Home Metamath Proof ExplorerTheorem List (p. 243 of 322) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-21498) Hilbert Space Explorer (21499-23021) Users' Mathboxes (23022-32154)

Theorem List for Metamath Proof Explorer - 24201-24300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremprednn 24201 The value of the predecessor class over the naturals. (Contributed by Scott Fenton, 6-Aug-2013.)

Theoremprednn0 24202 The value of the predecessor class over . (Contributed by Scott Fenton, 9-May-2014.)

Theorempredfz 24203 Calculate the predecessor of an integer under a finite set of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Proof shortened by Mario Carneiro, 3-May-2015.)

18.7.14  (Trans)finite Recursion Theorems

Theoremtfisg 24204* A closed form of tfis 4645. (Contributed by Scott Fenton, 8-Jun-2011.)

18.7.15  Well-founded induction

Theoremtz6.26 24205* All nonempty (possibly proper) subclasses of , which has a well-founded relation , have -minimal elements. Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremtz6.26i 24206* All nonempty (possibly proper) subclasses of , which has a well-founded relation , have -minimal elements. Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by Scott Fenton, 14-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremwfi 24207* The Principle of Well-Founded Induction. Theorem 6.27 of [TakeutiZaring] p. 32. This principle states that if is a subclass of a well-ordered class with the property that every element of whose inital segment is included in is itself equal to . (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremwfii 24208* The Principle of Well-Founded Induction. Theorem 6.27 of [TakeutiZaring] p. 32. This principle states that if is a subclass of a well-ordered class with the property that every element of whose inital segment is included in is itself equal to . (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremwfisg 24209* Well-Founded Induction Schema. If a property passes from all elements less than of a well founded class to itself (induction hypothesis), then the property holds for all elements of . (Contributed by Scott Fenton, 11-Feb-2011.)
Se

Theoremwfis 24210* Well-Founded Induction Schema. If all elements less than a given set of the well founded class have a property (induction hypothesis), then all elements of have that property. (Contributed by Scott Fenton, 29-Jan-2011.)
Se

Theoremwfis2fg 24211* Well-Founded Induction Schema, using implicit substitution. (Contributed by Scott Fenton, 11-Feb-2011.)
Se

Theoremwfis2f 24212* Well Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011.)
Se

Theoremwfis2g 24213* Well-Founded Induction Schema, using implicit substitution. (Contributed by Scott Fenton, 11-Feb-2011.)
Se

Theoremwfis2 24214* Well Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011.)
Se

Theoremwfis3 24215* Well Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011.)
Se

Theoremuzsinds 24216* Strong (or "total") induction principle over a set of upper integers. (Contributed by Scott Fenton, 16-May-2014.)

Theoremnnsinds 24217* Strong (or "total") induction principle over the naturals. (Contributed by Scott Fenton, 16-May-2014.)

Theoremnn0sinds 24218* Strong (or "total") induction principle over the non-negative integers. (Contributed by Scott Fenton, 16-May-2014.)

Theoremomsinds 24219* Strong (or "total") induction principle over the finite ordinals. (Contributed by Scott Fenton, 17-Jul-2015.)

18.7.16  Transitive closure under a relationship

Syntaxctrpred 24220 Define the transitive predecessor class as a class.

Definitiondf-trpred 24221* Define the transitive predecessors of a class under a relationship and a class . This class can be thought of as the "smallest" class containing all elements of that are linked to by a chain of relationships (see trpredtr 24233 and trpredmintr 24234). Definition based off of Lemma 4.2 of Don Monk's notes for Advanced Set Theory, which can be found at http://euclid.colorado.edu/~monkd/settheory (check The Internet Archive for it now as Prof. Monk appears to have rewritten his website). (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremdftrpred2 24222* A definition of the transitive predecessors of a class in terms of indexed union. (Contributed by Scott Fenton, 28-Apr-2012.)

Theoremtrpredeq1 24223 Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremtrpredeq2 24224 Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremtrpredeq3 24225 Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremtrpredeq1d 24226 Equality deduction for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremtrpredeq2d 24227 Equality deduction for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremtrpredeq3d 24228 Equality deduction for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)

Theoremeltrpred 24229* A class is a transitive predecessor iff it is in some value of the underlying function. This theorem is not really meant to be used directly: instead refer to trpredpred 24231 and trpredmintr 24234. (Contributed by Scott Fenton, 28-Apr-2012.)

