Home Metamath Proof ExplorerTheorem List (p. 356 of 411) < Previous  Next > Browser slow? Try the Unicode version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-26652) Hilbert Space Explorer (26653-28175) Users' Mathboxes (28176-41046)

Theorem List for Metamath Proof Explorer - 35501-35600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremhdmaprnlem10N 35501* Lemma for hdmaprnN 35506. Show is in the range of . (Contributed by NM, 29-May-2015.) (New usage is discouraged.)
LCDual              mapd       HDMap

Theoremhdmaprnlem11N 35502* Lemma for hdmaprnN 35506. Show is in the range of . (Contributed by NM, 29-May-2015.) (New usage is discouraged.)
LCDual              mapd       HDMap

Theoremhdmaprnlem15N 35503* Lemma for hdmaprnN 35506. Eliminate . (Contributed by NM, 30-May-2015.) (New usage is discouraged.)
LCDual                            mapd       HDMap

Theoremhdmaprnlem16N 35504 Lemma for hdmaprnN 35506. Eliminate . (Contributed by NM, 30-May-2015.) (New usage is discouraged.)
LCDual                            mapd       HDMap

Theoremhdmaprnlem17N 35505 Lemma for hdmaprnN 35506. Include zero. (Contributed by NM, 30-May-2015.) (New usage is discouraged.)
LCDual                            mapd       HDMap

TheoremhdmaprnN 35506 Part of proof of part 12 in [Baer] p. 49 line 21, As=B. (Contributed by NM, 30-May-2015.) (New usage is discouraged.)
LCDual              HDMap

Theoremhdmapf1oN 35507 Part 12 in [Baer] p. 49. The map from vectors to functionals with closed kernels maps one-to-one onto. Combined with hdmapadd 35485, this shows the map is an automorphism from the additive group of vectors to the additive group of functionals with closed kernels. (Contributed by NM, 30-May-2015.) (New usage is discouraged.)
LCDual              HDMap

Theoremhdmap14lem1a 35508 Prior to part 14 in [Baer] p. 49, line 25. (Contributed by NM, 31-May-2015.)
Scalar              LCDual                     Scalar              HDMap

Theoremhdmap14lem2a 35509* Prior to part 14 in [Baer] p. 49, line 25. TODO: fix to include so it can be used in hdmap14lem10 35519. (Contributed by NM, 31-May-2015.)
Scalar              LCDual                     Scalar              HDMap

Theoremhdmap14lem1 35510 Prior to part 14 in [Baer] p. 49, line 25. (Contributed by NM, 31-May-2015.)
Scalar                     LCDual                     Scalar                     HDMap

Theoremhdmap14lem2N 35511* Prior to part 14 in [Baer] p. 49, line 25. TODO: fix to include so it can be used in hdmap14lem10 35519. (Contributed by NM, 31-May-2015.) (New usage is discouraged.)
Scalar                     LCDual                     Scalar                     HDMap

Theoremhdmap14lem3 35512* Prior to part 14 in [Baer] p. 49, line 26. (Contributed by NM, 31-May-2015.)
Scalar                     LCDual                     Scalar                     HDMap

Theoremhdmap14lem4a 35513* Simplify in hdmap14lem3 35512 to provide a slightly simpler definition later. (Contributed by NM, 31-May-2015.)
Scalar                     LCDual                     Scalar                     HDMap

Theoremhdmap14lem4 35514* Simplify in hdmap14lem3 35512 to provide a slightly simpler definition later. TODO: Use hdmap14lem4a 35513 if that one is also used directly elsewhere. Otherwise, merge hdmap14lem4a 35513 into this one. (Contributed by NM, 31-May-2015.)
Scalar                     LCDual                     Scalar                     HDMap

Theoremhdmap14lem6 35515* Case where is zero. (Contributed by NM, 1-Jun-2015.)
Scalar                     LCDual                     Scalar                     HDMap

Theoremhdmap14lem7 35516* Combine cases of . TODO: Can this be done at once in hdmap14lem3 35512, in order to get rid of hdmap14lem6 35515? Perhaps modify lspsneu 18424 to become instead of ? (Contributed by NM, 1-Jun-2015.)
Scalar              LCDual              Scalar              HDMap

Theoremhdmap14lem8 35517 Part of proof of part 14 in [Baer] p. 49 lines 33-35. (Contributed by NM, 1-Jun-2015.)
Scalar              LCDual                     Scalar              HDMap

Theoremhdmap14lem9 35518 Part of proof of part 14 in [Baer] p. 49 line 38. (Contributed by NM, 1-Jun-2015.)
Scalar              LCDual                     Scalar              HDMap

Theoremhdmap14lem10 35519 Part of proof of part 14 in [Baer] p. 49 line 38. (Contributed by NM, 3-Jun-2015.)
Scalar              LCDual                     Scalar              HDMap

