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Theorem List for Metamath Proof Explorer - 35701-35800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempellfundgt1 35701 Weak lower bound on the Pell fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.)
 |-  ( D  e.  ( NN  \NN )  ->  1  <  (PellFund `  D ) )
 
Theorempellfundlb 35702 A nontrivial first quadrant solution is at least as large as the fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Proof shortened by AV, 15-Sep-2020.)
 |-  (
 ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (PellFund `  D )  <_  A )
 
Theorempellfundglb 35703* If a real is larger than the fundamental solution, there is a nontrivial solution less than it. (Contributed by Stefan O'Rear, 18-Sep-2014.)
 |-  (
 ( D  e.  ( NN  \NN )  /\  A  e.  RR  /\  (PellFund `  D )  <  A )  ->  E. x  e.  (Pell1QR `  D )
 ( (PellFund `  D )  <_  x  /\  x  <  A ) )
 
Theorempellfundex 35704 The fundamental solution as an infimum is itself a solution, showing that the solution set is discrete.

Since the fundamental solution is an infimum, there must be an element ge to Fund and lt 2*Fund. If this element is equal to the fundamental solution we're done, otherwise use the infimum again to find another element which must be ge Fund and lt the first element; their ratio is a group element in (1,2), contradicting pell14qrgapw 35692. (Contributed by Stefan O'Rear, 18-Sep-2014.)

 |-  ( D  e.  ( NN  \NN )  ->  (PellFund `  D )  e.  (Pell1QR `  D )
 )
 
Theorempellfund14gap 35705 There are no solutions between 1 and the fundamental solution. (Contributed by Stefan O'Rear, 18-Sep-2014.)
 |-  (
 ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )  /\  (
 1  <_  A  /\  A  <  (PellFund `  D )
 ) )  ->  A  =  1 )
 
Theorempellfundrp 35706 The fundamental Pell solution is a positive real. (Contributed by Stefan O'Rear, 19-Sep-2014.)
 |-  ( D  e.  ( NN  \NN )  ->  (PellFund `  D )  e.  RR+ )
 
Theorempellfundne1 35707 The fundamental Pell solution is never 1. (Contributed by Stefan O'Rear, 19-Sep-2014.)
 |-  ( D  e.  ( NN  \NN )  ->  (PellFund `  D )  =/=  1 )
 
21.23.25  Logarithm laws generalized to an arbitrary base

Section should be obsolete because its contents are covered by section "Logarithms to an arbitrary base" now.

 
Theoremreglogcl 35708 General logarithm is a real number. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbcl 23708 instead.
 |-  (
 ( A  e.  RR+  /\  B  e.  RR+  /\  B  =/=  1 )  ->  (
 ( log `  A )  /  ( log `  B ) )  e.  RR )
 
Theoremreglogltb 35709 General logarithm preserves "less than". (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logblt 23719 instead.
 |-  (
 ( ( A  e.  RR+  /\  B  e.  RR+ )  /\  ( C  e.  RR+  /\  1  <  C ) )  ->  ( A  <  B  <->  ( ( log `  A )  /  ( log `  C ) )  <  ( ( log `  B )  /  ( log `  C ) ) ) )
 
Theoremreglogleb 35710 General logarithm preserves  <_. (Contributed by Stefan O'Rear, 19-Oct-2014.) (New usage is discouraged.) Use logbleb 23718 instead.
 |-  (
 ( ( A  e.  RR+  /\  B  e.  RR+ )  /\  ( C  e.  RR+  /\  1  <  C ) )  ->  ( A  <_  B  <->  ( ( log `  A )  /  ( log `  C ) ) 
 <_  ( ( log `  B )  /  ( log `  C ) ) ) )
 
Theoremreglogmul 35711 Multiplication law for general log. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbmul 23712 instead.
 |-  (
 ( A  e.  RR+  /\  B  e.  RR+  /\  ( C  e.  RR+  /\  C  =/=  1 ) )  ->  ( ( log `  ( A  x.  B ) ) 
 /  ( log `  C ) )  =  (
 ( ( log `  A )  /  ( log `  C ) )  +  (
 ( log `  B )  /  ( log `  C ) ) ) )
 
Theoremreglogexp 35712 Power law for general log. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbzexp 23711 instead.
 |-  (
 ( A  e.  RR+  /\  N  e.  ZZ  /\  ( C  e.  RR+  /\  C  =/=  1 ) )  ->  ( ( log `  ( A ^ N ) ) 
 /  ( log `  C ) )  =  ( N  x.  ( ( log `  A )  /  ( log `  C ) ) ) )
 
Theoremreglogbas 35713 General log of the base is 1. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logbid1 23703 instead.
 |-  (
 ( C  e.  RR+  /\  C  =/=  1 ) 
 ->  ( ( log `  C )  /  ( log `  C ) )  =  1
 )
 
