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

Theoremrege0subm 19101 The nonnegative reals form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 20-Jun-2015.)
SubMndfld

Theoremabsabv 19102 The regular absolute value function on the complex numbers is in fact an absolute value under our definition. (Contributed by Mario Carneiro, 4-Dec-2014.)
AbsValfld

Theoremzsssubrg 19103 The integers are a subset of any subring of the complex numbers. (Contributed by Mario Carneiro, 15-Oct-2015.)
SubRingfld

Theoremqsssubdrg 19104 The rational numbers are a subset of any subfield of the complex numbers. (Contributed by Mario Carneiro, 15-Oct-2015.)
SubRingfld flds

Theoremcnsubrg 19105 There are no subrings of the complex numbers strictly between and . (Contributed by Mario Carneiro, 15-Oct-2015.)
SubRingfld

Theoremcnmgpabl 19106 The unit group of the complex numbers is an abelian group. (Contributed by Mario Carneiro, 21-Jun-2015.)
mulGrpflds

Theoremcnmsubglem 19107* Lemma for rpmsubg 19108 and friends. (Contributed by Mario Carneiro, 21-Jun-2015.)
mulGrpflds                                           SubGrp

Theoremrpmsubg 19108 The positive reals form a multiplicative subgroup of the complex numbers. (Contributed by Mario Carneiro, 21-Jun-2015.)
mulGrpflds        SubGrp

Theoremgzrngunitlem 19109 Lemma for gzrngunit 19110. (Contributed by Mario Carneiro, 4-Dec-2014.)
flds        Unit

Theoremgzrngunit 19110 The units on are the gaussian integers with norm . (Contributed by Mario Carneiro, 4-Dec-2014.)
flds        Unit

Theoremgsumfsum 19111* Relate a group sum on ℂfld to a finite sum on the complex numbers. (Contributed by Mario Carneiro, 28-Dec-2014.)
fld g

Theoremregsumfsum 19112* Relate a group sum on flds to a finite sum on the reals. Cf. gsumfsum 19111. (Contributed by Thierry Arnoux, 7-Sep-2018.)
flds g

Theoremexpmhm 19113* Exponentiation is a monoid homomorphism from addition to multiplication. (Contributed by Mario Carneiro, 18-Jun-2015.)
flds        mulGrpfld       MndHom

Theoremnn0srg 19114 The nonnegative integers form a semiring (commutative by subcmn 17555). (Contributed by Thierry Arnoux, 1-May-2018.)
flds SRing

Theoremrge0srg 19115 The nonnegative real numbers form a semiring (commutative by subcmn 17555). (Contributed by Thierry Arnoux, 6-Sep-2018.)
flds SRing

10.11.2  Ring of integers

According to Wikipedia ("Integer", 25-May-2019, https://en.wikipedia.org/wiki/Integer) "The integers form a unital ring which is the most basic one, in the following sense: for any unital ring, there is a unique ring homomorphism from the integers into this ring. This universal property, namely to be an initial object in the category of [unital] rings, characterizes the ring ." In set.mm, there was no explicit definition for the ring of integers until June 2019, but it was denoted by flds , the field of complex numbers restricted to the integers. In zringring 19119 it is shown that this restriction is a ring (it is actually a principal ideal ring as shown in zringlpir 19135), and zringbas 19122 shows that its base set is the integers. As of June 2019, there is an abbreviation of this expression as definition df-zring 19117 of the ring of integers.

Remark: Instead of using the symbol "ZZrng" analogous to ℂfld used for the field of complex numbers, we have chosen the version with an "i" to indicate that the ring of integers is a unital ring, see also Wikipedia ("Rng (algebra)", 9-Jun-2019, https://en.wikipedia.org/wiki/Rng_(algebra)).