Theoremtrpredlem1 24230* Technical lemma for transitive predecessors properties. All values of the transitive predecessors' underlying function are subsets of the base set. (Contributed by Scott Fenton, 28-Apr-2012.)

Theoremtrpredpred 24231 Assuming it exists, the predecessor class is a subset of the transitive predecessors. (Contributed by Scott Fenton, 18-Feb-2011.)

Theoremtrpredss 24232 The transitive predecessors form a subset of the base class. (Contributed by Scott Fenton, 20-Feb-2011.)

Theoremtrpredtr 24233 The transitive predecessors are transitive in and (Contributed by Scott Fenton, 20-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremtrpredmintr 24234* The transitive predecessors form the smallest class transitive in and . That is, if is another , transitive class containing , then (Contributed by Scott Fenton, 25-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremtrpredelss 24235 Given a transitive predecessor of , the transitive predecessors of are a subset of the transitive predecessors of . (Contributed by Scott Fenton, 25-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremdftrpred3g 24236* The transitive predecessors of are equal to the predecessors of together with their transitive predecessors. (Contributed by Scott Fenton, 26-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremdftrpred4g 24237* Another recursive expression for the transitive predecessors. (Contributed by Scott Fenton, 27-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremtrpredpo 24238 If partially orders , then the transitive predecessors are the same as the immediate predecessors . (Contributed by Scott Fenton, 28-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremtrpred0 24239 The class of transitive predecessors is empty when is empty. (Contributed by Scott Fenton, 30-Apr-2012.)

Theoremtrpredex 24240 The transitive predecessors of a relation form a set (NOTE: this is the first theorem in the transitive predecessor series that requires infinity). (Contributed by Scott Fenton, 18-Feb-2011.)

Theoremtrpredrec 24241* If is an , transitive predecessor, then it is either an immediate predecessor or there is a transitive predecessor between and (Contributed by Scott Fenton, 9-May-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

18.7.17  Founded Induction

Theoremfrmin 24242* Every (possibly proper) subclass of a class with a founded, set-like relation has a minimal element. Lemma 4.3 of Don Monk's notes for Advanced Set Theory, which can be found at http://euclid.colorado.edu/~monkd/settheory. This is a very strong generalization of tz6.26 24205 and tz7.5 4413. (Contributed by Scott Fenton, 4-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremfrind 24243* The principle of founded induction. Theorem 4.4 of Don Monk's notes (see frmin 24242). This principle states that if is a subclass of a founded class with the property that every element of whose initial segment is included in is is itself equal to . Compare wfi 24207 and tfi 4644, which are special cases of this theorem that do not require the axiom of infinity to prove. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremfrindi 24244* The principle of founded induction. Theorem 4.4 of Don Monk's notes (see frmin 24242). This principle states that if is a subclass of a founded class with the property that every element of whose initial segment is included in is is itself equal to . (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremfrinsg 24245* Founded Induction Schema. If a property passes from all elements less than of a founded class to itself (induction hypothesis), then the property holds for all elements of . (Contributed by Scott Fenton, 7-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremfrins 24246* Founded Induction Schema. If a property passes from all elements less than of a founded class to itself (induction hypothesis), then the property holds for all elements of . (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremfrins2fg 24247* Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 7-Feb-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
Se

Theoremfrins2f 24248* Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
Se

Theoremfrins2g 24249* Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 8-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremfrins2 24250* Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremfrins3 24251* Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

18.7.18  Ordering Ordinal Sequences

Theoremorderseqlem 24252* Lemma for poseq 24253 and soseq 24254. The function value of a sequene is either in or null. (Contributed by Scott Fenton, 8-Jun-2011.)

Theoremposeq 24253* A partial ordering of sequences of ordinals. (Contributed by Scott Fenton, 8-Jun-2011.)

Theoremsoseq 24254* A linear ordering of sequences of ordinals. (Contributed by Scott Fenton, 8-Jun-2011.)

18.7.19  Well-founded recursion

Theoremwfr3g 24255* Functions defined by well founded recursion are identical up to relation, domain, and characteristic function. (Contributed by Scott Fenton, 11-Feb-2011.)
Se

Theoremwfrlem1 24256* Lemma for well-founded recursion. The final item we are interested in is the union of acceptable functions . This lemma just changes bound variables for later use. (Contributed by Scott Fenton, 21-Apr-2011.)

