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Theorem List for Metamath Proof Explorer - 35601-35700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremhlhilphllem 35601* Lemma for hlhil 22475. (Contributed by NM, 23-Jun-2015.)
HLHil              Scalar                                   Scalar                                                 HDMap       HGMap

Theoremhlhilhillem 35602* Lemma for hlhil 22475. (Contributed by NM, 23-Jun-2015.)
HLHil              Scalar                                   Scalar                                                 HDMap       HGMap

Theoremhlathil 35603 Construction of a Hilbert space (df-hil 19344) from a Hilbert lattice (df-hlat 32988) , where is a fixed but arbitrary hyperplane (co-atom) in .

The Hilbert space is identical to the vector space (see dvhlvec 34748) except that it is extended with involution and inner product components. The construction of these two components is provided by Theorem 3.6 in [Holland95] p. 13, whose proof we follow loosely.

An example of involution is the complex conjugate when the division ring is the field of complex numbers. The nature of the division ring we constructed is indeterminate, however, until we specialize the initial Hilbert lattice with additional conditions found by Maria Solèr in 1995 and refined by René Mayet in 1998 that result in a division ring isomorphic to . See additional discussion at http://us.metamath.org/qlegif/mmql.html#what.

corresponds to the w in the proof of Theorem 13.4 of [Crawley] p. 111. Such a always exists since has lattice rank of at least 4 by df-hil 19344. It can be eliminated if we just want to show the existence of a Hilbert space, as is done in the literature. (Contributed by NM, 23-Jun-2015.)

HLHil

21.22  Mathbox for OpenAI

TheoremrntrclfvOAI 35604 The range of the transitive closure is equal to the range of the relation. (Contributed by OpenAI, 7-Jul-2020.)

21.23  Mathbox for Stefan O'Rear

21.23.1  Additional elementary logic and set theory

Theoremmoxfr 35605* Transfer at-most-one between related expressions. (Contributed by Stefan O'Rear, 12-Feb-2015.)

21.23.2  Additional theory of functions

Theoremimaiinfv 35606* Indexed intersection of an image. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremelrfi 35607* Elementhood in a set of relative finite intersections. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremelrfirn 35608* Elementhood in a set of relative finite intersections of an indexed family of sets. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremelrfirn2 35609* Elementhood in a set of relative finite intersections of an indexed family of sets (implicit). (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremcmpfiiin 35610* In a compact topology, a system of closed sets with nonempty finite intersections has a nonempty intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)

21.23.4  Characterization of closure operators. Kuratowski closure axioms

Theoremismrcd1 35611* Any function from the subsets of a set to itself, which is extensive (satisfies mrcssid 15601), isotone (satisfies mrcss 15600), and idempotent (satisfies mrcidm 15603) has a collection of fixed points which is a Moore collection, and itself is the closure operator for that collection. This can be taken as an alternate definition for the closure operators. This is the first half, ismrcd2 35612 is the second. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Moore

Theoremismrcd2 35612* Second half of ismrcd1 35611. (Contributed by Stefan O'Rear, 1-Feb-2015.)
mrCls

Theoremistopclsd 35613* A closure function which satisfies sscls 20148, clsidm 20160, cls0 20173, and clsun 31055 defines a (unique) topology which it is the closure function on. (Contributed by Stefan O'Rear, 1-Feb-2015.)
TopOn

Theoremismrc 35614* A function is a Moore closure operator iff it satisfies mrcssid 15601, mrcss 15600, and mrcidm 15603. (Contributed by Stefan O'Rear, 1-Feb-2015.)
mrClsMoore

21.23.5  Algebraic closure systems

Syntaxcnacs 35615 Class of Noetherian closure systems.
NoeACS

Definitiondf-nacs 35616* Define a closure system of Noetherian type (not standard terminology) as an algebraic system where all closed sets are finitely generated. (Contributed by Stefan O'Rear, 4-Apr-2015.)
NoeACS ACS mrCls

Theoremisnacs 35617* Expand definition of Noetherian-type closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
mrCls       NoeACS ACS

Theoremnacsfg 35618* In a Noetherian-type closure system, all closed sets are finitely generated. (Contributed by Stefan O'Rear, 4-Apr-2015.)
mrCls       NoeACS

Theoremisnacs2 35619 Express Noetherian-type closure system with fewer quantifiers. (Contributed by Stefan O'Rear, 4-Apr-2015.)
mrCls       NoeACS ACS

Theoremmrefg2 35620* Slight variation on finite genration for closure systems. (Contributed by Stefan O'Rear, 4-Apr-2015.)
mrCls       Moore

Theoremmrefg3 35621* Slight variation on finite genration for closure systems. (Contributed by Stefan O'Rear, 4-Apr-2015.)
mrCls       Moore

Theoremnacsacs 35622 A closure system of Noetherian type is algebraic. (Contributed by Stefan O'Rear, 4-Apr-2015.)
NoeACS ACS

Theoremisnacs3 35623* A choice-free order equivalent to the Noetherian condition on a closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
NoeACS Moore toInc Dirset

Theoremincssnn0 35624* Transitivity induction of subsets, lemma for nacsfix 35625. (Contributed by Stefan O'Rear, 4-Apr-2015.)