Theoremreglog1 35714 General log of 1 is 0. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logb1 23704 instead.
 |-  (
 ( C  e.  RR+  /\  C  =/=  1 ) 
 ->  ( ( log `  1
 )  /  ( log `  C ) )  =  0 )
 
Theoremreglogexpbas 35715 General log of a power of the base is the exponent. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbexp 23715 instead.
 |-  (
 ( N  e.  ZZ  /\  ( C  e.  RR+  /\  C  =/=  1 ) )  ->  ( ( log `  ( C ^ N ) )  /  ( log `  C )
 )  =  N )
 
21.23.26  Pell equations 4: the positive solution group is infinite cyclic
 
Theorempellfund14 35716* Every positive Pell solution is a power of the fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.)
 |-  (
 ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D ) )  ->  E. x  e.  ZZ  A  =  ( (PellFund `  D ) ^ x ) )
 
Theorempellfund14b 35717* The positive Pell solutions are precisely the integer powers of the fundamental solution. To get the general solution set (which we will not be using), throw in a copy of Z/2Z. (Contributed by Stefan O'Rear, 19-Sep-2014.)
 |-  ( D  e.  ( NN  \NN )  ->  ( A  e.  (Pell14QR `  D )  <->  E. x  e.  ZZ  A  =  ( (PellFund `  D ) ^ x ) ) )
 
21.23.27  X and Y sequences 1: Definition and recurrence laws
 
Syntaxcrmx 35718 Extend class notation to include the Robertson-Matiyasevich X sequence.
 class Xrm
 
Syntaxcrmy 35719 Extend class notation to include the Robertson-Matiyasevich Y sequence.
 class Yrm
 
Definitiondf-rmx 35720* Define the X sequence as the rational part of some solution of a special Pell equation. See frmx 35731 and rmxyval 35733 for a more useful but non-eliminable definition. (Contributed by Stefan O'Rear, 21-Sep-2014.)
 |- Xrm  =  ( a  e.  ( ZZ>= `  2 ) ,  n  e.  ZZ  |->  ( 1st `  ( `' ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b
 )  +  ( ( sqr `  ( (
 a ^ 2 )  -  1 ) )  x.  ( 2nd `  b
 ) ) ) ) `
  ( ( a  +  ( sqr `  (
 ( a ^ 2
 )  -  1 ) ) ) ^ n ) ) ) )
 
Definitiondf-rmy 35721* Define the X sequence as the irrational part of some solution of a special Pell equation. See frmy 35732 and rmxyval 35733 for a more useful but non-eliminable definition. (Contributed by Stefan O'Rear, 21-Sep-2014.)
 |- Yrm  =  ( a  e.  ( ZZ>= `  2 ) ,  n  e.  ZZ  |->  ( 2nd `  ( `' ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b
 )  +  ( ( sqr `  ( (
 a ^ 2 )  -  1 ) )  x.  ( 2nd `  b
 ) ) ) ) `
  ( ( a  +  ( sqr `  (
 ( a ^ 2
 )  -  1 ) ) ) ^ n ) ) ) )
 
Theoremrmxfval 35722* Value of the X sequence. Not used after rmxyval 35733 is proved. (Contributed by Stefan O'Rear, 21-Sep-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  ZZ )  ->  ( A Xrm  N )  =  ( 1st `  ( `' ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b
 )  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
 ) ) ) ) `
  ( ( A  +  ( sqr `  (
 ( A ^ 2
 )  -  1 ) ) ) ^ N ) ) ) )
 
Theoremrmyfval 35723* Value of the Y sequence. Not used after rmxyval 35733 is proved. (Contributed by Stefan O'Rear, 21-Sep-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  ZZ )  ->  ( A Yrm  N )  =  ( 2nd `  ( `' ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b
 )  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
 ) ) ) ) `
  ( ( A  +  ( sqr `  (
 ( A ^ 2
 )  -  1 ) ) ) ^ N ) ) ) )
 
Theoremrmspecsqrtnq 35724 The discriminant used to define the X and Y sequences has an irrational square root. (Contributed by Stefan O'Rear, 21-Sep-2014.)
 |-  ( A  e.  ( ZZ>= `  2 )  ->  ( sqr `  ( ( A ^
 2 )  -  1
 ) )  e.  ( CC  \  QQ ) )
 
Theoremrmspecnonsq 35725 The discriminant used to define the X and Y sequences is a nonsquare positive integer and thus a valid Pell equation discriminant. (Contributed by Stefan O'Rear, 21-Sep-2014.)
 |-  ( A  e.  ( ZZ>= `  2 )  ->  ( ( A ^ 2 )  -  1 )  e.  ( NN  \NN ) )
 
Theoremqirropth 35726 This lemma implements the concept of "equate rational and irrational parts", used to prove many arithmetical properties of the X and Y sequences. (Contributed by Stefan O'Rear, 21-Sep-2014.)
 |-  (
 ( A  e.  ( CC  \  QQ )  /\  ( B  e.  QQ  /\  C  e.  QQ )  /\  ( D  e.  QQ  /\  E  e.  QQ )
 )  ->  ( ( B  +  ( A  x.  C ) )  =  ( D  +  ( A  x.  E ) )  <-> 
 ( B  =  D  /\  C  =  E ) ) )
 
