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Theorem List for Metamath Proof Explorer - 26301-26400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempell14qrmulcl 26301 Positive Pell solutions are closed under multiplication. (Contributed by Stefan O'Rear, 17-Sep-2014.)
 |-  (
 ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )  /\  B  e.  (Pell14QR `  D )
 )  ->  ( A  x.  B )  e.  (Pell14QR `  D ) )
 
Theorempell14qrreccl 26302 Positive Pell solutions are closed under reciprocal. (Contributed by Stefan O'Rear, 18-Sep-2014.)
 |-  (
 ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D ) )  ->  ( 1  /  A )  e.  (Pell14QR `  D ) )
 
Theorempell14qrdivcl 26303 Positive Pell solutions are closed under division. (Contributed by Stefan O'Rear, 18-Sep-2014.)
 |-  (
 ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )  /\  B  e.  (Pell14QR `  D )
 )  ->  ( A  /  B )  e.  (Pell14QR `  D ) )
 
Theorempell14qrexpclnn0 26304 Lemma for pell14qrexpcl 26305. (Contributed by Stefan O'Rear, 18-Sep-2014.)
 |-  (
 ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )  /\  B  e.  NN0 )  ->  ( A ^ B )  e.  (Pell14QR `  D )
 )
 
Theorempell14qrexpcl 26305 Positive Pell solutions are closed under integer powers. (Contributed by Stefan O'Rear, 18-Sep-2014.)
 |-  (
 ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )  /\  B  e.  ZZ )  ->  ( A ^ B )  e.  (Pell14QR `  D )
 )
 
Theorempell1qrss14 26306 First-quadrant Pell solutions are a subset of the positive solutions. (Contributed by Stefan O'Rear, 18-Sep-2014.)
 |-  ( D  e.  ( NN  \NN )  ->  (Pell1QR `  D )  C_  (Pell14QR `  D )
 )
 
Theorempell14qrdich 26307 A positive Pell solution is either in the first quadrant, or its reciprocal is. (Contributed by Stefan O'Rear, 18-Sep-2014.)
 |-  (
 ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D ) )  ->  ( A  e.  (Pell1QR `  D )  \/  (
 1  /  A )  e.  (Pell1QR `  D )
 ) )
 
Theorempell1qrge1 26308 A Pell solution in the first quadrant is at least 1. (Contributed by Stefan O'Rear, 17-Sep-2014.)
 |-  (
 ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1QR `  D ) )  -> 
 1  <_  A )
 
Theorempell1qr1 26309 1 is a Pell solution and in the first quadrant as one. (Contributed by Stefan O'Rear, 17-Sep-2014.)
 |-  ( D  e.  ( NN  \NN )  ->  1  e.  (Pell1QR `  D ) )
 
Theoremelpell1qr2 26310 The first quadrant solutions are precisely the positive Pell solutions which are at least one. (Contributed by Stefan O'Rear, 18-Sep-2014.)
 |-  ( D  e.  ( NN  \NN )  ->  ( A  e.  (Pell1QR `  D )  <->  ( A  e.  (Pell14QR `  D )  /\  1  <_  A ) ) )
 
Theorempell1qrgaplem 26311 Lemma for pell1qrgap 26312. (Contributed by Stefan O'Rear, 18-Sep-2014.)
 |-  (
 ( ( D  e.  NN  /\  ( A  e.  NN0  /\  B  e.  NN0 )
 )  /\  ( 1  <  ( A  +  (
 ( sqr `  D )  x.  B ) )  /\  ( ( A ^
 2 )  -  ( D  x.  ( B ^
 2 ) ) )  =  1 ) ) 
 ->  ( ( sqr `  ( D  +  1 )
 )  +  ( sqr `  D ) )  <_  ( A  +  (
 ( sqr `  D )  x.  B ) ) )
 
Theorempell1qrgap 26312 First-quadrant Pell solutions are bounded away from 1. (This particular bound allows us to prove exact values for the fundamental solution later.) (Contributed by Stefan O'Rear, 18-Sep-2014.)
 |-  (
 ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1QR `  D )  /\  1  <  A )  ->  (
 ( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) )  <_  A )
 
Theorempell14qrgap 26313 Positive Pell solutions are bounded away from 1. (Contributed by Stefan O'Rear, 18-Sep-2014.)
 |-  (
 ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (
 ( sqr `  ( D  +  1 ) )  +  ( sqr `  D ) )  <_  A )
 
Theorempell14qrgapw 26314 Positive Pell solutions are bounded away from 1, with a friendlier bound. (Contributed by Stefan O'Rear, 18-Sep-2014.)
 |-  (
 ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  2  <  A )
 