Theoremwfrlem2 24257* Lemma for well-founded recursion. An acceptable function is a function. (Contributed by Scott Fenton, 21-Apr-2011.)

Theoremwfrlem3 24258* Lemma for well-founded recursion. An acceptable function's domain is a subset of . (Contributed by Scott Fenton, 21-Apr-2011.)

Theoremwfrlem4 24259* Lemma for well-founded recursion. Properties of the restriction of an acceptable function to the domain of another one. (Contributed by Scott Fenton, 21-Apr-2011.)

Theoremwfrlem5 24260* Lemma for well-founded recursion. The values of two acceptable functions agree within their domains. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremwfrlem6 24261* Lemma for well-founded recursion. The union of all acceptable functions is a relationship. (Contributed by Scott Fenton, 21-Apr-2011.)

Theoremwfrlem7 24262* Lemma for well-founded recursion. The domain of is a subclass of . (Contributed by Scott Fenton, 21-Apr-2011.)

Theoremwfrlem8 24263* Lemma for well-founded recursion. Compute the prececessor class for an minimal element of . (Contributed by Scott Fenton, 21-Apr-2011.)

Theoremwfrlem9 24264* Lemma for well-founded recursion. If , then its predecessor class is a subset of . (Contributed by Scott Fenton, 21-Apr-2011.)

Theoremwfrlem10 24265* Lemma for well-founded recursion. When is an minimal element of , then its predecessor class is equal to . (Contributed by Scott Fenton, 21-Apr-2011.)

Theoremwfrlem11 24266* Lemma for well-founded recursion. The union of all acceptable functions is a function. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremwfrlem12 24267* Lemma for well-founded recursion. Here, we compute the value of (the union of all acceptable functions). (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremwfrlem13 24268* Lemma for well-founded recursion. From here through wfrlem16 24271, we aim to prove that . We do this by supposing that there is an element of that is not in . We then define by extending with the appropriate value at . We then show that cannot be an minimal element of , meaning that must be empty, so . Here, we show that is a function extending the domain of by one. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremwfrlem14 24269* Lemma for well-founded recursion. Compute the value of . (Contributed by Scott Fenton, 21-Apr-2011.)
Se

Theoremwfrlem15 24270* Lemma for well-founded recursion. When is minimal, is an acceptable function. (Contributed by Scott Fenton, 21-Apr-2011.)
Se

Theoremwfrlem16 24271* Lemma for well-founded recursion. If is minimal in , then is acceptable and thus a subset of , but is bigger than . Thus, cannot be minimal, so must be empty, and (due to wfrlem7 24262), . (Contributed by Scott Fenton, 21-Apr-2011.)
Se

Theoremwfr1 24272* The Principle of Well-Founded Recursion, part 1 of 3. We start with an arbitrary function and a class of "acceptable" functions . Then, using a base class and a well-ordering of , we define a function . This function is said to be defined by "well-founded recursion." The purpose of these three theorems is to demonstrate the properties of . We begin by showing that is a function over . (Contributed by Scott Fenton, 22-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremwfr2 24273* The Principle of Well-Founded Recursion, part 2 of 3. Next, we show that the value of at any is recursively applied to all "previous" values of . (Contributed by Scott Fenton, 18-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremwfr2c 24274* Generalize wfr2 24273 to class arguments. (Contributed by Scott Fenton, 6-Aug-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremwfr3 24275* The principle of Well-Founded Recursion, part 3 of 3. Finally, we show that is unique. We do this by showing that any function with the same properties we proved of in wfr1 24272 and wfr2 24273 is identical to . (Contributed by Scott Fenton, 18-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

18.7.20  Transfinite recursion via Well-founded recursion

TheoremtfrALTlem 24276* Lemma for deriving transfinite recursion from well-founded recursion. (Contributed by Scott Fenton, 22-Apr-2011.)

Theoremtfr1ALT 24277* tfr1 6413 via well-founded recursion. (Contributed by Scott Fenton, 17-Aug-1994.) (Revised by Mario Carneiro, 26-Jun-2015.) (Proof modification is discouraged.)

Theoremtfr2ALT 24278* tfr2 6414 via well-founded recursion. (Contributed by Scott Fenton, 22-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Proof modification is discouraged.)