Theoremrmspecfund 35727 The base of exponent used to define the X and Y sequences is the fundamental solution of the corresponding Pell equation. (Contributed by Stefan O'Rear, 21-Sep-2014.)
 |-  ( A  e.  ( ZZ>= `  2 )  ->  (PellFund `  (
 ( A ^ 2
 )  -  1 ) )  =  ( A  +  ( sqr `  (
 ( A ^ 2
 )  -  1 ) ) ) )
 
Theoremrmxyelqirr 35728* The solutions used to construct the X and Y sequences are quadratic irrationals. (Contributed by Stefan O'Rear, 21-Sep-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  ZZ )  ->  ( ( A  +  ( sqr `  ( ( A ^ 2 )  -  1 ) ) ) ^ N )  e. 
 { a  |  E. c  e.  NN0  E. d  e.  ZZ  a  =  ( c  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  d ) ) }
 )
 
Theoremrmxypairf1o 35729* The function used to extract rational and irrational parts in df-rmx 35720 and df-rmy 35721 in fact achieves a one-to-one mapping from the quadratic irrationals to pairs of integers. (Contributed by Stefan O'Rear, 21-Sep-2014.)
 |-  ( A  e.  ( ZZ>= `  2 )  ->  ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b )  +  (
 ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
 ) ) ) ) : ( NN0  X.  ZZ )
 -1-1-onto-> { a  |  E. c  e.  NN0  E. d  e. 
 ZZ  a  =  ( c  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  d ) ) }
 )
 
Theoremrmxyelxp 35730* Lemma for frmx 35731 and frmy 35732. (Contributed by Stefan O'Rear, 22-Sep-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  ZZ )  ->  ( `' ( b  e.  ( NN0  X.  ZZ )  |->  ( ( 1st `  b )  +  (
 ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  ( 2nd `  b
 ) ) ) ) `
  ( ( A  +  ( sqr `  (
 ( A ^ 2
 )  -  1 ) ) ) ^ N ) )  e.  ( NN0  X.  ZZ ) )
 
Theoremfrmx 35731 The X sequence is a nonnegative integer. See rmxnn 35771 for a strengthening. (Contributed by Stefan O'Rear, 22-Sep-2014.)
 |- Xrm  : ( ( ZZ>= `  2 )  X.  ZZ ) --> NN0
 
Theoremfrmy 35732 The Y sequence is an integer. (Contributed by Stefan O'Rear, 22-Sep-2014.)
 |- Yrm  : ( ( ZZ>= `  2 )  X.  ZZ ) --> ZZ
 
Theoremrmxyval 35733 Main definition of the X and Y sequences. Compare definition 2.3 of [JonesMatijasevic] p. 694. (Contributed by Stefan O'Rear, 19-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  ZZ )  ->  ( ( A Xrm  N )  +  ( ( sqr `  ( ( A ^
 2 )  -  1
 ) )  x.  ( A Yrm 
 N ) ) )  =  ( ( A  +  ( sqr `  (
 ( A ^ 2
 )  -  1 ) ) ) ^ N ) )
 
Theoremrmspecpos 35734 The discriminant used to define the X and Y sequences is a positive real. (Contributed by Stefan O'Rear, 22-Sep-2014.)
 |-  ( A  e.  ( ZZ>= `  2 )  ->  ( ( A ^ 2 )  -  1 )  e.  RR+ )
 
Theoremrmxycomplete 35735* The X and Y sequences taken together enumerate all solutions to the corresponding Pell equation in the right half-plane. This is Metamath 100 proof #39. (Contributed by Stefan O'Rear, 22-Sep-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  X  e.  NN0  /\  Y  e.  ZZ )  ->  (
 ( ( X ^
 2 )  -  (
 ( ( A ^
 2 )  -  1
 )  x.  ( Y ^ 2 ) ) )  =  1  <->  E. n  e.  ZZ  ( X  =  ( A Xrm 
 n )  /\  Y  =  ( A Yrm  n ) ) ) )
 
Theoremrmxynorm 35736 The X and Y sequences define a solution to the corresponding Pell equation. (Contributed by Stefan O'Rear, 22-Sep-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  ZZ )  ->  ( ( ( A Xrm  N ) ^ 2 )  -  ( ( ( A ^ 2 )  -  1 )  x.  ( ( A Yrm  N ) ^ 2 ) ) )  =  1 )
 
Theoremrmbaserp 35737 The base of exponentiation for the X and Y sequences is a positive real. (Contributed by Stefan O'Rear, 22-Sep-2014.)
 |-  ( A  e.  ( ZZ>= `  2 )  ->  ( A  +  ( sqr `  (
 ( A ^ 2
 )  -  1 ) ) )  e.  RR+ )
 