Theorempellqrexplicit 26315 Condition for a calculated real to be a Pell solution. (Contributed by Stefan O'Rear, 19-Sep-2014.)
 |-  (
 ( ( D  e.  ( NN  \NN )  /\  A  e.  NN0  /\  B  e.  NN0 )  /\  ( ( A ^
 2 )  -  ( D  x.  ( B ^
 2 ) ) )  =  1 )  ->  ( A  +  (
 ( sqr `  D )  x.  B ) )  e.  (Pell1QR `  D )
 )
 
16.16.26  Pell equations 3: characterizing fundamental solution
 
Theoreminfmrgelbi 26316* Any lower bound of a nonempty set of real numbers is less than or equal to its infimum, one-direction version. (Contributed by Stefan O'Rear, 1-Sep-2013.)
 |-  (
 ( ( A  C_  RR  /\  A  =/=  (/)  /\  B  e.  RR )  /\  A. x  e.  A  B  <_  x )  ->  B  <_  sup ( A ,  RR ,  `'  <  )
 )
 
Theorempellqrex 26317* There is a nontrivial solution of a Pell equation in the first quadrant. (Contributed by Stefan O'Rear, 18-Sep-2014.)
 |-  ( D  e.  ( NN  \NN )  ->  E. x  e.  (Pell1QR `  D ) 1  < 
 x )
 
Theorempellfundval 26318* Value of the fundamental solution of a Pell equation. (Contributed by Stefan O'Rear, 18-Sep-2014.)
 |-  ( D  e.  ( NN  \NN )  ->  (PellFund `  D )  =  sup ( { x  e.  (Pell14QR `  D )  |  1  <  x } ,  RR ,  `'  <  ) )
 
Theorempellfundre 26319 The fundamental solution of a Pell equation exists as a real number. (Contributed by Stefan O'Rear, 18-Sep-2014.)
 |-  ( D  e.  ( NN  \NN )  ->  (PellFund `  D )  e.  RR )
 
Theorempellfundge 26320 Lower bound on the fundamental solution of a Pell equation. (Contributed by Stefan O'Rear, 19-Sep-2014.)
 |-  ( D  e.  ( NN  \NN )  ->  ( ( sqr `  ( D  +  1 )
 )  +  ( sqr `  D ) )  <_  (PellFund `  D ) )
 
Theorempellfundgt1 26321 Weak lower bound on the Pell fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.)
 |-  ( D  e.  ( NN  \NN )  ->  1  <  (PellFund `  D ) )
 
Theorempellfundlb 26322 A nontrivial first quadrant solution is at least as large as the fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.)
 |-  (
 ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )  /\  1  <  A )  ->  (PellFund `  D )  <_  A )
 
Theorempellfundglb 26323* 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 26324 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 26314. (Contributed by Stefan O'Rear, 18-Sep-2014.)

 |-  ( D  e.  ( NN  \NN )  ->  (PellFund `  D )  e.  (Pell1QR `  D )
 )
 
Theorempellfund14gap 26325 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 26326 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 26327 The fundamental Pell solution is never 1. (Contributed by Stefan O'Rear, 19-Sep-2014.)
 |-  ( D  e.  ( NN  \NN )  ->  (PellFund `  D )  =/=  1 )
 
16.16.27  Logarithm laws generalized to an arbitrary base
 
Theoremreglogcl 26328 General logarithm is a real number. (Contributed by Stefan O'Rear, 19-Sep-2014.)
 |-  (
 ( A  e.  RR+  /\  B  e.  RR+  /\  B  =/=  1 )  ->  (
 ( log `  A )  /  ( log `  B ) )  e.  RR )
 
Theoremreglogltb 26329 General logarithm preserves "less than". (Contributed by Stefan O'Rear, 19-Sep-2014.)
 |-  (
 ( ( A  e.  RR+  /\  B  e.  RR+ )  /\  ( C  e.  RR+  /\  1  <  C ) )  ->  ( A  <  B  <->  ( ( log `  A )  /  ( log `  C ) )  <  ( ( log `  B )  /  ( log `  C ) ) ) )
 
Theoremreglogleb 26330 General logarithm preserves  <_. (Contributed by Stefan O'Rear, 19-Oct-2014.)
 |-  (
 ( ( A  e.  RR+  /\  B  e.  RR+ )  /\  ( C  e.  RR+  /\  1  <  C ) )  ->  ( A  <_  B  <->  ( ( log `  A )  /  ( log `  C ) ) 
 <_  ( ( log `  B )  /  ( log `  C ) ) ) )
 
