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Theorem List for Metamath Proof Explorer - 35701-35800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeldioph4i 35701* Forward-only version of eldioph4b 35700. (Contributed by Stefan O'Rear, 16-Oct-2014.)
 |-  W  e.  _V   &    |-  -.  W  e.  Fin   &    |-  ( W  i^i  NN )  =  (/)   =>    |-  ( ( N  e.  NN0  /\  P  e.  (mzPoly `  ( W  u.  ( 1 ...
 N ) ) ) )  ->  { t  e.  ( NN0  ^m  (
 1 ... N ) )  |  E. w  e.  ( NN0  ^m  W ) ( P `  ( t  u.  w ) )  =  0 }  e.  (Dioph `  N ) )
 
Theoremdiophren 35702* Change variables in a Diophantine set, using class notation. This allows already proved Diophantine sets to be reused in contexts with more variables. (Contributed by Stefan O'Rear, 16-Oct-2014.) (Revised by Stefan O'Rear, 5-Jun-2015.)
 |-  (
 ( S  e.  (Dioph `  N )  /\  M  e.  NN0  /\  F :
 ( 1 ... N )
 --> ( 1 ... M ) )  ->  { a  e.  ( NN0  ^m  (
 1 ... M ) )  |  ( a  o.  F )  e.  S }  e.  (Dioph `  M ) )
 
Theoremrabrenfdioph 35703* Change variable numbers in a Diophantine class abstraction using explicit substitution. (Contributed by Stefan O'Rear, 17-Oct-2014.)
 |-  (
 ( B  e.  NN0  /\  F : ( 1
 ... A ) --> ( 1
 ... B )  /\  { a  e.  ( NN0  ^m  ( 1 ... A ) )  |  ph }  e.  (Dioph `  A ) ) 
 ->  { b  e.  ( NN0  ^m  ( 1 ...
 B ) )  | 
 [. ( b  o.  F )  /  a ]. ph }  e.  (Dioph `  B ) )
 
Theoremrabren3dioph 35704* Change variable numbers in a 3-variable Diophantine class abstraction. (Contributed by Stefan O'Rear, 17-Oct-2014.)
 |-  (
 ( ( a `  1 )  =  (
 b `  X )  /\  ( a `  2
 )  =  ( b `
  Y )  /\  ( a `  3
 )  =  ( b `
  Z ) ) 
 ->  ( ph  <->  ps ) )   &    |-  X  e.  ( 1 ... N )   &    |-  Y  e.  ( 1
 ... N )   &    |-  Z  e.  ( 1 ... N )   =>    |-  ( ( N  e.  NN0  /\  { a  e.  ( NN0  ^m  ( 1 ... 3 ) )  | 
 ph }  e.  (Dioph `  3 ) )  ->  { b  e.  ( NN0  ^m  ( 1 ...
 N ) )  |  ps }  e.  (Dioph `  N ) )
 
21.23.18  Pigeonhole Principle and cardinality helpers
 
Theoremfphpd 35705* Pigeonhole principle expressed with implicit substitution. If the range is smaller than the domain, two inputs must be mapped to the same output. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  ( ph  ->  B  ~<  A )   &    |-  ( ( ph  /\  x  e.  A )  ->  C  e.  B )   &    |-  ( x  =  y  ->  C  =  D )   =>    |-  ( ph  ->  E. x  e.  A  E. y  e.  A  ( x  =/=  y  /\  C  =  D ) )
 
Theoremfphpdo 35706* Pigeonhole principle for sets of real numbers with implicit output reordering. (Contributed by Stefan O'Rear, 12-Sep-2014.)
 |-  ( ph  ->  A  C_  RR )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  B 
 ~<  A )   &    |-  ( ( ph  /\  z  e.  A ) 
 ->  C  e.  B )   &    |-  ( z  =  x  ->  C  =  D )   &    |-  ( z  =  y  ->  C  =  E )   =>    |-  ( ph  ->  E. x  e.  A  E. y  e.  A  ( x  < 
 y  /\  D  =  E ) )
 
Theoremctbnfien 35707 An infinite subset of a countable set is countable, without using choice. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  (
 ( ( X  ~~  om 
 /\  Y  ~~  om )  /\  ( A  C_  X  /\  -.  A  e.  Fin ) )  ->  A  ~~  Y )
 
