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**Theorem List for Metamath Proof Explorer -** 19501-19600 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	cpmidpmat 19501*	Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as polynomial over the ring of matrices. (Contributed by AV, 14-Nov-2019.) (Revised by AV, 7-Dec-2019.)
Mat Poly₁ Mat var₁ ._gmulGrp algSc CharPlyMat matToPolyMat Poly₁ var₁ ._gmulGrp pMatToMatPoly $/\$ $/\$ _g coe₁

Theorem	cpmadugsumlemB 19502*	Lemma B for cpmadugsum 19506. (Contributed by AV, 2-Nov-2019.)
Mat Poly₁ Mat matToPolyMat var₁ ._gmulGrp $/\$ $/\$ $/\$ $/\$ _g _g

Theorem	cpmadugsumlemC 19503*	Lemma C for cpmadugsum 19506. (Contributed by AV, 2-Nov-2019.)
Mat Poly₁ Mat matToPolyMat var₁ ._gmulGrp $/\$ $/\$ $/\$ $/\$ _g _g

Theorem	cpmadugsumlemF 19504*	Lemma F for cpmadugsum 19506. (Contributed by AV, 7-Nov-2019.)
Mat Poly₁ Mat matToPolyMat var₁ ._gmulGrp $/\$ $/\$ $/\$ $/\$ _g _g _g

Theorem	cpmadugsumfi 19505*	The product of the characteristic matrix of a given matrix and its adjunct represented as finite sum. (Contributed by AV, 7-Nov-2019.) (Proof shortened by AV, 29-Nov-2019.)
Mat Poly₁ Mat matToPolyMat var₁ ._gmulGrp maAdju $/\$ $/\$ _g

Theorem	cpmadugsum 19506*	The product of the characteristic matrix of a given matrix and its adjunct represented as an infinite sum. (Contributed by AV, 10-Nov-2019.)
Mat Poly₁ Mat matToPolyMat var₁ ._gmulGrp maAdju $/\$ $/\$ _g

Theorem	cpmidgsum2 19507*	Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as another group sum. (Contributed by AV, 10-Nov-2019.)
Mat Poly₁ Mat matToPolyMat var₁ ._gmulGrp maAdju CharPlyMat $/\$ $/\$ _g

Theorem	cpmidg2sum 19508*	Equality of two sums representing the identity matrix multiplied with the characteristic polynomial of a matrix. (Contributed by AV, 11-Nov-2019.)
Mat Poly₁ Mat matToPolyMat var₁ ._gmulGrp maAdju CharPlyMat algSc $/\$ $/\$ _g coe₁ _g

Theorem	cpmadumatpolylem1 19509*	Lemma 1 for cpmadumatpoly 19511. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 15-Dec-2019.)
Mat Poly₁ Mat matToPolyMat ConstPolyMat var₁ maAdju Poly₁ var₁ ._gmulGrp cPolyMatToMat $/\$ $/\$ $/\$ $/\$

Theorem	cpmadumatpolylem2 19510*	Lemma 2 for cpmadumatpoly 19511. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 15-Dec-2019.)
Mat Poly₁ Mat matToPolyMat ConstPolyMat var₁ maAdju Poly₁ var₁ ._gmulGrp cPolyMatToMat $/\$ $/\$ $/\$ $/\$ finSupp

Theorem	cpmadumatpoly 19511*	The product of the characteristic matrix of a given matrix and its adjunct represented as a polynomial over matrices. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 7-Dec-2019.)
Mat Poly₁ Mat matToPolyMat ConstPolyMat var₁ maAdju Poly₁ var₁ ._gmulGrp cPolyMatToMat pMatToMatPoly $/\$ $/\$ _g

Theorem	cayhamlem2 19512	Lemma for cayhamlem3 19515. (Contributed by AV, 24-Nov-2019.)
Mat ._gmulGrp $/\$ $/\$ $/\$ $/\$

Theorem	chcoeffeqlem 19513*	Lemma for chcoeffeq 19514. (Contributed by AV, 21-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.) (Revised by AV, 15-Dec-2019.)
Mat Poly₁ Mat matToPolyMat CharPlyMat cPolyMatToMat $/\$ $/\$ $/\$ $/\$ Poly₁ _g Poly₁._gmulGrpPoly₁var₁ Poly₁ _g coe₁ Poly₁._gmulGrpPoly₁var₁ coe₁

Theorem	chcoeffeq 19514*	The coefficients of the characteristic polynomial multiplied with the identity matrix represented by (transformed) ring elements obtained from the adjunct of the characteristic matrix. (Contributed by AV, 21-Nov-2019.) (Proof shortened by AV, 8-Dec-2019.) (Revised by AV, 15-Dec-2019.)
Mat Poly₁ Mat matToPolyMat CharPlyMat cPolyMatToMat $/\$ $/\$ coe₁