Theoremreglogmul 26331 Multiplication law for general log. (Contributed by Stefan O'Rear, 19-Sep-2014.)
 |-  (
 ( 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 26332 Power law for general log. (Contributed by Stefan O'Rear, 19-Sep-2014.)
 |-  (
 ( A  e.  RR+  /\  N  e.  ZZ  /\  ( C  e.  RR+  /\  C  =/=  1 ) )  ->  ( ( log `  ( A ^ N ) ) 
 /  ( log `  C ) )  =  ( N  x.  ( ( log `  A )  /  ( log `  C ) ) ) )
 
Theoremreglogbas 26333 General log of the base is 1. (Contributed by Stefan O'Rear, 19-Sep-2014.)
 |-  (
 ( C  e.  RR+  /\  C  =/=  1 ) 
 ->  ( ( log `  C )  /  ( log `  C ) )  =  1
 )
 
Theoremreglog1 26334 General log of 1 is 0. (Contributed by Stefan O'Rear, 19-Sep-2014.)
 |-  (
 ( C  e.  RR+  /\  C  =/=  1 ) 
 ->  ( ( log `  1
 )  /  ( log `  C ) )  =  0 )
 
Theoremreglogexpbas 26335 General log of a power of the base is the exponent. (Contributed by Stefan O'Rear, 19-Sep-2014.)
 |-  (
 ( N  e.  ZZ  /\  ( C  e.  RR+  /\  C  =/=  1 ) )  ->  ( ( log `  ( C ^ N ) )  /  ( log `  C )
 )  =  N )
 
16.16.28  Pell equations 4: the positive solution group is infinite cyclic
 
Theorempellfund14 26336* 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 26337* 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 ) ) )
 
16.16.29  X and Y sequences 1: Definition and recurrence laws
 
Syntaxcrmx 26338 Extend class notation to include the Robertson-Matiyasevich X sequence.
 class Xrm
 
Syntaxcrmy 26339 Extend class notation to include the Robertson-Matiyasevich Y sequence.
 class Yrm
 
Definitiondf-rmx 26340* Define the X sequence as the rational part of some solution of a special Pell equation. See frmx 26351 and rmxyval 26353 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 26341* Define the X sequence as the irrational part of some solution of a special Pell equation. See frmy 26352 and rmxyval 26353 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 26342* Value of the X sequence. Not used after rmxyval 26353 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 26343* Value of the Y sequence. Not used after rmxyval 26353 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 ) ) ) )
 
Theoremrmspecsqrnq 26344 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 26345 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 26346 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 26347 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 26348* 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 26349* The function used to extract rational and irrational parts in df-rmx 26340 and df-rmy 26341 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 26350* Lemma for frmx 26351 and frmy 26352. (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 26351 The X sequence is a nonnegative integer. See rmxnn 26391 for a strengthening. (Contributed by Stefan O'Rear, 22-Sep-2014.)
 |- Xrm  : ( ( ZZ>= `  2 )  X.  ZZ ) --> NN0
 
Theoremfrmy 26352 The Y sequence is an integer. (Contributed by Stefan O'Rear, 22-Sep-2014.)
 |- Yrm  : ( ( ZZ>= `  2 )  X.  ZZ ) --> ZZ
 
Theoremrmxyval 26353 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 26354 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 26355* The X and Y sequences taken together enumerate all solutions to the corresponding Pell equation in the right half-plane. (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 26356 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 26357 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 26358 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 26359 Addition formula for X and Y sequences. See rmxadd 26365 and rmyadd 26369 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 26360 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 26361 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 26362 Negation law (even function) for the X sequence. The method of proof used for the previous four theorems rmxyneg 26358, rmxyadd 26359, rmxy0 26361, and rmxy1 26360 via qirropth 26346 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 26363 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 26364 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 26365 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 26366 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 26367 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 26368 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 26369 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 26370 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 26371 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 26372 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 26373 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 26374 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 26375 The Y sequence is a Lucas sequence, definable via this second-order recurrence with rmy0 26367 and rmy1 26368. 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 26376 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 26377 "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 26378 "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 ) ) )
 
16.16.30  Ordering and induction lemmas for the integers
 
Theoremmonotuz 26379* A function defined on a set of upper 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 26380* 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 26381* 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 26382* 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 26383* Induction on natural numbers 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 26384* 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 26385 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 26386 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 ) ) )
 
16.16.31  X and Y sequences 2: Order properties
 
Theoremrmxypos 26387 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 26388 The Y-sequence is strictly monotonic on  NN0. Strengthened by ltrmy 26392. (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 26389 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 26390 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 26391 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 26392 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 26393 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 26394 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 26395 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 26396 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 26397 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 26398 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 26399 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 26400 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 ) ) )
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