Syntaxzring 19116 Extend class notation with the (unital) ring of integers.
ring

Definitiondf-zring 19117 The (unital) ring of integers. (Contributed by Alexander van der Vekens, 9-Jun-2019.)
ring flds

Theoremzringcrng 19118 The ring of integers is a commutative ring. (Contributed by AV, 13-Jun-2019.)
ring

Theoremzringring 19119 The ring of integers is a ring. (Contributed by AV, 20-May-2019.) (Revised by AV, 9-Jun-2019.) (Proof shortened by AV, 13-Jun-2019.)
ring

Theoremzringabl 19120 The ring of integers is an (additive) abelian group. (Contributed by AV, 13-Jun-2019.)
ring

Theoremzringgrp 19121 The ring of integers is an (additive) group. (Contributed by AV, 10-Jun-2019.)
ring

Theoremzringbas 19122 The integers are the base of the ring of integers. (Contributed by Thierry Arnoux, 31-Oct-2017.) (Revised by AV, 9-Jun-2019.)
ring

Theoremzringplusg 19123 The addition operation of the ring of integers. (Contributed by Thierry Arnoux, 8-Nov-2017.) (Revised by AV, 9-Jun-2019.)
ring

Theoremzringmulg 19124 The multiplication (group power) operation of the group of integers. (Contributed by Thierry Arnoux, 31-Oct-2017.) (Revised by AV, 9-Jun-2019.)
.gring

Theoremzringmulr 19125 The multiplication operation of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.)
ring

Theoremzring0 19126 The neutral element of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.)
ring

Theoremzring1 19127 The multiplicative neutral element of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.)
ring

Theoremzringnzr 19128 The ring of integers is a nonzero ring. (Contributed by AV, 18-Apr-2020.)
ring NzRing

Theoremdvdsrzring 19129 Ring divisibility in the ring of integers corresponds to ordinary divisibility in . (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.)
rring

Theoremzringlpirlem1 19130 Lemma for zringlpir 19135. A nonzero ideal of integers contains some positive integers. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.)
LIdealring

Theoremzringlpirlem2OLD 19131 Lemma for zringlpir 19135. A nonzero ideal of integers contains the least positive element. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.) Obsolete version of zringlpirlem2 19133 as of 27-Sep-2020. (Proof modification is discouraged.) (New usage is discouraged.)
LIdealring

Theoremzringlpirlem3OLD 19132 Lemma for zringlpir 19135. All elements of a nonzero ideal of integers are divided by the least one. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.) Obsolete version of zringlpirlem3 19134 as of 27-Sep-2020. (Proof modification is discouraged.) (New usage is discouraged.)
LIdealring

Theoremzringlpirlem2 19133 Lemma for zringlpir 19135. A nonzero ideal of integers contains the least positive element. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.) (Revised by AV, 27-Sep-2020.)
LIdealring              inf

Theoremzringlpirlem3 19134 Lemma for zringlpir 19135. All elements of a nonzero ideal of integers are divided by the least one. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.) (Proof shortened by AV, 27-Sep-2020.)
LIdealring              inf

Theoremzringlpir 19135 The integers are a principal ideal ring but not a field. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.) (Proof shortened by AV, 27-Sep-2020.)
ring LPIR

TheoremzringlpirOLD 19136 The integers are a principal ideal ring but not a field. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.) Obsolete version of zringlpir 19135 as of 27-Sep-2020. (Proof modification is discouraged.) (New usage is discouraged.)
ring LPIR

Theoremzringcyg 19137 The integers are a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 9-Jun-2019.)
ring CycGrp

Theoremzringinvg 19138 The additive inverse of an element of the ring of integers. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
ring

Theoremzringunit 19139 The units of are the integers with norm , i.e. and . (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by AV, 10-Jun-2019.)
Unitring

Theoremzringmpg 19140 The multiplication group of the ring of integers is the restriction of the multiplication group of the complex numbers to the integers. (Contributed by AV, 15-Jun-2019.)
mulGrpflds mulGrpring

Theoremprmirredlem 19141 A positive integer is irreducible over iff it is a prime number. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by AV, 10-Jun-2019.)
Irredring

Theoremdfprm2 19142 The positive irreducible elements of are the prime numbers. This is an alternative way to define . (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by AV, 10-Jun-2019.)
Irredring

Theoremprmirred 19143 The irreducible elements of are exactly the prime numbers (and their negatives). (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by AV, 10-Jun-2019.)
Irredring

Theoremexpghm 19144* Exponentiation is a group homomorphism from addition to multiplication. (Contributed by Mario Carneiro, 18-Jun-2015.) (Revised by AV, 10-Jun-2019.)
mulGrpfld       s        ring

Theoremmulgghm2 19145* The powers of a group element give a homomorphism from to a group. (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.)
.g                     ring

Theoremmulgrhm 19146* The powers of the element give a ring homomorphism from to a ring. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 12-Jun-2019.)
.g                     ring RingHom

Theoremmulgrhm2 19147* The powers of the element give the unique ring homomorphism from to a ring. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 12-Jun-2019.)
.g                     ring RingHom