Theoremfiphp3d 35708* Infinite pigeonhole principle for partitioning an infinite set between finitely many buckets. (Contributed by Stefan O'Rear, 18-Oct-2014.)
 |-  ( ph  ->  A  ~~  NN )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  D  e.  B )   =>    |-  ( ph  ->  E. y  e.  B  { x  e.  A  |  D  =  y }  ~~  NN )
 
21.23.19  A non-closed set of reals is infinite
 
Theoremrencldnfilem 35709* Lemma for rencldnfi 35710. (Contributed by Stefan O'Rear, 18-Oct-2014.)
 |-  (
 ( ( A  C_  RR  /\  B  e.  RR  /\  ( A  =/=  (/)  /\  -.  B  e.  A )
 )  /\  A. x  e.  RR+  E. y  e.  A  ( abs `  ( y  -  B ) )  < 
 x )  ->  -.  A  e.  Fin )
 
Theoremrencldnfi 35710* A set of real numbers which comes arbitrarily close to some target yet excludes it is infinite. The work is done in rencldnfilem 35709 using infima; this theorem removes the requirement that A be nonempty. (Contributed by Stefan O'Rear, 19-Oct-2014.)
 |-  (
 ( ( A  C_  RR  /\  B  e.  RR  /\ 
 -.  B  e.  A )  /\  A. x  e.  RR+  E. y  e.  A  ( abs `  ( y  -  B ) )  < 
 x )  ->  -.  A  e.  Fin )
 
21.23.20  Miscellanea for Lagrange's theorem
 
Theoremmodelico 35711 Modular reduction produces a half-open interval. (Contributed by Stefan O'Rear, 12-Sep-2014.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR+ )  ->  ( A  mod  B )  e.  ( 0 [,) B ) )
 
21.23.21  Lagrange's rational approximation theorem
 
Theoremirrapxlem1 35712* Lemma for irrapx1 35718. Divides the unit interval into  B half-open sections and using the pigeonhole principle fphpdo 35706 finds two multiples of  A in the same section mod 1. (Contributed by Stefan O'Rear, 12-Sep-2014.)
 |-  (
 ( A  e.  RR+  /\  B  e.  NN )  ->  E. x  e.  (
 0 ... B ) E. y  e.  ( 0 ... B ) ( x  <  y  /\  ( |_ `  ( B  x.  ( ( A  x.  x )  mod  1 ) ) )  =  ( |_ `  ( B  x.  ( ( A  x.  y )  mod  1 ) ) ) ) )
 
Theoremirrapxlem2 35713* Lemma for irrapx1 35718. Two multiples in the same bucket means they are very close mod 1. (Contributed by Stefan O'Rear, 12-Sep-2014.)
 |-  (
 ( A  e.  RR+  /\  B  e.  NN )  ->  E. x  e.  (
 0 ... B ) E. y  e.  ( 0 ... B ) ( x  <  y  /\  ( abs `  ( ( ( A  x.  x ) 
 mod  1 )  -  ( ( A  x.  y )  mod  1 ) ) )  <  (
 1  /  B )
 ) )
 
Theoremirrapxlem3 35714* Lemma for irrapx1 35718. By subtraction, there is a multiple very close to an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  (
 ( A  e.  RR+  /\  B  e.  NN )  ->  E. x  e.  (
 1 ... B ) E. y  e.  NN0  ( abs `  ( ( A  x.  x )  -  y
 ) )  <  (
 1  /  B )
 )
 
Theoremirrapxlem4 35715* Lemma for irrapx1 35718. Eliminate ranges, use positivity of the input to force positivity of the output by increasing  B as needed. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  (
 ( A  e.  RR+  /\  B  e.  NN )  ->  E. x  e.  NN  E. y  e.  NN  ( abs `  ( ( A  x.  x )  -  y ) )  < 
 ( 1  /  if ( x  <_  B ,  B ,  x )
 ) )
 
Theoremirrapxlem5 35716* Lemma for irrapx1 35718. Switching to real intervals and fraction syntax. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  (
 ( A  e.  RR+  /\  B  e.  RR+ )  ->  E. x  e.  QQ  ( 0  <  x  /\  ( abs `  ( x  -  A ) )  <  B  /\  ( abs `  ( x  -  A ) )  < 
 ( (denom `  x ) ^ -u 2 ) ) )
 