Theoremrmxyneg 35738 Negation law for X and Y sequences. JonesMatijasevic is inconsistent on whether the X and Y sequences have domain  NN0 or  ZZ; we use  ZZ consistently to avoid the need for a separate subtraction law. (Contributed by Stefan O'Rear, 22-Sep-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  ZZ )  ->  ( ( A Xrm  -u N )  =  ( A Xrm  N )  /\  ( A Yrm  -u N )  =  -u ( A Yrm  N ) ) )
 
Theoremrmxyadd 35739 Addition formula for X and Y sequences. See rmxadd 35745 and rmyadd 35749 for most uses. (Contributed by Stefan O'Rear, 22-Sep-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
 ( A Xrm  ( M  +  N ) )  =  ( ( ( A Xrm  M )  x.  ( A Xrm  N ) )  +  (
 ( ( A ^
 2 )  -  1
 )  x.  ( ( A Yrm  M )  x.  ( A Yrm 
 N ) ) ) )  /\  ( A Yrm  ( M  +  N ) )  =  ( ( ( A Yrm  M )  x.  ( A Xrm  N ) )  +  ( ( A Xrm  M )  x.  ( A Yrm  N ) ) ) ) )
 
Theoremrmxy1 35740 Value of the X and Y sequences at 1. (Contributed by Stefan O'Rear, 22-Sep-2014.)
 |-  ( A  e.  ( ZZ>= `  2 )  ->  ( ( A Xrm  1 )  =  A  /\  ( A Yrm  1 )  =  1 ) )
 
Theoremrmxy0 35741 Value of the X and Y sequences at 0. (Contributed by Stefan O'Rear, 22-Sep-2014.)
 |-  ( A  e.  ( ZZ>= `  2 )  ->  ( ( A Xrm  0 )  =  1 
 /\  ( A Yrm  0 )  =  0 ) )
 
Theoremrmxneg 35742 Negation law (even function) for the X sequence. The method of proof used for the previous four theorems rmxyneg 35738, rmxyadd 35739, rmxy0 35741, and rmxy1 35740 via qirropth 35726 results in two theorems at once, but typical use requires only one, so this group of theorems serves to separate the cases. (Contributed by Stefan O'Rear, 22-Sep-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  ZZ )  ->  ( A Xrm  -u N )  =  ( A Xrm  N ) )
 
Theoremrmx0 35743 Value of X sequence at 0. Part 1 of equation 2.11 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.)
 |-  ( A  e.  ( ZZ>= `  2 )  ->  ( A Xrm  0 )  =  1 )
 
Theoremrmx1 35744 Value of X sequence at 1. Part 2 of equation 2.11 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.)
 |-  ( A  e.  ( ZZ>= `  2 )  ->  ( A Xrm  1 )  =  A )
 
Theoremrmxadd 35745 Addition formula for X sequence. Equation 2.7 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A Xrm 
 ( M  +  N ) )  =  (
 ( ( A Xrm  M )  x.  ( A Xrm  N ) )  +  ( ( ( A ^ 2
 )  -  1 )  x.  ( ( A Yrm  M )  x.  ( A Yrm  N ) ) ) ) )
 
Theoremrmyneg 35746 Negation formula for Y sequence (odd function). (Contributed by Stefan O'Rear, 22-Sep-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  ZZ )  ->  ( A Yrm  -u N )  =  -u ( A Yrm  N ) )
 
Theoremrmy0 35747 Value of Y sequence at 0. Part 1 of equation 2.12 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.)
 |-  ( A  e.  ( ZZ>= `  2 )  ->  ( A Yrm  0 )  =  0 )
 
Theoremrmy1 35748 Value of Y sequence at 1. Part 2 of equation 2.12 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.)
 |-  ( A  e.  ( ZZ>= `  2 )  ->  ( A Yrm  1 )  =  1 )
 
Theoremrmyadd 35749 Addition formula for Y sequence. Equation 2.8 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( A Yrm 
 ( M  +  N ) )  =  (
 ( ( A Yrm  M )  x.  ( A Xrm  N ) )  +  ( ( A Xrm  M )  x.  ( A Yrm 
 N ) ) ) )
 
Theoremrmxp1 35750 Special addition-of-1 formula for X sequence. Part 1 of equation 2.9 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 19-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  ZZ )  ->  ( A Xrm  ( N  +  1 ) )  =  ( ( ( A Xrm  N )  x.  A )  +  ( ( ( A ^ 2 )  -  1 )  x.  ( A Yrm  N ) ) ) )
 
Theoremrmyp1 35751 Special addition of 1 formula for Y sequence. Part 2 of equation 2.9 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 24-Sep-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  ZZ )  ->  ( A Yrm  ( N  +  1 ) )  =  ( ( ( A Yrm  N )  x.  A )  +  ( A Xrm  N ) ) )
 