Theoremhdmap14lem11 35520 Part of proof of part 14 in [Baer] p. 50 line 3. (Contributed by NM, 3-Jun-2015.)
Scalar              LCDual                     Scalar              HDMap

Theoremhdmap14lem12 35521* Lemma for proof of part 14 in [Baer] p. 50. (Contributed by NM, 6-Jun-2015.)
Scalar              LCDual              HDMap                     Scalar

Theoremhdmap14lem13 35522* Lemma for proof of part 14 in [Baer] p. 50. (Contributed by NM, 6-Jun-2015.)
Scalar              LCDual              HDMap                     Scalar

Theoremhdmap14lem14 35523* Part of proof of part 14 in [Baer] p. 50 line 3. (Contributed by NM, 6-Jun-2015.)
Scalar              LCDual              HDMap                     Scalar

Theoremhdmap14lem15 35524* Part of proof of part 14 in [Baer] p. 50 line 3. Convert scalar base of dual to scalar base of vector space. (Contributed by NM, 6-Jun-2015.)
Scalar              LCDual              HDMap

Syntaxchg 35525 Extend class notation with g-map.
HGMap

Definitiondf-hgmap 35526* Define map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. (Contributed by NM, 25-Mar-2015.)
HGMap Scalar HDMap LCDual

Theoremhgmapffval 35527* Map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. (Contributed by NM, 25-Mar-2015.)
HGMap Scalar HDMap LCDual

Theoremhgmapfval 35528* Map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. (Contributed by NM, 25-Mar-2015.)
Scalar              LCDual              HDMap       HGMap

Theoremhgmapval 35529* Value of map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. Function sigma of scalar f in part 14 of [Baer] p. 50 line 4. TODO: variable names are inherited from older version. Maybe make more consistent with hdmap14lem15 35524. (Contributed by NM, 25-Mar-2015.)
Scalar              LCDual              HDMap       HGMap

TheoremhgmapfnN 35530 Functionality of scalar sigma map. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
Scalar              HGMap

Theoremhgmapcl 35531 Closure of scalar sigma map i.e. the map from the vector space scalar base to the dual space scalar base. (Contributed by NM, 6-Jun-2015.)
Scalar              HGMap

Theoremhgmapdcl 35532 Closure of the vector space to dual space scalar map, in the scalar sigma map. (Contributed by NM, 6-Jun-2015.)
Scalar              LCDual       Scalar              HGMap

Theoremhgmapvs 35533 Part 15 of [Baer] p. 50 line 6. Also line 15 in [Holland95] p. 14. (Contributed by NM, 6-Jun-2015.)
Scalar              LCDual              HDMap       HGMap

Theoremhgmapval0 35534 Value of the scalar sigma map at zero. (Contributed by NM, 12-Jun-2015.)
Scalar              HGMap

Theoremhgmapval1 35535 Value of the scalar sigma map at one. (Contributed by NM, 12-Jun-2015.)
Scalar              HGMap

Theoremhgmapadd 35536 Part 15 of [Baer] p. 50 line 13. (Contributed by NM, 6-Jun-2015.)
Scalar                     HGMap

Theoremhgmapmul 35537 Part 15 of [Baer] p. 50 line 16. The multiplication is reversed after converting to the dual space scalar to the vector space scalar. (Contributed by NM, 7-Jun-2015.)
Scalar                     HGMap

Theoremhgmaprnlem1N 35538 Lemma for hgmaprnN 35543. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
Scalar                            LCDual              Scalar                            HDMap       HGMap

Theoremhgmaprnlem2N 35539 Lemma for hgmaprnN 35543. Part 15 of [Baer] p. 50 line 20. We only require a subset relation, rather than equality, so that the case of zero is taken care of automatically. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
Scalar                            LCDual              Scalar                            HDMap       HGMap                                          mapd

Theoremhgmaprnlem3N 35540* Lemma for hgmaprnN 35543. Eliminate . (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
Scalar                            LCDual              Scalar                            HDMap       HGMap                                          mapd

Theoremhgmaprnlem4N 35541* Lemma for hgmaprnN 35543. Eliminate . (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
Scalar                            LCDual              Scalar                            HDMap       HGMap

Theoremhgmaprnlem5N 35542 Lemma for hgmaprnN 35543. Eliminate . (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
Scalar                            LCDual              Scalar                            HDMap       HGMap

TheoremhgmaprnN 35543 Part of proof of part 16 in [Baer] p. 50 line 23, Fs=G, except that we use the original vector space scalars for the range. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
Scalar              HGMap

Theoremhgmap11 35544 The scalar sigma map is one-to-one. (Contributed by NM, 7-Jun-2015.)
Scalar              HGMap

Theoremhgmapf1oN 35545 The scalar sigma map is a one-to-one onto function. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
Scalar              HGMap