Theoremirrapxlem6 35717* Lemma for irrapx1 35718. Explicit description of a non-closed set. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 |-  (
 ( A  e.  RR+  /\  B  e.  RR+ )  ->  E. x  e.  {
 y  e.  QQ  |  ( 0  <  y  /\  ( abs `  (
 y  -  A ) )  <  ( (denom `  y ) ^ -u 2
 ) ) }  ( abs `  ( x  -  A ) )  <  B )
 
Theoremirrapx1 35718* Dirichlet's approximation theorem. Every positive irrational number has infinitely many rational approximations which are closer than the inverse squares of their reduced denominators. Lemma 61 in [vandenDries] p. 42. (Contributed by Stefan O'Rear, 14-Sep-2014.)
 |-  ( A  e.  ( RR+  \  QQ )  ->  { y  e.  QQ  |  ( 0  <  y  /\  ( abs `  ( y  -  A ) )  < 
 ( (denom `  y
 ) ^ -u 2
 ) ) }  ~~  NN )
 
21.23.22  Pell equations 1: A nontrivial solution always exists
 
Theorempellexlem1 35719 Lemma for pellex 35725. Arithmetical core of pellexlem3, norm lower bound. This begins Dirichlet's proof of the Pell equation solution existence; the proof here follows theorem 62 of [vandenDries] p. 43. (Contributed by Stefan O'Rear, 14-Sep-2014.)
 |-  (
 ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  /\  -.  ( sqr `  D )  e.  QQ )  ->  ( ( A ^
 2 )  -  ( D  x.  ( B ^
 2 ) ) )  =/=  0 )
 
Theorempellexlem2 35720 Lemma for pellex 35725. Arithmetical core of pellexlem3, norm upper bound. (Contributed by Stefan O'Rear, 14-Sep-2014.)
 |-  (
 ( ( D  e.  NN  /\  A  e.  NN  /\  B  e.  NN )  /\  ( abs `  (
 ( A  /  B )  -  ( sqr `  D ) ) )  < 
 ( B ^ -u 2
 ) )  ->  ( abs `  ( ( A ^ 2 )  -  ( D  x.  ( B ^ 2 ) ) ) )  <  (
 1  +  ( 2  x.  ( sqr `  D ) ) ) )
 
Theorempellexlem3 35721* Lemma for pellex 35725. To each good rational approximation of  ( sqr `  D
), there exists a near-solution. (Contributed by Stefan O'Rear, 14-Sep-2014.)
 |-  (
 ( D  e.  NN  /\ 
 -.  ( sqr `  D )  e.  QQ )  ->  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  < 
 ( (denom `  x ) ^ -u 2 ) ) }  ~<_  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  (
 ( ( y ^
 2 )  -  ( D  x.  ( z ^
 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^ 2 )  -  ( D  x.  (
 z ^ 2 ) ) ) )  < 
 ( 1  +  (
 2  x.  ( sqr `  D ) ) ) ) ) } )
 
Theorempellexlem4 35722* Lemma for pellex 35725. Invoking irrapx1 35718, we have infinitely many near-solutions. (Contributed by Stefan O'Rear, 14-Sep-2014.)
 |-  (
 ( D  e.  NN  /\ 
 -.  ( sqr `  D )  e.  QQ )  ->  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  (
 ( ( y ^
 2 )  -  ( D  x.  ( z ^
 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^ 2 )  -  ( D  x.  (
 z ^ 2 ) ) ) )  < 
 ( 1  +  (
 2  x.  ( sqr `  D ) ) ) ) ) }  ~~  NN )
 
Theorempellexlem5 35723* Lemma for pellex 35725. Invoking fiphp3d 35708, we have infinitely many near-solutions for some specific norm. (Contributed by Stefan O'Rear, 19-Oct-2014.)
 |-  (
 ( D  e.  NN  /\ 
 -.  ( sqr `  D )  e.  QQ )  ->  E. x  e.  ZZ  ( x  =/=  0  /\  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  (
 ( y ^ 2
 )  -  ( D  x.  ( z ^
 2 ) ) )  =  x ) }  ~~  NN ) )
 