Theoremrmxm1 35752 Subtraction of 1 formula for X sequence. Part 1 of equation 2.10 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 14-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  ZZ )  ->  ( A Xrm  ( N  -  1 ) )  =  ( ( A  x.  ( A Xrm  N ) )  -  ( ( ( A ^ 2 )  -  1 )  x.  ( A Yrm  N ) ) ) )
 
Theoremrmym1 35753 Subtraction of 1 formula for Y sequence. Part 2 of equation 2.10 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 19-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  ZZ )  ->  ( A Yrm  ( N  -  1 ) )  =  ( ( ( A Yrm  N )  x.  A )  -  ( A Xrm  N ) ) )
 
Theoremrmxluc 35754 The X sequence is a Lucas (second-order integer recurrence) sequence. Part 3 of equation 2.11 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 14-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  ZZ )  ->  ( A Xrm  ( N  +  1 ) )  =  ( ( ( 2  x.  A )  x.  ( A Xrm  N ) )  -  ( A Xrm  ( N  -  1 ) ) ) )
 
Theoremrmyluc 35755 The Y sequence is a Lucas sequence, definable via this second-order recurrence with rmy0 35747 and rmy1 35748. Part 3 of equation 2.12 of [JonesMatijasevic] p. 695. JonesMatijasevic uses this theorem to redefine the X and Y sequences to have domain  ( ZZ  X.  ZZ ), which simplifies some later theorems. It may shorten the derivation to use this as our initial definition. Incidentally, the X sequence satisfies the exact same recurrence. (Contributed by Stefan O'Rear, 1-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  ZZ )  ->  ( A Yrm  ( N  +  1 ) )  =  ( ( 2  x.  ( ( A Yrm  N )  x.  A ) )  -  ( A Yrm  ( N  -  1 ) ) ) )
 
Theoremrmyluc2 35756 Lucas sequence property of Y with better output ordering. (Contributed by Stefan O'Rear, 16-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  ZZ )  ->  ( A Yrm  ( N  +  1 ) )  =  ( ( ( 2  x.  A )  x.  ( A Yrm  N ) )  -  ( A Yrm  ( N  -  1 ) ) ) )
 
Theoremrmxdbl 35757 "Double-angle formula" for X-values. Equation 2.13 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 2-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  ZZ )  ->  ( A Xrm  ( 2  x.  N ) )  =  ( ( 2  x.  ( ( A Xrm  N ) ^ 2 ) )  -  1 ) )
 
Theoremrmydbl 35758 "Double-angle formula" for Y-values. Equation 2.14 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 2-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  ZZ )  ->  ( A Yrm  ( 2  x.  N ) )  =  ( ( 2  x.  ( A Xrm  N ) )  x.  ( A Yrm  N ) ) )
 
21.23.28  Ordering and induction lemmas for the integers
 
Theoremmonotuz 35759* A function defined on an upper set of integers which increases at every adjacent pair is globally strictly monotonic by induction. (Contributed by Stefan O'Rear, 24-Sep-2014.)
 |-  (
 ( ph  /\  y  e.  H )  ->  F  <  G )   &    |-  ( ( ph  /\  x  e.  H ) 
 ->  C  e.  RR )   &    |-  H  =  ( ZZ>= `  I )   &    |-  ( x  =  ( y  +  1 )  ->  C  =  G )   &    |-  ( x  =  y  ->  C  =  F )   &    |-  ( x  =  A  ->  C  =  D )   &    |-  ( x  =  B  ->  C  =  E )   =>    |-  ( ( ph  /\  ( A  e.  H  /\  B  e.  H ) )  ->  ( A  <  B  <->  D  <  E ) )
 
Theoremmonotoddzzfi 35760* A function which is odd and monotonic on  NN0 is monotonic on  ZZ. This proof is far too long. (Contributed by Stefan O'Rear, 25-Sep-2014.)
 |-  (
 ( ph  /\  x  e. 
 ZZ )  ->  ( F `  x )  e. 
 RR )   &    |-  ( ( ph  /\  x  e.  ZZ )  ->  ( F `  -u x )  =  -u ( F `
  x ) )   &    |-  ( ( ph  /\  x  e.  NN0  /\  y  e.  NN0 )  ->  ( x  <  y  ->  ( F `  x )  <  ( F `  y ) ) )   =>    |-  ( ( ph  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  <  B  <->  ( F `  A )  <  ( F `
  B ) ) )
 
Theoremmonotoddzz 35761* A function (given implicitly) which is odd and monotonic on  NN0 is monotonic on  ZZ. This proof is far too long. (Contributed by Stefan O'Rear, 25-Sep-2014.)
 |-  (
 ( ph  /\  x  e. 
 NN0  /\  y  e.  NN0 )  ->  ( x  < 
 y  ->  E  <  F ) )   &    |-  ( ( ph  /\  x  e.  ZZ )  ->  E  e.  RR )   &    |-  (
 ( ph  /\  y  e. 
 ZZ )  ->  G  =  -u F )   &    |-  ( x  =  A  ->  E  =  C )   &    |-  ( x  =  B  ->  E  =  D )   &    |-  ( x  =  y  ->  E  =  F )   &    |-  ( x  =  -u y  ->  E  =  G )   =>    |-  (
 ( ph  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  <  B  <->  C  <  D ) )
 