Theoremtfr3ALT 24279* tfr3 6415 via well-founded recursion. (Contributed by Scott Fenton, 22-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Proof modification is discouraged.)

18.7.21  Founded Recursion

Theoremfrr3g 24280* Functions defined by founded recursion are identical up to relation, domain, and characteristic function. General version of frr3 (Contributed by Scott Fenton, 10-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremfrrlem1 24281* Lemma for founded recursion. The final item we are interested in is the union of acceptable functions . This lemma just changes bound variables for later use. (Contributed by Paul Chapman, 21-Apr-2012.)

Theoremfrrlem2 24282* Lemma for founded recursion. An acceptable function is a function. (Contributed by Paul Chapman, 21-Apr-2012.)

Theoremfrrlem3 24283* Lemma for founded recursion. An acceptable function's domain is a subset of . (Contributed by Paul Chapman, 21-Apr-2012.)

Theoremfrrlem4 24284* Lemma for founded recursion. Properties of the restriction of an acceptable function to the domain of another acceptable function. (Contributed by Paul Chapman, 21-Apr-2012.)

Theoremfrrlem5 24285* Lemma for founded recursion. The values of two acceptable functions agree within their domains. (Contributed by Paul Chapman, 21-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
Se

Theoremfrrlem5b 24286* Lemma for founded recursion. The union of a subclass of is a relationship. (Contributed by Paul Chapman, 29-Apr-2012.)
Se

Theoremfrrlem5c 24287* Lemma for founded recursion. The union of a subclass of is a function. (Contributed by Paul Chapman, 29-Apr-2012.)
Se

Theoremfrrlem5d 24288* Lemma for founded recursion. The domain of the union of a subset of is a subset of . (Contributed by Paul Chapman, 29-Apr-2012.)
Se

Theoremfrrlem5e 24289* Lemma for founded recursion. The domain of the union of a subset of is closed under predecessors. (Contributed by Paul Chapman, 1-May-2012.)
Se

Theoremfrrlem6 24290* Lemma for founded recursion. The union of all acceptable functions is a relationship. (Contributed by Paul Chapman, 21-Apr-2012.)

Theoremfrrlem7 24291* Lemma for founded recursion. The domain of is a subclass of . (Contributed by Paul Chapman, 21-Apr-2012.)

Theoremfrrlem10 24292* Lemma for founded recursion. The union of all acceptable functions is a function. (Contributed by Paul Chapman, 21-Apr-2012.)
Se

Theoremfrrlem11 24293* Lemma for founded recursion. Here, we calculate the value of (the union of all acceptable functions). (Contributed by Paul Chapman, 21-Apr-2012.)
Se

18.7.22  Surreal Numbers

Syntaxcsur 24294 Declare the class of all surreal numbers (see df-no 24297).

Syntaxcslt 24295 Declare the less than relationship over surreal numbers (see df-slt 24298).

Syntaxcbday 24296 Declare the birthday function for surreal numbers (see df-bday 24299).

Definitiondf-no 24297* Define the class of surreal numbers. The surreal numbers are a proper class of numbers developed by John H. Conway and introduced by Donald Knuth in 1975. They form a proper class into which all ordered fields can be embedded. The approach we take to defining them was first introduced by Hary Goshnor, and is based on the conception of a "sign expansion" of a surreal number. We define the surreals as ordinal-indexed sequences of and , analagous to Goshnor's and .

After introducing this definition, we will abstract away from it using axioms that Norman Alling developed in "Foundations of Analysis over Surreal Number Fields." This is done in a effort to be agnostic towards the exact implementation of surreals. (Contributed by Scott Fenton, 9-Jun-2011.)

Definitiondf-slt 24298* Next, we introduce surreal less-than, a comparison relationship over the surreals by lexicographically ordering them. (Contributed by Scott Fenton, 9-Jun-2011.)

Definitiondf-bday 24299 Finally, we introduce the birthday function. This function maps each surreal to an ordinal. In our implementation, this is the domain of the sign function. The important properties of this function are established later. (Contributed by Scott Fenton, 11-Jun-2011.)

Theoremelno 24300* Membership in the surreals. (Shortened proof on 2012-Apr-14, SF). (Contributed by Scott Fenton, 11-Jun-2011.)

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 32101-32154
 Copyright terms: Public domain < Previous  Next >