Theorempellexlem6 35724* Lemma for pellex 35725. Doing a field division between near solutions get us to norm 1, and the modularity constraint ensures we still have an integer. Returning NN guarantees that we are not returning the trivial solution (1,0). We are not explicitly defining the Pell-field, Pell-ring, and Pell-norm explicitly because after this construction is done we will never use them. This is mostly basic algebraic number theory and could be simplified if a generic framework for that were in place. (Contributed by Stefan O'Rear, 19-Oct-2014.)
 |-  ( ph  ->  A  e.  NN )   &    |-  ( ph  ->  B  e.  NN )   &    |-  ( ph  ->  C  e.  ZZ )   &    |-  ( ph  ->  D  e.  NN )   &    |-  ( ph  ->  -.  ( sqr `  D )  e. 
 QQ )   &    |-  ( ph  ->  E  e.  NN )   &    |-  ( ph  ->  F  e.  NN )   &    |-  ( ph  ->  -.  ( A  =  E  /\  B  =  F )
 )   &    |-  ( ph  ->  C  =/=  0 )   &    |-  ( ph  ->  ( ( A ^ 2
 )  -  ( D  x.  ( B ^
 2 ) ) )  =  C )   &    |-  ( ph  ->  ( ( E ^ 2 )  -  ( D  x.  ( F ^ 2 ) ) )  =  C )   &    |-  ( ph  ->  ( A  mod  ( abs `  C ) )  =  ( E  mod  ( abs `  C ) ) )   &    |-  ( ph  ->  ( B  mod  ( abs `  C )
 )  =  ( F 
 mod  ( abs `  C ) ) )   =>    |-  ( ph  ->  E. a  e.  NN  E. b  e.  NN  (
 ( a ^ 2
 )  -  ( D  x.  ( b ^
 2 ) ) )  =  1 )
 
Theorempellex 35725* Every Pell equation has a nontrivial solution. Theorem 62 in [vandenDries] p. 43. (Contributed by Stefan O'Rear, 19-Oct-2014.)
 |-  (
 ( D  e.  NN  /\ 
 -.  ( sqr `  D )  e.  QQ )  ->  E. x  e.  NN  E. y  e.  NN  (
 ( x ^ 2
 )  -  ( D  x.  ( y ^
 2 ) ) )  =  1 )
 
21.23.23  Pell equations 2: Algebraic number theory of the solution set
 
Syntaxcsquarenn 35726 Extend class notation to include the set of square positive integers.
 classNN
 
Syntaxcpell1qr 35727 Extend class notation to include the class of quadrant-1 Pell solutions.
 class Pell1QR
 
Syntaxcpell1234qr 35728 Extend class notation to include the class of any-quadrant Pell solutions.
 class Pell1234QR
 
Syntaxcpell14qr 35729 Extend class notation to include the class of positive Pell solutions.
 class Pell14QR
 
Syntaxcpellfund 35730 Extend class notation to include the Pell-equation fundamental solution function.
 class PellFund
 
Syntaxcpellfundold 35731 Extend class notation to include the Pell-equation fundamental solution function (old version).
 class PellFund
 
Definitiondf-squarenn 35732 Define the set of square positive integers. (Contributed by Stefan O'Rear, 18-Sep-2014.)
 |-NN  =  { x  e. 
 NN  |  ( sqr `  x )  e.  QQ }
 
Definitiondf-pell1qr 35733* Define the solutions of a Pell equation in the first quadrant. To avoid pair pain, we represent this via the canonical embedding into the reals. (Contributed by Stefan O'Rear, 17-Sep-2014.)
 |- Pell1QR  =  ( x  e.  ( NN  \NN ) 
 |->  { y  e.  RR  |  E. z  e.  NN0  E. w  e.  NN0  (
 y  =  ( z  +  ( ( sqr `  x )  x.  w ) )  /\  ( ( z ^ 2 )  -  ( x  x.  ( w ^ 2 ) ) )  =  1 ) } )
 
Definitiondf-pell14qr 35734* Define the positive solutions of a Pell equation. (Contributed by Stefan O'Rear, 17-Sep-2014.)
 |- Pell14QR  =  ( x  e.  ( NN  \NN ) 
 |->  { y  e.  RR  |  E. z  e.  NN0  E. w  e.  ZZ  (
 y  =  ( z  +  ( ( sqr `  x )  x.  w ) )  /\  ( ( z ^ 2 )  -  ( x  x.  ( w ^ 2 ) ) )  =  1 ) } )
 