Theoremnacsfix 35625* An increasing sequence of closed sets in a Noetherian-type closure system eventually fixates. (Contributed by Stefan O'Rear, 4-Apr-2015.)
NoeACS

21.23.6  Miscellanea 1. Map utilities

Theoremconstmap 35626 A constant (represented without dummy variables) is an element of a function set.

_Note: In the following development, we will be quite often quantifying over functions and points in N-dimensional space (which are equivalent to functions from an "index set"). Many of the following theorems exist to transfer standard facts about functions to elements of function sets._ (Contributed by Stefan O'Rear, 30-Aug-2014.) (Revised by Stefan O'Rear, 5-May-2015.)

Theoremmapco2g 35627 Renaming indexes in a tuple, with sethood as antecedents. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Mario Carneiro, 5-May-2015.)

Theoremmapco2 35628 Post-composition (renaming indexes) of a mapping viewed as a point. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)

Theoremmapfzcons 35629 Extending a one-based mapping by adding a tuple at the end results in another mapping. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)

Theoremmapfzcons1 35630 Recover prefix mapping from an extended mapping. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)

Theoremmapfzcons1cl 35631 A nonempty mapping has a prefix. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)

Theoremmapfzcons2 35632 Recover added element from an extended mapping. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)

21.23.7  Miscellanea for polynomials

Theoremmptfcl 35633* Interpret range of a maps-to notation as a constraint on the definition. (Contributed by Stefan O'Rear, 10-Oct-2014.)

21.23.8  Multivariate polynomials over the integers

Syntaxcmzpcl 35634 Extend class notation to include pre-polynomial rings.
mzPolyCld

Syntaxcmzp 35635 Extend class notation to include polynomial rings.
mzPoly

Definitiondf-mzpcl 35636* Define the polynomially closed function rings over an arbitrary index set . The set mzPolyCld contains all sets of functions from to which include all constants and projections and are closed under addition and multiplication. This is a "temporary" set used to define the polynomial function ring itself mzPoly; see df-mzp 35637. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPolyCld

Definitiondf-mzp 35637 Polynomials over with an arbitrary index set, that is, the smallest ring of functions containing all constant functions and all projections. This is almost the most general reasonable definition; to reach full generality, we would need to be able to replace ZZ with an arbitrary (semi-)ring (and a coordinate subring), but rings have not been defined yet. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPoly mzPolyCld

Theoremmzpclval 35638* Substitution lemma for mzPolyCld. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPolyCld

Theoremelmzpcl 35639* Double substitution lemma for mzPolyCld. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPolyCld

Theoremmzpclall 35640 The set of all functions with the signature of a polynomial is a polynomially closed set. This is a lemma to show that the intersection in df-mzp 35637 is well-defined. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPolyCld

Theoremmzpcln0 35641 Corrolary of mzpclall 35640: polynomially closed function sets are not empty. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPolyCld

Theoremmzpcl1 35642 Defining property 1 of a polynomially closed function set : it contains all constant functions. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPolyCld

Theoremmzpcl2 35643* Defining property 2 of a polynomially closed function set : it contains all projections. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPolyCld

Theoremmzpcl34 35644 Defining properties 3 and 4 of a polynomially closed function set : it is closed under pointwise addition and multiplication. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPolyCld

Theoremmzpval 35645 Value of the mzPoly function. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPoly mzPolyCld

Theoremdmmzp 35646 mzPoly is defined for all index sets which are sets. This is used with elfvdm 5905 to eliminate sethood antecedents. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPoly

Theoremmzpincl 35647 Polynomial closedness is a universal first-order property and passes to intersections. This is where the closure properties of the polynomial ring itself are proved. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPoly mzPolyCld

Theoremmzpconst 35648 Constant functions are polynomial. See also mzpconstmpt 35653. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPoly

Theoremmzpf 35649 A polynomial function is a function from the coordinate space to the integers. (Contributed by Stefan O'Rear, 5-Oct-2014.)
mzPoly

Theoremmzpproj 35650* A projection function is polynomial. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPoly

Theoremmzpadd 35651 The pointwise sum of two polynomial functions is a polynomial function. See also mzpaddmpt 35654. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPoly mzPoly mzPoly