Theoremoddcomabszz 35762* An odd function which takes nonnegative values on nonnegative arguments commutes with  abs. (Contributed by Stefan O'Rear, 26-Sep-2014.)
 |-  (
 ( ph  /\  x  e. 
 ZZ )  ->  A  e.  RR )   &    |-  ( ( ph  /\  x  e.  ZZ  /\  0  <_  x )  -> 
 0  <_  A )   &    |-  (
 ( ph  /\  y  e. 
 ZZ )  ->  C  =  -u B )   &    |-  ( x  =  y  ->  A  =  B )   &    |-  ( x  =  -u y  ->  A  =  C )   &    |-  ( x  =  D  ->  A  =  E )   &    |-  ( x  =  ( abs `  D )  ->  A  =  F )   =>    |-  ( ( ph  /\  D  e.  ZZ )  ->  ( abs `  E )  =  F )
 
Theorem2nn0ind 35763* Induction on nonnegative integers with two base cases, for use with Lucas-type sequences. (Contributed by Stefan O'Rear, 1-Oct-2014.)
 |-  ps   &    |-  ch   &    |-  (
 y  e.  NN  ->  ( ( th  /\  ta )  ->  et ) )   &    |-  ( x  =  0  ->  ( ph  <->  ps ) )   &    |-  ( x  =  1  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  ( y  -  1 )  ->  ( ph  <->  th ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ta ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  et ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  rh ) )   =>    |-  ( A  e.  NN0 
 ->  rh )
 
Theoremzindbi 35764* Inductively transfer a property to the integers if it holds for zero and passes between adjacent integers in either direction. (Contributed by Stefan O'Rear, 1-Oct-2014.)
 |-  (
 y  e.  ZZ  ->  ( ps  <->  ch ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  ch ) )   &    |-  ( x  =  0  ->  (
 ph 
 <-> 
 th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   =>    |-  ( A  e.  ZZ  ->  ( th  <->  ta ) )
 
Theoremexpmordi 35765 Mantissa ordering relationship for exponentiation. (Contributed by Stefan O'Rear, 16-Oct-2014.)
 |-  (
 ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  A  <  B )  /\  N  e.  NN )  ->  ( A ^ N )  <  ( B ^ N ) )
 
Theoremrpexpmord 35766 Mantissa ordering relationship for exponentiation of positive reals. (Contributed by Stefan O'Rear, 16-Oct-2014.)
 |-  (
 ( N  e.  NN  /\  A  e.  RR+  /\  B  e.  RR+ )  ->  ( A  <  B  <->  ( A ^ N )  <  ( B ^ N ) ) )
 
21.23.29  X and Y sequences 2: Order properties
 
Theoremrmxypos 35767 For all nonnegative indices, X is positive and Y is nonnegative. (Contributed by Stefan O'Rear, 24-Sep-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  NN0 )  ->  ( 0  <  ( A Xrm 
 N )  /\  0  <_  ( A Yrm  N ) ) )
 
Theoremltrmynn0 35768 The Y-sequence is strictly monotonic on  NN0. Strengthened by ltrmy 35772. (Contributed by Stefan O'Rear, 24-Sep-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  M  e.  NN0  /\  N  e.  NN0 )  ->  ( M  <  N  <->  ( A Yrm  M )  <  ( A Yrm  N ) ) )
 
Theoremltrmxnn0 35769 The X-sequence is strictly monotonic on  NN0. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  M  e.  NN0  /\  N  e.  NN0 )  ->  ( M  <  N  <->  ( A Xrm  M )  <  ( A Xrm  N ) ) )
 
Theoremlermxnn0 35770 The X-sequence is monotonic on 
NN0. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  M  e.  NN0  /\  N  e.  NN0 )  ->  ( M  <_  N  <->  ( A Xrm  M ) 
 <_  ( A Xrm  N ) ) )
 
Theoremrmxnn 35771 The X-sequence is defined to range over  NN0 but never actually takes the value 0. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  ZZ )  ->  ( A Xrm  N )  e. 
 NN )
 
Theoremltrmy 35772 The Y-sequence is strictly monotonic over  ZZ. (Contributed by Stefan O'Rear, 25-Sep-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <  N  <->  ( A Yrm  M )  <  ( A Yrm  N ) ) )
 
Theoremrmyeq0 35773 Y is zero only at zero. (Contributed by Stefan O'Rear, 26-Sep-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  ZZ )  ->  ( N  =  0  <-> 
 ( A Yrm  N )  =  0 ) )
 
Theoremrmyeq 35774 Y is one-to-one. (Contributed by Stefan O'Rear, 3-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  =  N  <->  ( A Yrm  M )  =  ( A Yrm  N ) ) )
 