Definitiondf-pell1234qr 35735* Define the general solutions of a Pell equation. (Contributed by Stefan O'Rear, 17-Sep-2014.)
 |- Pell1234QR  =  ( x  e.  ( NN  \NN ) 
 |->  { y  e.  RR  |  E. z  e.  ZZ  E. w  e.  ZZ  (
 y  =  ( z  +  ( ( sqr `  x )  x.  w ) )  /\  ( ( z ^ 2 )  -  ( x  x.  ( w ^ 2 ) ) )  =  1 ) } )
 
Definitiondf-pellfund 35736* A function mapping Pell discriminants to the corresponding fundamental solution. (Contributed by Stefan O'Rear, 18-Sep-2014.) (Revised by AV, 17-Sep-2020.)
 |- PellFund  =  ( x  e.  ( NN  \NN ) 
 |-> inf ( { z  e.  (Pell14QR `  x )  |  1  <  z } ,  RR ,  <  )
 )
 
Definitiondf-pellfundOLD 35737* A function mapping Pell discriminants to the corresponding fundamental solution. (Contributed by Stefan O'Rear, 18-Sep-2014.) Obsolete version of df-pellfund 35736 as of 17-Sep-2020. (New usage is discouraged.)
 |- PellFund  =  ( x  e.  ( NN  \NN ) 
 |->  sup ( { z  e.  (Pell14QR `  x )  |  1  <  z } ,  RR ,  `'  <  ) )
 
Theorempell1qrval 35738* Value of the set of first-quadrant Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
 |-  ( D  e.  ( NN  \NN )  ->  (Pell1QR `  D )  =  { y  e.  RR  |  E. z  e.  NN0  E. w  e.  NN0  (
 y  =  ( z  +  ( ( sqr `  D )  x.  w ) )  /\  ( ( z ^ 2 )  -  ( D  x.  ( w ^ 2 ) ) )  =  1 ) } )
 
Theoremelpell1qr 35739* Membership in a first-quadrant Pell solution set. (Contributed by Stefan O'Rear, 17-Sep-2014.)
 |-  ( D  e.  ( NN  \NN )  ->  ( A  e.  (Pell1QR `  D )  <->  ( A  e.  RR  /\  E. z  e. 
 NN0  E. w  e.  NN0  ( A  =  (
 z  +  ( ( sqr `  D )  x.  w ) )  /\  ( ( z ^
 2 )  -  ( D  x.  ( w ^
 2 ) ) )  =  1 ) ) ) )
 
Theorempell14qrval 35740* Value of the set of positive Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
 |-  ( D  e.  ( NN  \NN )  ->  (Pell14QR `  D )  =  { y  e.  RR  |  E. z  e.  NN0  E. w  e.  ZZ  (
 y  =  ( z  +  ( ( sqr `  D )  x.  w ) )  /\  ( ( z ^ 2 )  -  ( D  x.  ( w ^ 2 ) ) )  =  1 ) } )
 
Theoremelpell14qr 35741* Membership in the set of positive Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
 |-  ( D  e.  ( NN  \NN )  ->  ( A  e.  (Pell14QR `  D )  <->  ( A  e.  RR  /\  E. z  e. 
 NN0  E. w  e.  ZZ  ( A  =  (
 z  +  ( ( sqr `  D )  x.  w ) )  /\  ( ( z ^
 2 )  -  ( D  x.  ( w ^
 2 ) ) )  =  1 ) ) ) )
 
Theorempell1234qrval 35742* Value of the set of general Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
 |-  ( D  e.  ( NN  \NN )  ->  (Pell1234QR `  D )  =  { y  e.  RR  |  E. z  e.  ZZ  E. w  e.  ZZ  (
 y  =  ( z  +  ( ( sqr `  D )  x.  w ) )  /\  ( ( z ^ 2 )  -  ( D  x.  ( w ^ 2 ) ) )  =  1 ) } )
 
Theoremelpell1234qr 35743* Membership in the set of general Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
 |-  ( D  e.  ( NN  \NN )  ->  ( A  e.  (Pell1234QR `  D )  <->  ( A  e.  RR  /\  E. z  e. 
 ZZ  E. w  e.  ZZ  ( A  =  (
 z  +  ( ( sqr `  D )  x.  w ) )  /\  ( ( z ^
 2 )  -  ( D  x.  ( w ^
 2 ) ) )  =  1 ) ) ) )
 
Theorempell1234qrre 35744 General Pell solutions are (coded as) real numbers. (Contributed by Stefan O'Rear, 17-Sep-2014.)
 |-  (
 ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  ->  A  e.  RR )
 