Theoremhgmapeq0 35546 The scalar sigma map is zero iff its argument is zero. (Contributed by NM, 12-Jun-2015.)
Scalar                     HGMap

Theoremhdmapipcl 35547 The inner product (Hermitian form) will be defined as . Show closure. (Contributed by NM, 7-Jun-2015.)
Scalar              HDMap

Theoremhdmapln1 35548 Linearity property that will be used for inner product. TODO: try to combine hypotheses in hdmap*ln* series. (Contributed by NM, 7-Jun-2015.)
Scalar                            HDMap

Theoremhdmaplna1 35549 Additive property of first (inner product) argument. (Contributed by NM, 11-Jun-2015.)
Scalar              HDMap

Theoremhdmaplns1 35550 Subtraction property of first (inner product) argument. (Contributed by NM, 12-Jun-2015.)
Scalar              HDMap

Theoremhdmaplnm1 35551 Multiplicative property of first (inner product) argument. (Contributed by NM, 11-Jun-2015.)
Scalar                     HDMap

Theoremhdmaplna2 35552 Additive property of second (inner product) argument. (Contributed by NM, 10-Jun-2015.)
Scalar              HDMap

Theoremhdmapglnm2 35553 g-linear property of second (inner product) argument. Line 19 in [Holland95] p. 14. (Contributed by NM, 10-Jun-2015.)
Scalar                     HDMap       HGMap

Theoremhdmapgln2 35554 g-linear property that will be used for inner product. (Contributed by NM, 14-Jun-2015.)
Scalar                            HDMap       HGMap

Theoremhdmaplkr 35555 Kernel of the vector to dual map. Line 16 in [Holland95] p. 14. TODO: eliminate hypothesis. (Contributed by NM, 9-Jun-2015.)
LFnl       LKer       HDMap

Theoremhdmapellkr 35556 Membership in the kernel (as shown by hdmaplkr 35555) of the vector to dual map. Line 17 in [Holland95] p. 14. (Contributed by NM, 16-Jun-2015.)
Scalar              HDMap

Theoremhdmapip0 35557 Zero property that will be used for inner product. (Contributed by NM, 9-Jun-2015.)
Scalar              HDMap

Theoremhdmapip1 35558 Construct a proportional vector whose inner product with the original equals one. (Contributed by NM, 13-Jun-2015.)
Scalar                     HDMap

Theoremhdmapip0com 35559 Commutation property of Baer's sigma map (Holland's A map). Line 20 of [Holland95] p. 14. Also part of Lemma 1 of [Baer] p. 110 line 7. (Contributed by NM, 9-Jun-2015.)
Scalar              HDMap

Theoremhdmapinvlem1 35560 Line 27 in [Baer] p. 110. We use for Baer's u. Our unit vector has the required properties for his w by hdmapevec2 35478. Our means the inner product i.e. his f(u,w) (note argument reversal). (Contributed by NM, 11-Jun-2015.)
Scalar                            HDMap

Theoremhdmapinvlem2 35561 Line 28 in [Baer] p. 110, 0 = f(w,u). (Contributed by NM, 11-Jun-2015.)
Scalar                            HDMap

Theoremhdmapinvlem3 35562 Line 30 in [Baer] p. 110, f(sw + u, tw - v) = 0. (Contributed by NM, 12-Jun-2015.)
Scalar                            HDMap       HGMap

Theoremhdmapinvlem4 35563 Part 1.1 of Proposition 1 of [Baer] p. 110. We use , , , and for Baer's u, v, s, and t. Our unit vector has the required properties for his w by hdmapevec2 35478. Our means his f(u,v) (note argument reversal). (Contributed by NM, 12-Jun-2015.)
Scalar                            HDMap       HGMap

Theoremhdmapglem5 35564 Part 1.2 in [Baer] p. 110 line 34, f(u,v) alpha = f(v,u). (Contributed by NM, 12-Jun-2015.)
Scalar                            HDMap       HGMap

Theoremhgmapvvlem1 35565 Involution property of scalar sigma map. Line 10 in [Baer] p. 111, t sigma squared = t. Our , , , , correspond to Baer's w, h, k, s, t. (Contributed by NM, 13-Jun-2015.)
Scalar                                          HDMap       HGMap

Theoremhgmapvvlem2 35566 Lemma for hgmapvv 35568. Eliminate (Baer's s). (Contributed by NM, 13-Jun-2015.)
Scalar                                          HDMap       HGMap

Theoremhgmapvvlem3 35567 Lemma for hgmapvv 35568. Eliminate (Baer's f(h,k)=1). (Contributed by NM, 13-Jun-2015.)
Scalar                                          HDMap       HGMap