Theoremlermy 35775 Y is monotonic (non-strict). (Contributed by Stefan O'Rear, 3-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <_  N  <->  ( A Yrm  M ) 
 <_  ( A Yrm  N ) ) )
 
Theoremrmynn 35776 Yrm is positive for positive arguments. (Contributed by Stefan O'Rear, 16-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  NN )  ->  ( A Yrm  N )  e. 
 NN )
 
Theoremrmynn0 35777 Yrm is nonnegative for nonnegative arguments. (Contributed by Stefan O'Rear, 16-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  NN0 )  ->  ( A Yrm  N )  e. 
 NN0 )
 
Theoremrmyabs 35778 Yrm commutes with  abs. (Contributed by Stefan O'Rear, 26-Sep-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  B  e.  ZZ )  ->  ( abs `  ( A Yrm 
 B ) )  =  ( A Yrm  ( abs `  B ) ) )
 
Theoremjm2.24nn 35779 X(n) is strictly greater than Y(n) + Y(n-1). Lemma 2.24 of [JonesMatijasevic] p. 697 restricted to  NN. (Contributed by Stefan O'Rear, 3-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  NN )  ->  ( ( A Yrm  ( N  -  1 ) )  +  ( A Yrm  N ) )  <  ( A Xrm  N ) )
 
Theoremjm2.17a 35780 First half of lemma 2.17 of [JonesMatijasevic] p. 696. (Contributed by Stefan O'Rear, 14-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  NN0 )  ->  ( ( ( 2  x.  A )  -  1 ) ^ N )  <_  ( A Yrm  ( N  +  1 ) ) )
 
Theoremjm2.17b 35781 Weak form of the second half of lemma 2.17 of [JonesMatijasevic] p. 696, allowing induction to start lower. (Contributed by Stefan O'Rear, 15-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  NN0 )  ->  ( A Yrm  ( N  +  1 ) )  <_  ( ( 2  x.  A ) ^ N ) )
 
Theoremjm2.17c 35782 Second half of lemma 2.17 of [JonesMatijasevic] p. 696. (Contributed by Stefan O'Rear, 15-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  NN )  ->  ( A Yrm  ( ( N  +  1 )  +  1 ) )  < 
 ( ( 2  x.  A ) ^ ( N  +  1 )
 ) )
 
Theoremjm2.24 35783 Lemma 2.24 of [JonesMatijasevic] p. 697 extended to  ZZ. Could be eliminated with a more careful proof of jm2.26lem3 35826. (Contributed by Stefan O'Rear, 3-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  ZZ )  ->  ( ( A Yrm  ( N  -  1 ) )  +  ( A Yrm  N ) )  <  ( A Xrm  N ) )
 
Theoremrmygeid 35784 Y(n) increases faster than n. Used implicitly without proof or comment in lemma 2.27 of [JonesMatijasevic] p. 697. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  (
 ( A  e.  ( ZZ>=
 `  2 )  /\  N  e.  NN0 )  ->  N  <_  ( A Yrm  N ) )
 
21.23.30  Congruential equations
 
Theoremcongtr 35785 A wff of the form  A  ||  ( B  -  C ) is interpreted as a congruential equation. This is similar to  ( B  mod  A
)  =  ( C  mod  A ), but is defined such that behavior is regular for zero and negative values of  A. To use this concept effectively, we need to show that congruential equations behave similarly to normal equations; first a transitivity law. Idea for the future: If there was a congruential equation symbol, it could incorporate type constraints, so that most of these would not need them. (Contributed by Stefan O'Rear, 1-Oct-2014.)
 |-  (
 ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( C  -  D ) ) )  ->  A  ||  ( B  -  D ) )
 
Theoremcongadd 35786 If two pairs of numbers are componentwise congruent, so are their sums. (Contributed by Stefan O'Rear, 1-Oct-2014.)
 |-  (
 ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( D  -  E ) ) ) 
 ->  A  ||  ( ( B  +  D )  -  ( C  +  E ) ) )
 
Theoremcongmul 35787 If two pairs of numbers are componentwise congruent, so are their products. (Contributed by Stefan O'Rear, 1-Oct-2014.)
 |-  (
 ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( D  -  E ) ) ) 
 ->  A  ||  ( ( B  x.  D )  -  ( C  x.  E ) ) )
 
Theoremcongsym 35788 Congruence mod  A is a symmetric/commutative relation. (Contributed by Stefan O'Rear, 1-Oct-2014.)
 |-  (
 ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  A  ||  ( B  -  C ) ) )  ->  A  ||  ( C  -  B ) )
 
Theoremcongneg 35789 If two integers are congruent mod 
A, so are their negatives. (Contributed by Stefan O'Rear, 1-Oct-2014.)
 |-  (
 ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  A  ||  ( B  -  C ) ) )  ->  A  ||  ( -u B  -  -u C ) )
 
