recs is a set function, then is acceptable, and thus a subset of recs, but is bigger than recs. This is a contradiction, so recs must be a proper class function. (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2014.)
recs

Theoremtfrlem14 6361* Lemma for transfinite recursion. Assuming ax-rep 4091, recs recs , so since recs is an ordinal, it must be equal to . (Contributed by NM, 14-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
recs

Theoremtfrlem15 6362* Lemma for transfinite recursion. Without assuming ax-rep 4091, we can show that all proper initial subsets of recs are sets, while nothing larger is a set. (Contributed by Mario Carneiro, 14-Nov-2014.)
recs recs

Theoremtfrlem16 6363* Lemma for finite recursion. Without assuming ax-rep 4091, we can show that the domain of the constructed function is a limit ordinal, and hence contains all the finite ordinals. (Contributed by Mario Carneiro, 14-Nov-2014.)
recs

Theoremtfr1a 6364 A weak version of tfr1 6367 which is useful for proofs that avoid the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
recs

Theoremtfr2a 6365 A weak version of tfr2 6368 which is useful for proofs that avoid the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
recs

Theoremtfr2b 6366 Without assuming ax-rep 4091, we can show that all proper initial subsets of recs are sets, while nothing larger is a set. (Contributed by Mario Carneiro, 24-Jun-2015.)
recs

Theoremtfr1 6367 Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of [TakeutiZaring] p. 47. We start with an arbitrary class , normally a function, and define a class of all "acceptable" functions. The final function we're interested in is the union recs of them. is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of . In this first part we show that is a function whose domain is all ordinal numbers. (Contributed by NM, 17-Aug-1994.) (Revised by Mario Carneiro, 18-Jan-2015.)
recs

Theoremtfr2 6368 Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47. Here we show that the function has the property that for any function whatsoever, the "next" value of is recursively applied to all "previous" values of . (Contributed by NM, 9-Apr-1995.) (Revised by Stefan O'Rear, 18-Jan-2015.)
recs

Theoremtfr3 6369* Principle of Transfinite Recursion, part 3 of 3. Theorem 7.41(3) of [TakeutiZaring] p. 47. Finally we show that is unique. We do this by showing that any class with the same properties of that we showed in parts 1 and 2 is identical to . (Contributed by NM, 18-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.)
recs

Theoremrecsfnon 6370 Strong transfinite recursion defines a function on ordinals. (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs

Theoremrecsval 6371 Strong transfinite recursion in terms of all previous values. (Contributed by Stefan O'Rear, 18-Jan-2015.)
recs recs

Theoremtz7.44lem1 6372* is a function. Lemma for tz7.44-1 6373, tz7.44-2 6374, and tz7.44-3 6375. (Contributed by NM, 23-Apr-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremtz7.44-1 6373* The value of at . Part 1 of Theorem 7.44 of [TakeutiZaring] p. 49. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Theoremtz7.44-2 6374* The value of at a successor ordinal. Part 2 of Theorem 7.44 of [TakeutiZaring] p. 49. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Theoremtz7.44-3 6375* The value of at a limit ordinal. Part 3 of Theorem 7.44 of [TakeutiZaring] p. 49. (Contributed by NM, 23-Apr-1995.) (Revised by David Abernethy, 19-Jun-2012.)

2.4.20  Recursive definition generator

Syntaxcrdg 6376 Extend class notation with the recursive definition generator, with characteristic function and initial value .

