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Theorem List for Metamath Proof Explorer - 33301-33400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembdaydm 33301 The birthday function's domain is No . (Contributed by Scott Fenton, 14-Jun-2011.)
dom bday = No

Theorembdayrn 33302 The birthday function's range is On. (Contributed by Scott Fenton, 14-Jun-2011.)
ran bday = On

Theorembdayelon 33303 The value of the birthday function is always an ordinal. (Contributed by Scott Fenton, 14-Jun-2011.) (Proof shortened by Scott Fenton, 8-Dec-2021.)
( bday 𝐴) ∈ On

Theoremnocvxminlem 33304* Lemma for nocvxmin 33305. Given two birthday-minimal elements of a convex class of surreals, they are not comparable. (Contributed by Scott Fenton, 30-Jun-2011.)
((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → (((𝑋𝐴𝑌𝐴) ∧ (( bday 𝑋) = ( bday 𝐴) ∧ ( bday 𝑌) = ( bday 𝐴))) → ¬ 𝑋 <s 𝑌))

Theoremnocvxmin 33305* Given a nonempty convex class of surreals, there is a unique birthday-minimal element of that class. (Contributed by Scott Fenton, 30-Jun-2011.)
((𝐴 ≠ ∅ ∧ 𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → ∃!𝑤𝐴 ( bday 𝑤) = ( bday 𝐴))

Theoremnoprc 33306 The surreal numbers are a proper class. (Contributed by Scott Fenton, 16-Jun-2011.)
¬ No ∈ V

20.9.28  Surreal numbers: Conway cuts

Syntaxcsslt 33307 Declare the syntax for surreal set less than.
class <<s

Definitiondf-sslt 33308* Define the relationship that holds iff one set of surreals completely precedes another. (Contributed by Scott Fenton, 7-Dec-2021.)
<<s = {⟨𝑎, 𝑏⟩ ∣ (𝑎 No 𝑏 No ∧ ∀𝑥𝑎𝑦𝑏 𝑥 <s 𝑦)}

Syntaxcscut 33309 Declare the syntax for the surreal cut operator.
class |s

Definitiondf-scut 33310* Define the cut operator on surreal numbers. This operator, which Conway takes as the primitive operator over surreals, picks the surreal lying between two sets of surreals of minimal birthday. (Contributed by Scott Fenton, 7-Dec-2021.)
|s = (𝑎 ∈ 𝒫 No , 𝑏 ∈ ( <<s “ {𝑎}) ↦ (𝑥 ∈ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})))

Theorembrsslt 33311* Binary relation form of the surreal set less-than relation. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 No 𝐵 No ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)))

Theoremssltex1 33312 The first argument of surreal set less than exists. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵𝐴 ∈ V)

Theoremssltex2 33313 The second argument of surreal set less than exists. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵𝐵 ∈ V)

Theoremssltss1 33314 The first argument of surreal set is a set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵𝐴 No )

Theoremssltss2 33315 The second argument of surreal set is a set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵𝐵 No )

Theoremssltsep 33316* The separation property of surreal set less than. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵 → ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦)

Theoremsssslt1 33317 Relationship between surreal set less than and subset. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐵𝐶𝐴) → 𝐶 <<s 𝐵)

Theoremsssslt2 33318 Relationship between surreal set less than and subset. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐵𝐶𝐵) → 𝐴 <<s 𝐶)

Theoremnulsslt 33319 The empty set is less than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 ∈ 𝒫 No → ∅ <<s 𝐴)

Theoremnulssgt 33320 The empty set is greater than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 ∈ 𝒫 No 𝐴 <<s ∅)

Theoremconway 33321* Conway's Simplicity Theorem. Given 𝐴 preceeding 𝐵, there is a unique surreal of minimal length separating them. This is a fundamental property of surreals and will be used (via surreal cuts) to prove many properties later on. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵 → ∃!𝑥 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))

Theoremscutval 33322* The value of the surreal cut operation. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵 → (𝐴 |s 𝐵) = (𝑥 ∈ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday 𝑥) = ( bday “ {𝑦 No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})))

Theoremscutcut 33323 Cut properties of the surreal cut operation. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))

Theoremscutbday 33324* The birthday of the surreal cut is equal to the minimum birthday in the gap. (Contributed by Scott Fenton, 8-Dec-2021.)
(𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) = ( bday “ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}))

Theoremsslttr 33325 Transitive law for surreal set less than. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐵𝐵 <<s 𝐶𝐵 ≠ ∅) → 𝐴 <<s 𝐶)