Theorem	cayhamlem3 19515*	Lemma for cayhamlem4 19516. (Contributed by AV, 24-Nov-2019.) (Revised by AV, 15-Dec-2019.)
Mat Poly₁ Mat matToPolyMat CharPlyMat cPolyMatToMat ._gmulGrp $/\$ $/\$ _g coe₁ _g

Theorem	cayhamlem4 19516*	Lemma for cayleyhamilton 19518. (Contributed by AV, 25-Nov-2019.) (Revised by AV, 15-Dec-2019.)
Mat Poly₁ Mat matToPolyMat CharPlyMat cPolyMatToMat ._gmulGrp ._gmulGrp $/\$ $/\$ _g coe₁ _g

Theorem	cayleyhamilton0 19517*	The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation". This version of cayleyhamilton 19518 provides definitions not used in the theorem itself, but in its proof to make it clearer, more readable and shorter compared with a proof without them (see cayleyhamiltonALT 19519)! (Contributed by AV, 25-Nov-2019.) (Revised by AV, 15-Dec-2019.)
Mat ._gmulGrp CharPlyMat coe₁ Poly₁ Mat ._gmulGrp matToPolyMat cPolyMatToMat $/\$ $/\$ _g

Theorem	cayleyhamilton 19518*	The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation", see theorem 7.8 in [Roman] p. 170 (without proof!), or theorem 3.1 in [Lang] p. 561. In other words, a matrix over a commutative ring "inserted" into its characteristic polynomial results in zero. This is Metamath 100 proof #49. (Contributed by Alexander van der Vekens, 25-Nov-2019.)
Mat CharPlyMat coe₁ ._gmulGrp $/\$ $/\$ _g

Theorem	cayleyhamiltonALT 19519*	Alternate proof of cayleyhamilton 19518, the Cayley-Hamilton theorem. This proof does not use cayleyhamilton0 19517 directly, but has the same structure as the proof of cayleyhamilton0 19517. In contrast to the proof of cayleyhamilton0 19517, only the definitions required to formulate the theorem itself are used, causing the definitions used in the lemmas being expanded, which makes the proof longer and more difficult to read. (Contributed by AV, 25-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
Mat CharPlyMat coe₁ ._gmulGrp $/\$ $/\$ _g

Theorem	cayleyhamilton1 19520*	The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation", or, in other words, a matrix over a commutative ring "inserted" into its characteristic polynomial results in zero. In this variant of cayleyhamilton 19518, the meaning of "inserted" is made more transparent: If the characteristic polynomial is a polynomial with coefficients , then a matrix over a commutative ring "inserted" into its characteristic polynomial is the sum of these coefficients multiplied with the corresponding power of the matrix. (Contributed by AV, 25-Nov-2019.)
Mat CharPlyMat coe₁ ._gmulGrp var₁ Poly₁ ._gmulGrp $/\$ $/\$ $/\$ $/\$ finSupp _g _g

PART 12 BASIC TOPOLOGY

12.1 Topology

12.1.1 Topological spaces

Syntax	ctop 19521	Extend class notation with the class of all topologies.


Syntax	ctopon 19522	The class function of all topologies over a base set.
TopOn

Syntax	ctpsOLD 19523	Extend class notation with the class of all topological spaces. (New usage is discouraged.)


Syntax	ctps 19524	Extend class notation with the class of all topological spaces.


Syntax	ctb 19525	Extend class notation with the class of all topological bases.


Definition	df-top 19526*	Define the (proper) class of all topologies. See istop2g 19532 for an alternate way to express finite intersection and istps5OLD 19552 for a standard definition as an ordered pair of a set and a topology on it. The final form of the definition is due to Bourbaki (Def. 1 of [BourbakiTop1] p. I.1), while the idea of defining a topology in terms of its open sets is due to Aleksandrov. For the convoluted history of the definitions of these notions, see Gregory H. Moore, The emergence of open sets, closed sets, and limit points in analysis and topology, Historia Mathematica 35 (2008) 220--241. (Contributed by NM, 3-Mar-2006.) (Revised by BJ, 20-Oct-2018.)
$/\$

Definition	df-topspOLD 19527*	Define the class of all topological spaces, each of which is an ordered pair the second of which is a topology on the first. See istps5OLD 19552 for a standard way to express a topological space. (Contributed by NM, 8-Mar-2006.) (New usage is discouraged.)
$/\$

Definition	df-bases 19528*	Define the class of all topological bases. Equivalent to definition of basis in [Munkres] p. 78 (see isbasis2g 19576). Note that "bases" is the plural of "basis." (Contributed by NM, 17-Jul-2006.)