10.11.3  Algebraic constructions based on the complex numbers

Syntaxczrh 19148 Map the rationals into a field, or the integers into a ring.
RHom

Syntaxczlm 19149 Augment an abelian group with vector space operations to turn it into a -module.
Mod

Syntaxcchr 19150 Syntax for ring characteristic.
chr

Syntaxczn 19151 The ring of integers modulo .
ℤ/n

Definitiondf-zrh 19152 Define the unique homomorphism from the integers into a ring. This encodes the usual notation of for integers (see also df-mulg 16754). (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.)
RHom ring RingHom

Definitiondf-zlm 19153 Augment an abelian group with vector space operations to turn it into a -module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.)
Mod sSet Scalarring sSet .g

Definitiondf-chr 19154 The characteristic of a ring is the smallest positive integer which is equal to 0 when interpreted in the ring, or 0 if there is no such positive integer. (Contributed by Stefan O'Rear, 5-Sep-2015.)
chr

Definitiondf-zn 19155* Define the ring of integers . This is literally the quotient ring of by the ideal , but we augment it with a total order. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 12-Jun-2019.)
ℤ/n ring s ~QG RSpan sSet RHom ..^

Theoremzrhval 19156 Define the unique homomorphism from the integers to a ring or field. (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.)
RHom       ring RingHom

Theoremzrhval2 19157* Alternate value of the RHom homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
RHom       .g

Theoremzrhmulg 19158 Value of the RHom homomorphism. (Contributed by Mario Carneiro, 14-Jun-2015.)
RHom       .g

Theoremzrhrhmb 19159 The RHom homomorphism is the unique ring homomorphism from . (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 12-Jun-2019.)
RHom       ring RingHom

Theoremzrhrhm 19160 The RHom homomorphism is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.) (Revised by AV, 12-Jun-2019.)
RHom       ring RingHom

Theoremzrh1 19161 Interpretation of 1 in a ring. (Contributed by Stefan O'Rear, 6-Sep-2015.)
RHom

Theoremzrh0 19162 Interpretation of 0 in a ring. (Contributed by Stefan O'Rear, 6-Sep-2015.)
RHom

Theoremzrhpropd 19163* The ring homomorphism depends only on the ring attributes of a structure. (Contributed by Mario Carneiro, 15-Jun-2015.)
RHom RHom

Theoremzlmval 19164 Augment an abelian group with vector space operations to turn it into a -module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.)
Mod       .g       sSet Scalarring sSet

Theoremzlmlem 19165 Lemma for zlmbas 19166 and zlmplusg 19167. (Contributed by Mario Carneiro, 2-Oct-2015.)
Mod       Slot

Theoremzlmbas 19166 Base set of a -module. (Contributed by Mario Carneiro, 2-Oct-2015.)
Mod

Theoremzlmplusg 19167 Group operation of a -module. (Contributed by Mario Carneiro, 2-Oct-2015.)
Mod

Theoremzlmmulr 19168 Ring operation of a -module (if present). (Contributed by Mario Carneiro, 2-Oct-2015.)
Mod

Theoremzlmsca 19169 Scalar ring of a -module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.)
Mod       ring Scalar

Theoremzlmvsca 19170 Scalar multiplication operation of a -module. (Contributed by Mario Carneiro, 2-Oct-2015.)
Mod       .g

Theoremzlmlmod 19171 The -module operation turns an arbitrary abelian group into a left module over . (Contributed by Mario Carneiro, 2-Oct-2015.)
Mod

Theoremzlmassa 19172 The -module operation turns a ring into an associative algebra over . (Contributed by Mario Carneiro, 2-Oct-2015.)
Mod       AssAlg

Theoremchrval 19173 Definition substitution of the ring characteristic. (Contributed by Stefan O'Rear, 5-Sep-2015.)
chr

Theoremchrcl 19174 Closure of the characteristic. (Contributed by Mario Carneiro, 23-Sep-2015.)
chr

Theoremchrid 19175 The canonical ring homomorphism applied to a ring's characteristic is zero. (Contributed by Mario Carneiro, 23-Sep-2015.)
chr       RHom

Theoremchrdvds 19176 The ring homomorphism is zero only at multiples of the characteristic. (Contributed by Mario Carneiro, 23-Sep-2015.)
chr       RHom