Theoremssltun1 33326 Union law for surreal set less than. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐶𝐵 <<s 𝐶) → (𝐴𝐵) <<s 𝐶)

Theoremssltun2 33327 Union law for surreal set less than. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐵𝐴 <<s 𝐶) → 𝐴 <<s (𝐵𝐶))

Theoremscutun12 33328 Union law for surreal cuts. (Contributed by Scott Fenton, 9-Dec-2021.)
((𝐴 <<s 𝐵𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ((𝐴𝐶) |s (𝐵𝐷)) = (𝐴 |s 𝐵))

Theoremdmscut 33329 The domain of the surreal cut operation is all separated surreal sets. (Contributed by Scott Fenton, 8-Dec-2021.)
dom |s = <<s

Theoremscutf 33330 Functionhood statement for the surreal cut operator. (Contributed by Scott Fenton, 15-Dec-2021.)
|s : <<s ⟶ No

Theoremetasslt 33331* A restatement of noeta 33279 using set less than. (Contributed by Scott Fenton, 10-Dec-2021.)
(𝐴 <<s 𝐵 → ∃𝑥 No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday 𝑥) ⊆ suc ( bday “ (𝐴𝐵))))

Theoremscutbdaybnd 33332 An upper bound on the birthday of a surreal cut. (Contributed by Scott Fenton, 10-Dec-2021.)
(𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) ⊆ suc ( bday “ (𝐴𝐵)))

Theoremscutbdaylt 33333 If a surreal lies in a gap and is not equal to the cut, its birthday is greater than the cut's. (Contributed by Scott Fenton, 11-Dec-2021.)
((𝑋 No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) ∈ ( bday 𝑋))

Theoremslerec 33334* A comparison law for surreals considered as cuts of sets of surreals. In Conway's treatment, this is the definition of less than or equal. (Contributed by Scott Fenton, 11-Dec-2021.)
(((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑋 ≤s 𝑌 ↔ (∀𝑑𝐷 𝑋 <s 𝑑 ∧ ∀𝑎𝐴 𝑎 <s 𝑌)))

Theoremsltrec 33335* A comparison law for surreals considered as cuts of sets of surreals. (Contributed by Scott Fenton, 11-Dec-2021.)
(((𝐴 <<s 𝐵𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑋 <s 𝑌 ↔ (∃𝑐𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏𝐵 𝑏 ≤s 𝑌)))

20.9.29  Surreal numbers - cuts and options

Syntaxcmade 33336 Declare the symbol for the made by function.
class M

Syntaxcold 33337 Declare the symbol for the older than function.
class O

Syntaxcnew 33338 Declare the symbol for the new on function.
class N

Syntaxcleft 33339 Declare the symbol for the left option function.
class L

Syntaxcright 33340 Declare the symbol for the right option function.
class R

Definitiondf-made 33341 Define the made by function. This function carries an ordinal to all surreals made by sections of surreals older than it. (Contributed by Scott Fenton, 17-Dec-2021.)
M = recs((𝑓 ∈ V ↦ ( |s “ (𝒫 ran 𝑓 × 𝒫 ran 𝑓))))

Definitiondf-old 33342 Define the older than function. This function carries an ordinal to all surreals made by a previous ordinal. (Contributed by Scott Fenton, 17-Dec-2021.)
O = (𝑥 ∈ On ↦ ( M “ 𝑥))

Definitiondf-new 33343 Define the newer than function. This function carries an ordinal to all surreals made on that day. (Contributed by Scott Fenton, 17-Dec-2021.)
N = (𝑥 ∈ On ↦ (( O ‘𝑥) ∖ ( M ‘𝑥)))

Definitiondf-left 33344* Define the left options of a surreal. This is the set of surreals that are "closest" on the left to the given surreal. (Contributed by Scott Fenton, 17-Dec-2021.)
L = (𝑥 No ↦ {𝑦 ∈ ( O ‘( bday 𝑥)) ∣ ∀𝑧 No ((𝑦 <s 𝑧𝑧 <s 𝑥) → ( bday 𝑦) ∈ ( bday 𝑧))})

Definitiondf-right 33345* Define the left options of a surreal. This is the set of surreals that are "closest" on the right to the given surreal. (Contributed by Scott Fenton, 17-Dec-2021.)
R = (𝑥 No ↦ {𝑦 ∈ ( O ‘( bday 𝑥)) ∣ ∀𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → ( bday 𝑦) ∈ ( bday 𝑧))})