Theoremmzpmul 35652 The pointwise product of two polynomial functions is a polynomial function. See also mzpmulmpt 35655. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPoly mzPoly mzPoly

Theoremmzpconstmpt 35653* A constant function expressed in maps-to notation is polynomial. This theorem and the several that follow (mzpaddmpt 35654, mzpmulmpt 35655, mzpnegmpt 35657, mzpsubmpt 35656, mzpexpmpt 35658) can be used to build proofs that functions which are "manifestly polynomial", in the sense of being a maps-to containing constants, projections, and simple arithmetic operations, are actually polynomial functions. There is no mzpprojmpt because mzpproj 35650 is already expressed using maps-to notation. (Contributed by Stefan O'Rear, 5-Oct-2014.)
mzPoly

Theoremmzpaddmpt 35654* Sum of polynomial functions is polynomial. Maps-to version of mzpadd 35651. (Contributed by Stefan O'Rear, 5-Oct-2014.)
mzPoly mzPoly mzPoly

Theoremmzpmulmpt 35655* Product of polynomial functions is polynomial. Maps-to version of mzpmulmpt 35655. (Contributed by Stefan O'Rear, 5-Oct-2014.)
mzPoly mzPoly mzPoly

Theoremmzpsubmpt 35656* The difference of two polynomial functions is polynomial. (Contributed by Stefan O'Rear, 10-Oct-2014.)
mzPoly mzPoly mzPoly

Theoremmzpnegmpt 35657* Negation of a polynomial function. (Contributed by Stefan O'Rear, 11-Oct-2014.)
mzPoly mzPoly

Theoremmzpexpmpt 35658* Raise a polynomial function to a (fixed) exponent. (Contributed by Stefan O'Rear, 5-Oct-2014.)
mzPoly mzPoly

Theoremmzpindd 35659* "Structural" induction to prove properties of all polynomial functions. (Contributed by Stefan O'Rear, 4-Oct-2014.)
mzPoly

Theoremmzpmfp 35660 Relationship between multivariate Z-polynomials and general multivariate polynomial functions. (Contributed by Stefan O'Rear, 20-Mar-2015.) (Revised by AV, 13-Jun-2019.)
mzPoly eval ℤring

Theoremmzpsubst 35661* Substituting polynomials for the variables of a polynomial results in a polynomial. is expected to depend on and provide the polynomials which are being substituted. (Contributed by Stefan O'Rear, 5-Oct-2014.)
mzPoly mzPoly mzPoly

Theoremmzprename 35662* Simplified version of mzpsubst 35661 to simply relabel variables in a polynomial. (Contributed by Stefan O'Rear, 5-Oct-2014.)
mzPoly mzPoly

Theoremmzpresrename 35663* A polynomial is a polynomial over all larger index sets. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Stefan O'Rear, 5-Jun-2015.)
mzPoly mzPoly

Theoremmzpcompact2lem 35664* Lemma for mzpcompact2 35665. (Contributed by Stefan O'Rear, 9-Oct-2014.)
mzPoly mzPoly

Theoremmzpcompact2 35665* Polynomials are finitary objects and can only reference a finite number of variables, even if the index set is infinite. Thus, every polynomial can be expressed as a (uniquely minimal, although we do not prove that) polynomial on a finite number of variables, which is then extended by adding an arbitrary set of ignored variables. (Contributed by Stefan O'Rear, 9-Oct-2014.)
mzPoly mzPoly

21.23.9  Miscellanea for Diophantine sets 1

Theoremcoeq0i 35666 coeq0 5351 but without explicitly introducing domain and range symbols. (Contributed by Stefan O'Rear, 16-Oct-2014.)

Theoremfzsplit1nn0 35667 Split a finite 1-based set of integers in the middle, allowing either end to be empty (). (Contributed by Stefan O'Rear, 8-Oct-2014.)

21.23.10  Diophantine sets 1: definitions

Syntaxcdioph 35668 Extend class notation to include the family of Diophantine sets.
Dioph

Definitiondf-dioph 35669* A Diophantine set is a set of positive integers which is a projection of the zero set of some polynomial. This definition somewhat awkwardly mixes (via mzPoly) and (to define the zero sets); the former could be avoided by considering coincidence sets of polynomials at the cost of requiring two, and the second is driven by consistency with our mu-recursive functions and the requirements of the Davis-Putnam-Robinson-Matiyasevich proof. Both are avoidable at a complexity cost. In particular, it is a consequence of 4sq 14993 that implicitly restricting variables to adds no expressive power over allowing them to range over . While this definition stipulates a specific index set for the polynomials, there is actually flexibility here, see eldioph2b 35676. (Contributed by Stefan O'Rear, 5-Oct-2014.)
Dioph mzPoly

Theoremeldiophb 35670* Initial expression of Diophantine property of a set. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
Dioph mzPoly

Theoremeldioph 35671* Condition for a set to be Diophantine (unpacking existential quantifier). (Contributed by Stefan O'Rear, 5-Oct-2014.)
mzPoly Dioph

Theoremdiophrw 35672* Renaming and adding unused witness variables does not change the Diophantine set coded by a polynomial. (Contributed by Stefan O'Rear, 7-Oct-2014.)