Definition	df-topon 19529*	Define the set of topologies with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
TopOn

Definition	df-topsp 19530	Define the class of all topological spaces (structures). (Contributed by Stefan O'Rear, 13-Aug-2015.)
TopOn

Theorem	istopg 19531*	Express the predicate " is a topology." Note: In the literature, a topology is often represented by a script letter T, which resembles the letter J. This confusion may have led to J being used by some authors - e.g. K. D. Joshi, Introduction to General Topology (1983), p. 114 - and it is convenient for us since we later use to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
$/\$

Theorem	istop2g 19532*	Express the predicate " is a topology," using "the intersection of the elements of any finite subcollection" instead of the intersection of any two elements. (Contributed by NM, 19-Jul-2006.)
$/\$ $/\$ $/\$

Theorem	uniopn 19533	The union of a subset of a topology is an open set. (Contributed by Stefan Allan, 27-Feb-2006.)
$/\$

Theorem	iunopn 19534*	The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)
$/\$

Theorem	inopn 19535	The intersection of two open sets of a topology is also an open set. (Contributed by NM, 17-Jul-2006.)
$/\$ $/\$

Theorem	fitop 19536	A topology is closed under finite intersections. (Contributed by Jeff Hankins, 7-Oct-2009.)


Theorem	fiinopn 19537	The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.)
$/\$ $/\$

Theorem	iinopn 19538*	The intersection of a nonempty finite family of open sets is open. (Contributed by Mario Carneiro, 14-Sep-2014.)
$/\$ $/\$ $/\$

Theorem	unopn 19539	The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.)
$/\$ $/\$

Theorem	0opn 19540	The empty set is an open subset of a topology. (Contributed by Stefan Allan, 27-Feb-2006.)


Theorem	0ntop 19541	The empty set is not a topology. (Contributed by FL, 1-Jun-2008.)


Theorem	topopn 19542	The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.)


Theorem	eltopss 19543	A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.)
$/\$

Theorem	riinopn 19544*	A finite indexed relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.)
$/\$ $/\$

Theorem	rintopn 19545	A finite relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.)
$/\$ $/\$

Theorem	eltopspOLD 19546	Construct a topological space from a topology and vice-versa. We say that is a topology on . (This could be proved more efficiently from istpsOLD 19548, but the proof here does not require the Axiom of Regularity.) (Contributed by NM, 8-Mar-2006.) (Proof modification is discouraged.) (New usage is discouraged.)


Theorem	tpsexOLD 19547	Existence implied by membership in a topological space. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 30-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
$/\$

Theorem	istpsOLD 19548	Express the predicate "is a topological space." (Contributed by NM, 18-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
$/\$

Theorem	istps2OLD 19549	Express the predicate "is a topological space." (Contributed by NM, 18-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
$/\$ $/\$ $/\$

Theorem	istps3OLD 19550*	A standard textbook definition of a topological space. (Contributed by NM, 18-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
$/\$ $/\$ $/\$ $/\$

Theorem	istps4OLD 19551*	A standard textbook definition of a topological space. (Contributed by NM, 19-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	istps5OLD 19552*	A standard textbook definition of a topological space : a topology on is a collection of subsets of such that and are in and that is closed under union and finite intersection. Definition of topological space in [Munkres] p. 76. (Contributed by NM, 19-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
$/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	istopon 19553	Property of being a topology with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 13-Aug-2015.)
TopOn $/\$

Theorem	topontop 19554	A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theorem	toponuni 19555	The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theorem	toponmax 19556	The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theorem	toponss 19557	A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.)
TopOn $/\$

Theorem	toponcom 19558	If is a topology on the base set of topology , then is a topology on the base of . (Contributed by Mario Carneiro, 22-Aug-2015.)
$/\$ TopOn TopOn

Theorem	topontopi 19559	A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theorem	toponunii 19560	The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theorem	toptopon 19561	Alternative definition of in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theorem	topgele 19562	The topologies over the same set have the greatest element (the discrete topology) and the least element (the indiscrete topology). (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 16-Sep-2015.)
TopOn $/\$

Theorem	topsn 19563	The only topology on a singleton is the discrete topology (which is also the indiscrete topology by pwsn 4245). (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 16-Sep-2015.)
TopOn

Theorem	istps 19564	Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theorem	istps2 19565	Express the predicate "is a topological space." (Contributed by NM, 20-Oct-2012.)
$/\$

Theorem	tpsuni 19566	The base set of a topological space. (Contributed by FL, 27-Jun-2014.)


Theorem	tpstop 19567	The topology extractor on a topological space is a topology. (Contributed by FL, 27-Jun-2014.)


Theorem	tpspropd 19568	A topological space depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 13-Aug-2015.)