Theoremmadeval 33346 The value of the made by function. (Contributed by Scott Fenton, 17-Dec-2021.)
(𝐴 ∈ On → ( M ‘𝐴) = ( |s “ (𝒫 ( M “ 𝐴) × 𝒫 ( M “ 𝐴))))

Theoremmadeval2 33347* Alternative characterization of the made by function. (Contributed by Scott Fenton, 17-Dec-2021.)
(𝐴 ∈ On → ( M ‘𝐴) = {𝑥 No ∣ ∃𝑎 ∈ 𝒫 ( M “ 𝐴)∃𝑏 ∈ 𝒫 ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)})

20.9.30  Quantifier-free definitions

Syntaxctxp 33348 Declare the syntax for tail Cartesian product.
class (𝐴𝐵)

Syntaxcpprod 33349 Declare the syntax for the parallel product.
class pprod(𝑅, 𝑆)

Syntaxcsset 33350 Declare the subset relationship class.
class SSet

Syntaxctrans 33351 Declare the transitive set class.
class Trans

Syntaxcbigcup 33352 Declare the set union relationship.
class Bigcup

Syntaxcfix 33353 Declare the syntax for the fixpoints of a class.
class Fix 𝐴

Syntaxclimits 33354 Declare the class of limit ordinals.
class Limits

Syntaxcfuns 33355 Declare the syntax for the class of all function.
class Funs

Syntaxcsingle 33356 Declare the syntax for the singleton function.
class Singleton

Syntaxcsingles 33357 Declare the syntax for the class of all singletons.
class Singletons

Syntaxcimage 33358 Declare the syntax for the image functor.
class Image𝐴

Syntaxccart 33359 Declare the syntax for the cartesian function.
class Cart

Syntaxcimg 33360 Declare the syntax for the image function.
class Img

Syntaxcdomain 33361 Declare the syntax for the domain function.
class Domain

Syntaxcrange 33362 Declare the syntax for the range function.
class Range

Syntaxcapply 33363 Declare the syntax for the application function.
class Apply

Syntaxccup 33364 Declare the syntax for the cup function.
class Cup

Syntaxccap 33365 Declare the syntax for the cap function.
class Cap

Syntaxcsuccf 33366 Declare the syntax for the successor function.
class Succ

Syntaxcfunpart 33367 Declare the syntax for the functional part functor.
class Funpart𝐹

Syntaxcfullfn 33368 Declare the syntax for the full function functor.
class FullFun𝐹

Syntaxcrestrict 33369 Declare the syntax for the restriction function.
class Restrict

Syntaxcub 33370 Declare the syntax for the upper bound relationship functor.
class UB𝑅

Syntaxclb 33371 Declare the syntax for the lower bound relationship functor.
class LB𝑅

Definitiondf-txp 33372 Define the tail cross of two classes. Membership in this class is defined by txpss3v 33396 and brtxp 33398. (Contributed by Scott Fenton, 31-Mar-2012.)
(𝐴𝐵) = (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵))

Definitiondf-pprod 33373 Define the parallel product of two classes. Membership in this class is defined by pprodss4v 33402 and brpprod 33403. (Contributed by Scott Fenton, 11-Apr-2014.)
pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V))))

Definitiondf-sset 33374 Define the subset class. For the value, see brsset 33407. (Contributed by Scott Fenton, 31-Mar-2012.)
SSet = ((V × V) ∖ ran ( E ⊗ (V ∖ E )))

Definitiondf-trans 33375 Define the class of all transitive sets. (Contributed by Scott Fenton, 31-Mar-2012.)
Trans = (V ∖ ran (( E ∘ E ) ∖ E ))

Definitiondf-bigcup 33376 Define the Bigcup function, which, per fvbigcup 33420, carries a set to its union. (Contributed by Scott Fenton, 11-Apr-2012.)
Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))

Definitiondf-fix 33377 Define the class of all fixpoints of a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)
Fix 𝐴 = dom (𝐴 ∩ I )

Definitiondf-limits 33378 Define the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.)
Limits = ((On ∩ Fix Bigcup ) ∖ {∅})

Definitiondf-funs 33379 Define the class of all functions. See elfuns 33433 for membership. (Contributed by Scott Fenton, 18-Feb-2013.)
Funs = (𝒫 (V × V) ∖ Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘ E )))

Definitiondf-singleton 33380 Define the singleton function. See brsingle 33435 for its value. (Contributed by Scott Fenton, 4-Apr-2014.)
Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V)))