Theoremeldioph2lem1 35673* Lemma for eldioph2 35675. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.)

Theoremeldioph2lem2 35674* Lemma for eldioph2 35675. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.)

Theoremeldioph2 35675* Construct a Diophantine set from a polynomial with witness variables drawn from any set whatsoever, via mzpcompact2 35665. (Contributed by Stefan O'Rear, 8-Oct-2014.) (Revised by Stefan O'Rear, 5-Jun-2015.)
mzPoly Dioph

Theoremeldioph2b 35676* While Diophantine sets were defined to have a finite number of witness variables consequtively following the observable variables, this is not necessary; they can equivalently be taken to use any witness set . For instance, in diophin 35686 we use this to take the two input sets to have disjoint witness sets. (Contributed by Stefan O'Rear, 8-Oct-2014.)
Dioph mzPoly

Theoremeldiophelnn0 35677 Remove antecedent on from Diophantine set constructors. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Dioph

Theoremeldioph3b 35678* Define Diophantine sets in terms of polynomials with variables indexed by . This avoids a quantifier over the number of witness variables and will be easier to use than eldiophb 35670 in most cases. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Dioph mzPoly

Theoremeldioph3 35679* Inference version of eldioph3b 35678 with quantifier expanded. (Contributed by Stefan O'Rear, 10-Oct-2014.)
mzPoly Dioph

21.23.11  Diophantine sets 2 miscellanea

Theoremellz1 35680 Membership in a lower set of integers. (Contributed by Stefan O'Rear, 9-Oct-2014.)

Theoremlzunuz 35681 The union of a lower set of integers and an upper set of integers which abut or overlap is all of the integers. (Contributed by Stefan O'Rear, 9-Oct-2014.)

Theoremfz1eqin 35682 Express a one-based finite range as the intersection of lower integers with . (Contributed by Stefan O'Rear, 9-Oct-2014.)

Theoremlzenom 35683 Lower integers are countably infinite. (Contributed by Stefan O'Rear, 10-Oct-2014.)

Theoremelmapresaun 35684 fresaun 5766 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)

Theoremelmapresaunres2 35685 fresaunres2 5767 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014.)

21.23.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra

Theoremdiophin 35686 If two sets are Diophantine, so is their intersection. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Dioph Dioph Dioph

Theoremdiophun 35687 If two sets are Diophantine, so is their union. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Dioph Dioph Dioph

Theoremeldiophss 35688 Diophantine sets are sets of tuples of nonnegative integers. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Dioph

21.23.13  Diophantine sets 3: construction

Theoremdiophrex 35689* Projecting a Diophantine set by removing a coordinate results in a Diophantine set. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Dioph Dioph

Theoremeq0rabdioph 35690* This is the first of a number of theorems which allow sets to be proven Diophantine by syntactic induction, and models the correspondence between Diophantine sets and monotone existential first-order logic. This first theorem shows that the zero set of an implicit polynomial is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.)
mzPoly Dioph

Theoremeqrabdioph 35691* Diophantine set builder for equality of polynomial expressions. Note that the two expressions need not be nonnegative; only variables are so constrained. (Contributed by Stefan O'Rear, 10-Oct-2014.)
mzPoly mzPoly Dioph

Theorem0dioph 35692 The null set is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Dioph

Theoremvdioph 35693 The "universal" set (as large as possible given eldiophss 35688) is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Dioph

Theoremanrabdioph 35694* Diophantine set builder for conjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Dioph Dioph Dioph

Theoremorrabdioph 35695* Diophantine set builder for disjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Dioph Dioph Dioph

Theorem3anrabdioph 35696* Diophantine set builder for ternary conjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Dioph Dioph Dioph Dioph

Theorem3orrabdioph 35697* Diophantine set builder for ternary disjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Dioph Dioph Dioph Dioph

21.23.14  Diophantine sets 4 miscellanea

Theorem2sbcrex 35698* Exchange an existential quantifier with two substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by NM, 24-Aug-2018.)

TheoremsbcrexgOLD 35699* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) Obsolete as of 18-Aug-2018. Use sbcrex 3331 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Theorem2sbcrexOLD 35700* Exchange an existential quantifier with two substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) Obsolete as of 24-Aug-2018. Use csbov123 6342 instead. (New usage is discouraged.) (Proof modification is discouraged.)

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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 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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 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