Theoremhgmapvv 35568 Value of a double involution. Part 1.2 of [Baer] p. 110 line 37. (Contributed by NM, 13-Jun-2015.)
Scalar              HGMap

Theoremhdmapglem7a 35569* Lemma for hdmapg 35572. (Contributed by NM, 14-Jun-2015.)
Scalar

Theoremhdmapglem7b 35570 Lemma for hdmapg 35572. (Contributed by NM, 14-Jun-2015.)
Scalar                                                               HDMap       HGMap

Theoremhdmapglem7 35571 Lemma for hdmapg 35572. Line 15 in [Baer] p. 111, f(x,y) alpha = f(y,x). In the proof, our , , , , , , correspond to Baer's w, H, x, y, x', x'', y' , y'', and our corresponds to Baer's f(x,y). (Contributed by NM, 14-Jun-2015.)
Scalar                                                               HDMap       HGMap

Theoremhdmapg 35572 Apply the scalar sigma function (involution) to an inner product reverses the arguments. The inner product of and is represented by . Line 15 in [Baer] p. 111, f(x,y) alpha = f(y,x). (Contributed by NM, 14-Jun-2015.)
HDMap       HGMap

Theoremhdmapoc 35573* Express our constructed orthocomplement (polarity) in terms of the Hilbert space definition of orthocomplement. Lines 24 and 25 in [Holland95] p. 14. (Contributed by NM, 17-Jun-2015.)
Scalar                     HDMap

Syntaxchlh 35574 Extend class notation with the final constructed Hilbert space.
HLHil

Definitiondf-hlhil 35575* Define our final Hilbert space constructed from a Hilbert lattice. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
HLHil Scalar sSet HGMap HDMap

Theoremhlhilset 35576* The final Hilbert space constructed from a Hilbert lattice and an arbitrary hyperplane in . (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
HLHil                                   HGMap       sSet               HDMap                     Scalar

Theoremhlhilsca 35577 The scalar of the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
HLHil                     HGMap       sSet        Scalar

Theoremhlhilbase 35578 The base set of the final constructed Hilbert space. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
HLHil

Theoremhlhilplus 35579 The vector addition for the final constructed Hilbert space. (Contributed by NM, 21-Jun-2015.)
HLHil

Theoremhlhilslem 35580 Lemma for hlhilsbase2 35584. (Contributed by Mario Carneiro, 28-Jun-2015.)
HLHil       Scalar              Slot

Theoremhlhilsbase 35581 The scalar base set of the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
HLHil       Scalar

Theoremhlhilsplus 35582 Scalar addition for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
HLHil       Scalar

Theoremhlhilsmul 35583 Scalar multiplication for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
HLHil       Scalar

Theoremhlhilsbase2 35584 The scalar base set of the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
Scalar       HLHil       Scalar

Theoremhlhilsplus2 35585 Scalar addition for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
Scalar       HLHil       Scalar

Theoremhlhilsmul2 35586 Scalar multiplication for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
Scalar       HLHil       Scalar

Theoremhlhils0 35587 The scalar ring zero for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.)
Scalar       HLHil       Scalar

Theoremhlhils1N 35588 The scalar ring unity for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.) (New usage is discouraged.)
Scalar       HLHil       Scalar

Theoremhlhilvsca 35589 The scalar product for the final constructed Hilbert space. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
HLHil

Theoremhlhilip 35590* Inner product operation for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
HDMap       HLHil

Theoremhlhilipval 35591 Value of inner product operation for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
HDMap       HLHil

Theoremhlhilnvl 35592 The involution operation of the star division ring for the final constructed Hilbert space. (Contributed by NM, 20-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
HLHil       Scalar       HGMap

Theoremhlhillvec 35593 The final constructed Hilbert space is a vector space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.)
HLHil

Theoremhlhildrng 35594 The star division ring for the final constructed Hilbert space is a division ring. (Contributed by NM, 20-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
HLHil              Scalar

Theoremhlhilsrnglem 35595 Lemma for hlhilsrng 35596. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
HLHil              Scalar              Scalar                            HGMap

Theoremhlhilsrng 35596 The star division ring for the final constructed Hilbert space is a division ring. (Contributed by NM, 21-Jun-2015.)
HLHil              Scalar

Theoremhlhil0 35597 The zero vector for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.)
HLHil

Theoremhlhillsm 35598 The vector sum operation for the final constructed Hilbert space. (Contributed by NM, 23-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.)
HLHil

Theoremhlhilocv 35599 The orthocomplement for the final constructed Hilbert space. (Contributed by NM, 23-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.)
HLHil

Theoremhlhillcs 35600 The closed subspaces of the final constructed Hilbert space. TODO: hlhilbase 35578 is applied over and over to conclusion rather than applied once to antecedent - would compressed proof be shorter if applied once to antecedent? (Contributed by NM, 23-Jun-2015.)
HLHil

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