Definitiondf-singles 33381 Define the class of all singletons. See elsingles 33436 for membership. (Contributed by Scott Fenton, 19-Feb-2013.)
Singletons = ran Singleton

Definitiondf-image 33382 Define the image functor. This function takes a set 𝐴 to a function 𝑥 ↦ (𝐴𝑥), providing that the latter exists. See imageval 33448 for the derivation. (Contributed by Scott Fenton, 27-Mar-2014.)
Image𝐴 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ 𝐴) ⊗ V)))

Definitiondf-cart 33383 Define the cartesian product function. See brcart 33450 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
Cart = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (pprod( E , E ) ⊗ V)))

Definitiondf-img 33384 Define the image function. See brimg 33455 for its value. (Contributed by Scott Fenton, 12-Apr-2014.)
Img = (Image((2nd ∘ 1st ) ↾ (1st ↾ (V × V))) ∘ Cart)

Definitiondf-domain 33385 Define the domain function. See brdomain 33451 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
Domain = Image(1st ↾ (V × V))

Definitiondf-range 33386 Define the range function. See brrange 33452 for its value. (Contributed by Scott Fenton, 11-Apr-2014.)
Range = Image(2nd ↾ (V × V))

Definitiondf-cup 33387 Define the little cup function. See brcup 33457 for its value. (Contributed by Scott Fenton, 14-Apr-2014.)
Cup = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((1st ∘ E ) ∪ (2nd ∘ E )) ⊗ V)))

Definitiondf-cap 33388 Define the little cap function. See brcap 33458 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
Cap = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((1st ∘ E ) ∩ (2nd ∘ E )) ⊗ V)))

Definitiondf-restrict 33389 Define the restriction function. See brrestrict 33467 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
Restrict = (Cap ∘ (1st ⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st )))))

Definitiondf-succf 33390 Define the successor function. See brsuccf 33459 for its value. (Contributed by Scott Fenton, 14-Apr-2014.)
Succ = (Cup ∘ ( I ⊗ Singleton))

Definitiondf-apply 33391 Define the application function. See brapply 33456 for its value. (Contributed by Scott Fenton, 12-Apr-2014.)
Apply = (( Bigcup Bigcup ) ∘ (((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton))))

Definitiondf-funpart 33392 Define the functional part of a class 𝐹. This is the maximal part of 𝐹 that is a function. See funpartfun 33461 and funpartfv 33463 for the meaning of this statement. (Contributed by Scott Fenton, 16-Apr-2014.)
Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons )))

Definitiondf-fullfun 33393 Define the full function over 𝐹. This is a function with domain V that always agrees with 𝐹 for its value. (Contributed by Scott Fenton, 17-Apr-2014.)
FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) × {∅}))

Definitiondf-ub 33394 Define the upper bound relationship functor. See brub 33472 for value. (Contributed by Scott Fenton, 3-May-2018.)
UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ E ))

Definitiondf-lb 33395 Define the lower bound relationship functor. See brlb 33473 for value. (Contributed by Scott Fenton, 3-May-2018.)
LB𝑅 = UB𝑅

Theoremtxpss3v 33396 A tail Cartesian product is a subset of the class of ordered triples. (Contributed by Scott Fenton, 31-Mar-2012.)
(𝐴𝐵) ⊆ (V × (V × V))

Theoremtxprel 33397 A tail Cartesian product is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
Rel (𝐴𝐵)

Theorembrtxp 33398 Characterize a ternary relation over a tail Cartesian product. Together with txpss3v 33396, this completely defines membership in a tail cross. (Contributed by Scott Fenton, 31-Mar-2012.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
𝑋 ∈ V    &   𝑌 ∈ V    &   𝑍 ∈ V       (𝑋(𝐴𝐵)⟨𝑌, 𝑍⟩ ↔ (𝑋𝐴𝑌𝑋𝐵𝑍))

Theorembrtxp2 33399* The binary relation over a tail cross when the second argument is not an ordered pair. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 3-May-2015.)
𝐴 ∈ V       (𝐴(𝑅𝑆)𝐵 ↔ ∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥𝐴𝑆𝑦))

Theoremdfpprod2 33400 Expanded definition of parallel product. (Contributed by Scott Fenton, 3-May-2014.)
pprod(𝐴, 𝐵) = (((1st ↾ (V × V)) ∘ (𝐴 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝐵 ∘ (2nd ↾ (V × V)))))

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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 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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45259
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