Theorem	tpsprop2d 19569	A topological space depends only on the base and topology components. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopSet TopSet

Theorem	topontopn 19570	Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.)
TopSet TopOn

Theorem	tsettps 19571	If the topology component is already correctly truncated, then it forms a topological space (with the topology extractor function coming out the same as the component). (Contributed by Mario Carneiro, 13-Aug-2015.)
TopSet TopOn

Theorem	istpsi 19572	Properties that determine a topological space. (Contributed by NM, 20-Oct-2012.)


Theorem	eltpsg 19573	Properties that determine a topological space from a construction (using no explicit indices). (Contributed by Mario Carneiro, 13-Aug-2015.)
TopSet TopOn

Theorem	eltpsi 19574	Properties that determine a topological space from a construction (using no explicit indices). (Contributed by NM, 20-Oct-2012.) (Revised by Mario Carneiro, 13-Aug-2015.)
TopSet

12.1.2 TopBases for topologies

Theorem	isbasisg 19575*	Express the predicate " is a basis for a topology." (Contributed by NM, 17-Jul-2006.)


Theorem	isbasis2g 19576*	Express the predicate " is a basis for a topology." (Contributed by NM, 17-Jul-2006.)
$/\$

Theorem	isbasis3g 19577*	Express the predicate " is a basis for a topology." Definition of basis in [Munkres] p. 78. (Contributed by NM, 17-Jul-2006.)
$/\$ $/\$ $/\$

Theorem	basis1 19578	Property of a basis. (Contributed by NM, 16-Jul-2006.)
$/\$ $/\$

Theorem	basis2 19579*	Property of a basis. (Contributed by NM, 17-Jul-2006.)
$/\$ $/\$ $/\$ $/\$

Theorem	fiinbas 19580*	If a set is closed under finite intersection, then it is a basis for a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
$/\$

Theorem	basdif0 19581	A basis is not affected by the addition or removal of the empty set. (Contributed by Mario Carneiro, 28-Aug-2015.)
$\$

Theorem	baspartn 19582*	A disjoint system of sets is a basis for a topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)
$/\$ $\/$

Theorem	tgval 19583*	The topology generated by a basis. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)


Theorem	tgval2 19584*	Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 19598) that is indeed a topology (on ; see unitg 19595). (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
$/\$ $/\$

Theorem	eltg 19585	Membership in a topology generated by a basis. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)


Theorem	eltg2 19586*	Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
$/\$ $/\$

Theorem	eltg2b 19587*	Membership in a topology generated by a basis. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Mario Carneiro, 10-Jan-2015.)
$/\$

Theorem	eltg4i 19588	An open set in a topology generated by a basis is the union of all basic open sets contained in it. (Contributed by Stefan O'Rear, 22-Feb-2015.)


Theorem	eltg3i 19589	The union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 30-Aug-2015.)
$/\$

Theorem	eltg3 19590*	Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Mario Carneiro, 30-Aug-2015.)
$/\$

Theorem	tgval3 19591*	Alternate expression for the topology generated by a basis. Lemma 2.1 of [Munkres] p. 80. (Contributed by NM, 17-Jul-2006.) (Revised by Mario Carneiro, 30-Aug-2015.)
$/\$

Theorem	tg1 19592	Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)


Theorem	tg2 19593*	Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
$/\$ $/\$

Theorem	bastg 19594	A member of a basis is a subset of the topology it generates. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)


Theorem	unitg 19595	The topology generated by a basis is a topology on . Importantly, this theorem means that we don't have to specify separately the base set for the topological space generated by a basis. In other words, any member of the class completely specifies the basis it corresponds to. (Contributed by NM, 16-Jul-2006.) (Proof shortened by OpenAI, 30-Mar-2020.)


Theorem	unitgOLD 19596	The topology generated by a basis is a topology on . Importantly, this theorem means that we don't have to specify separately the base set for the topological space generated by a basis. In other words, any member of the class completely specifies the basis it corresponds to. (Contributed by NM, 16-Jul-2006.) Obsolete version of unitg 19595 as of 30-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.)


Theorem	tgss 19597	Subset relation for generated topologies. (Contributed by NM, 7-May-2007.)
$/\$

Theorem	tgcl 19598	Show that a basis generates a topology. Remark in [Munkres] p. 79. (Contributed by NM, 17-Jul-2006.)


Theorem	tgclb 19599	The property tgcl 19598 can be reversed: if the topology generated by is actually a topology, then must be a topological basis. This yields an alternative definition of . (Contributed by Mario Carneiro, 2-Sep-2015.)


Theorem	tgtopon 19600	A basis generates a topology on . (Contributed by Mario Carneiro, 14-Aug-2015.)
TopOn

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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 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 38201-38213

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