Theoremcongsub 35790 If two pairs of numbers are componentwise congruent, so are their differences. (Contributed by Stefan O'Rear, 2-Oct-2014.)
 |-  (
 ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( D  e.  ZZ  /\  E  e.  ZZ )  /\  ( A  ||  ( B  -  C )  /\  A  ||  ( D  -  E ) ) ) 
 ->  A  ||  ( ( B  -  D )  -  ( C  -  E ) ) )
 
Theoremcongid 35791 Every integer is congruent to itself mod every base. (Contributed by Stefan O'Rear, 1-Oct-2014.)
 |-  (
 ( A  e.  ZZ  /\  B  e.  ZZ )  ->  A  ||  ( B  -  B ) )
 
Theoremmzpcong 35792* Polynomials commute with congruences. (Does this characterize them?) (Contributed by Stefan O'Rear, 5-Oct-2014.)
 |-  (
 ( F  e.  (mzPoly `  V )  /\  ( X  e.  ( ZZ  ^m  V )  /\  Y  e.  ( ZZ  ^m  V ) )  /\  ( N  e.  ZZ  /\  A. k  e.  V  N  ||  ( ( X `  k )  -  ( Y `  k ) ) ) )  ->  N  ||  ( ( F `  X )  -  ( F `  Y ) ) )
 
Theoremcongrep 35793* Every integer is congruent to some number in the fundamental domain. (Contributed by Stefan O'Rear, 2-Oct-2014.)
 |-  (
 ( A  e.  NN  /\  N  e.  ZZ )  ->  E. a  e.  (
 0 ... ( A  -  1 ) ) A 
 ||  ( a  -  N ) )
 
Theoremcongabseq 35794 If two integers are congruent, they are either equal or separated by at least the congruence base. (Contributed by Stefan O'Rear, 4-Oct-2014.)
 |-  (
 ( ( A  e.  NN  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  A  ||  ( B  -  C ) )  ->  ( ( abs `  ( B  -  C ) )  <  A  <->  B  =  C ) )
 
21.23.31  Alternating congruential equations
 
Theoremacongid 35795 A wff like that in this theorem will be known as an "alternating congruence". A special symbol might be considered if more uses come up. They have many of the same properties as normal congruences, starting with reflexivity.

JonesMatijasevic uses "a ≡ ± b (mod c)" for this construction. The disjunction of divisibility constraints seems to adequately capture the concept, but it's rather verbose and somewhat inelegant. Use of an explicit equivalence relation might also work. (Contributed by Stefan O'Rear, 2-Oct-2014.)

 |-  (
 ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  ||  ( B  -  B )  \/  A  ||  ( B  -  -u B ) ) )
 
Theoremacongsym 35796 Symmetry of alternating congruence. (Contributed by Stefan O'Rear, 2-Oct-2014.)
 |-  (
 ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( A  ||  ( B  -  C )  \/  A  ||  ( B  -  -u C ) ) )  ->  ( A  ||  ( C  -  B )  \/  A  ||  ( C  -  -u B ) ) )
 
Theoremacongneg2 35797 Negate right side of alternating congruence. Makes essential use of the "alternating" part. (Contributed by Stefan O'Rear, 3-Oct-2014.)
 |-  (
 ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  /\  ( A  ||  ( B  -  -u C )  \/  A  ||  ( B  -  -u -u C ) ) )  ->  ( A  ||  ( B  -  C )  \/  A  ||  ( B  -  -u C ) ) )
 
Theoremacongtr 35798 Transitivity of alternating congruence. (Contributed by Stefan O'Rear, 2-Oct-2014.)
 |-  (
 ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  ZZ  /\  D  e.  ZZ )  /\  ( ( A 
 ||  ( B  -  C )  \/  A  ||  ( B  -  -u C ) )  /\  ( A 
 ||  ( C  -  D )  \/  A  ||  ( C  -  -u D ) ) ) ) 
 ->  ( A  ||  ( B  -  D )  \/  A  ||  ( B  -  -u D ) ) )
 
Theoremacongeq12d 35799 Substitution deduction for alternating congruence. (Contributed by Stefan O'Rear, 3-Oct-2014.)
 |-  ( ph  ->  B  =  C )   &    |-  ( ph  ->  D  =  E )   =>    |-  ( ph  ->  (
 ( A  ||  ( B  -  D )  \/  A  ||  ( B  -  -u D ) )  <-> 
 ( A  ||  ( C  -  E )  \/  A  ||  ( C  -  -u E ) ) ) )
 
Theoremacongrep 35800* Every integer is alternating-congruent to some number in the first half of the fundamental domain. (Contributed by Stefan O'Rear, 2-Oct-2014.)
 |-  (
 ( A  e.  NN  /\  N  e.  ZZ )  ->  E. a  e.  (
 0 ... A ) ( ( 2  x.  A )  ||  ( a  -  N )  \/  (
 2  x.  A ) 
 ||  ( a  -  -u N ) ) )
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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 38201-38300 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 40101-40161
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