Theoremchrcong 19177 If two integers are congruent relative to the ring characteristic, their images in the ring are the same. (Contributed by Mario Carneiro, 24-Sep-2015.)
chr       RHom

Theoremchrnzr 19178 Nonzero rings are precisely those with characteristic not 1. (Contributed by Stefan O'Rear, 6-Sep-2015.)
NzRing chr

Theoremchrrhm 19179 The characteristic restriction on ring homomorphisms. (Contributed by Stefan O'Rear, 6-Sep-2015.)
RingHom chr chr

Theoremdomnchr 19180 The characteristic of a domain can only be zero or a prime. (Contributed by Stefan O'Rear, 6-Sep-2015.)
Domn chr chr

Theoremznlidl 19181 The set is an ideal in . (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.)
RSpanring       LIdealring

Theoremzncrng2 19182 The value of the ℤ/nℤ structure. It is defined as the quotient ring , with an "artificial" ordering added to make it a Toset. (In other words, ℤ/nℤ is a ring with an order , but it is not an ordered ring , which as a term implies that the order is compatible with the ring operations in some way.) (Contributed by Mario Carneiro, 12-Jun-2015.) (Revised by AV, 13-Jun-2019.)
RSpanring       ring s ring ~QG

Theoremznval 19183 The value of the ℤ/nℤ structure. It is defined as the quotient ring , with an "artificial" ordering added to make it a Toset. (In other words, ℤ/nℤ is a ring with an order , but it is not an ordered ring , which as a term implies that the order is compatible with the ring operations in some way.) (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 2-May-2016.) (Revised by AV, 13-Jun-2019.)
RSpanring       ring s ring ~QG        ℤ/n       RHom        ..^              sSet

Theoremznle 19184 The value of the ℤ/nℤ structure. It is defined as the quotient ring , with an "artificial" ordering added to make it a Toset. (In other words, ℤ/nℤ is a ring with an order , but it is not an ordered ring , which as a term implies that the order is compatible with the ring operations in some way.) (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.)
RSpanring       ring s ring ~QG        ℤ/n       RHom        ..^

Theoremznval2 19185 Self-referential expression for the ℤ/nℤ structure. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.)
RSpanring       ring s ring ~QG        ℤ/n              sSet

Theoremznbaslem 19186 Lemma for znbas 19191. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 13-Jun-2019.)
RSpanring       ring s ring ~QG        ℤ/n       Slot

Theoremznbas2 19187 The base set of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.)
RSpanring       ring s ring ~QG        ℤ/n

Theoremznadd 19188 The additive structure of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.)
RSpanring       ring s ring ~QG        ℤ/n

Theoremznmul 19189 The multiplicative structure of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.)
RSpanring       ring s ring ~QG        ℤ/n

Theoremznzrh 19190 The ring homomorphism of ℤ/nℤ is inherited from the quotient ring it is based on. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.)
RSpanring       ring s ring ~QG        ℤ/n       RHom RHom

Theoremznbas 19191 The base set of ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.)
RSpanring       ℤ/n       ring ~QG

Theoremzncrng 19192 ℤ/nℤ is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n

Theoremznzrh2 19193* The ring homomorphism maps elements to their equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.)
RSpanring       ring ~QG        ℤ/n       RHom

Theoremznzrhval 19194 The ring homomorphism maps elements to their equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.)
RSpanring       ring ~QG        ℤ/n       RHom

Theoremznzrhfo 19195 The ring homomorphism is a surjection onto . (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n              RHom

Theoremzncyg 19196 The group is cyclic for all (including ). (Contributed by Mario Carneiro, 21-Apr-2016.)
ℤ/n       CycGrp

Theoremzndvds 19197 Express equality of equivalence classes in in terms of divisibility. (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n       RHom

Theoremzndvds0 19198 Special case of zndvds 19197 when one argument is zero. (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n       RHom

Theoremznf1o 19199 The function enumerates all equivalence classes in ℤ/nℤ for each . When , so we let ; otherwise enumerates all the equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by Mario Carneiro, 2-May-2016.) (Revised by AV, 13-Jun-2019.)
ℤ/n              RHom        ..^

Theoremzzngim 19200 The ring homomorphism is an isomorphism for . (We only show group isomorphism here, but ring isomorphism follows, since it is a bijective ring homomorphism.) (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 13-Jun-2019.)
ℤ/n       RHom       ring GrpIso

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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 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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