Definitiondf-rdg 6377* Define a recursive definition generator on (the class of ordinal numbers) with characteristic function and initial value . This combines functions in tfr1 6367 and in tz7.44-1 6373 into one definition. This rather amazing operation allows us to define, with compact direct definitions, functions that are usually defined in textbooks only with indirect self-referencing recursive definitions. A recursive definition requires advanced metalogic to justify - in particular, eliminating a recursive definition is very difficult and often not even shown in textbooks. On the other hand, the elimination of a direct definition is a matter of simple mechanical substitution. The price paid is the daunting complexity of our operation. But once we get past this hurdle, otherwise recursive definitions become relatively simple, as in for example oav 6464, from which we prove the recursive textbook definition as theorems oa0 6469, oasuc 6477, and oalim 6485 (with the help of theorems rdg0 6388, rdgsuc 6391, and rdglim2a 6400). We can also restrict the operation to define otherwise recursive functions on the natural numbers ; see fr0g 6402 and frsuc 6403. Our operation apparently does not appear in published literature, although closely related is Definition 25.2 of [Quine] p. 177, which he uses to "turn...a recursion into a genuine or direct definition" (p. 174). Note that the operations (see df-if 3526) select cases based on whether the domain of is zero, a successor, or a limit ordinal.

An important use of this definition is in the recursive sequence generator df-seq 10999 on the natural numbers (as a subset of the complex numbers), allowing us to define, with direct definitions, recursive infinite sequences such as the factorial function df-fac 11241 and integer powers df-exp 11057.

Note: We introduce with the philosophical goal of being able to eliminate all definitions with direct mechanical substitution and to verify easily the soundness of definitions. Metamath itself has no built-in technical limitation that prevents multiple-part recursive definitions in the traditional textbook style. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)

recs

Theoremrdgeq1 6378 Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)

Theoremrdgeq2 6379 Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)

Theoremrdgeq12 6380 Equality theorem for the recursive definition generator. (Contributed by Scott Fenton, 28-Apr-2012.)

Theoremnfrdg 6381 Bound-variable hypothesis builder for the recursive definition generator. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremrdglem1 6382* Lemma used with the recursive definition generator. This is a trivial lemma that just changes bound variables for later use. (Contributed by NM, 9-Apr-1995.)

Theoremrdgfun 6383 The recursive definition generator is a function. (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremrdgdmlim 6384 The domain of the recursive definition generator is a limit ordinal. (Contributed by NM, 16-Nov-2014.)

Theoremrdgfnon 6385 The recursive definition generator is a function on ordinal numbers. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)

Theoremrdgvalg 6386* Value of the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremrdgval 6387* Value of the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)

Theoremrdg0 6388 The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Theoremrdgseg 6389 The initial segments of the recursive definition generator are sets. (Contributed by Mario Carneiro, 16-Nov-2014.)

Theoremrdgsucg 6390 The value of the recursive definition generator at a successor. (Contributed by NM, 16-Nov-2014.)

Theoremrdgsuc 6391 The value of the recursive definition generator at a successor. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Theoremrdglimg 6392 The value of the recursive definition generator at a limit ordinal. (Contributed by NM, 16-Nov-2014.)

Theoremrdglim 6393 The value of the recursive definition generator at a limit ordinal. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Theoremrdg0g 6394 The initial value of the recursive definition generator. (Contributed by NM, 25-Apr-1995.)

Theoremrdgsucmptf 6395 The value of the recursive definition generator at a successor (special case where the characteristic function uses the map operation). (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremrdgsucmptnf 6396 The value of the recursive definition generator at a successor (special case where the characteristic function is an ordered-pair class abstraction and where the mapping class is a proper class). This is a technical lemma that can be used together with rdgsucmptf 6395 to help eliminate redundant sethood antecedents. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremrdgsucmpt2 6397* This version of rdgsucmpt 6398 avoids the not-free hypothesis of rdgsucmptf 6395 by using two substitutions instead of one. (Contributed by Mario Carneiro, 11-Sep-2015.)

Theoremrdgsucmpt 6398* The value of the recursive definition generator at a successor (special case where the characteristic function uses the map operation). (Contributed by Mario Carneiro, 9-Sep-2013.)

Theoremrdglim2 6399* The value of the recursive definition generator at a limit ordinal, in terms of the union of all smaller values. (Contributed by NM, 23-Apr-1995.)

Theoremrdglim2a 6400* The value of the recursive definition generator at a limit ordinal, in terms of indexed union of all smaller values. (Contributed by NM, 28-Jun-1998.)

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 31301-31305
 Copyright terms: Public domain < Previous  Next >