Theorempell1234qrne0 35745 No solution to a Pell equation is zero. (Contributed by Stefan O'Rear, 17-Sep-2014.)
 |-  (
 ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  ->  A  =/=  0 )
 
Theorempell1234qrreccl 35746 General solutions of the Pell equation are closed under reciprocals. (Contributed by Stefan O'Rear, 18-Sep-2014.)
 |-  (
 ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  ->  ( 1  /  A )  e.  (Pell1234QR `  D ) )
 
Theorempell1234qrmulcl 35747 General solutions of the Pell equation are closed under multiplication. (Contributed by Stefan O'Rear, 18-Sep-2014.)
 |-  (
 ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D )  /\  B  e.  (Pell1234QR `  D ) ) 
 ->  ( A  x.  B )  e.  (Pell1234QR `  D ) )
 
Theorempell14qrss1234 35748 A positive Pell solution is a general Pell solution. (Contributed by Stefan O'Rear, 18-Sep-2014.)
 |-  ( D  e.  ( NN  \NN )  ->  (Pell14QR `  D )  C_  (Pell1234QR `  D ) )
 
Theorempell14qrre 35749 A positive Pell solution is a real number. (Contributed by Stefan O'Rear, 18-Sep-2014.)
 |-  (
 ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR )
 
Theorempell14qrne0 35750 A positive Pell solution is a nonzero number. (Contributed by Stefan O'Rear, 17-Sep-2014.)
 |-  (
 ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  =/=  0 )
 
Theorempell14qrgt0 35751 A positive Pell solution is a positive number. (Contributed by Stefan O'Rear, 18-Sep-2014.)
 |-  (
 ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
 0  <  A )
 
Theorempell14qrrp 35752 A positive Pell solution is a positive real. (Contributed by Stefan O'Rear, 19-Sep-2014.)
 |-  (
 ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR+ )
 
Theorempell1234qrdich 35753 A general Pell solution is either a positive solution, or its negation is. (Contributed by Stefan O'Rear, 18-Sep-2014.)
 |-  (
 ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  ->  ( A  e.  (Pell14QR `  D )  \/  -u A  e.  (Pell14QR `  D )
 ) )
 
Theoremelpell14qr2 35754 A number is a positive Pell solution iff it is positive and a Pell solution, justifying our name choice. (Contributed by Stefan O'Rear, 19-Oct-2014.)
 |-  ( D  e.  ( NN  \NN )  ->  ( A  e.  (Pell14QR `  D )  <->  ( A  e.  (Pell1234QR `
  D )  /\  0  <  A ) ) )
 
Theorempell14qrmulcl 35755 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 35756 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 35757 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 35758 Lemma for pell14qrexpcl 35759. (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 35759 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 35760 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 35761 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 35762 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 35763 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 35764 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 35765 Lemma for pell1qrgap 35766. (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 35766 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 35767 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 35768 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 35769 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 )
 )
 
21.23.24  Pell equations 3: characterizing fundamental solution
 
Theoreminfmrgelbi 35770* 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.) (Revised by AV, 17-Sep-2020.)
 |-  (
 ( ( A  C_  RR  /\  A  =/=  (/)  /\  B  e.  RR )  /\  A. x  e.  A  B  <_  x )  ->  B  <_ inf ( A ,  RR ,  <  ) )
 
TheoreminfmrgelbiOLD 35771* 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.) Obsolete version of infmrgelbi 35770 as of 17-Sep-2020. ( (New usage is discouraged.) (Proof modification is discouraged.)
 |-  (
 ( ( A  C_  RR  /\  A  =/=  (/)  /\  B  e.  RR )  /\  A. x  e.  A  B  <_  x )  ->  B  <_  sup ( A ,  RR ,  `'  <  )
 )
 
Theorempellqrex 35772* 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 35773* Value of the fundamental solution of a Pell equation. (Contributed by Stefan O'Rear, 18-Sep-2014.) (Revised by AV, 17-Sep-2020.)
 |-  ( D  e.  ( NN  \NN )  ->  (PellFund `  D )  = inf ( { x  e.  (Pell14QR `  D )  |  1  <  x } ,  RR ,  <  )
 )
 
TheorempellfundvalOLD 35774* Value of the fundamental solution of a Pell equation. (Contributed by Stefan O'Rear, 18-Sep-2014.) Obsolete version of pellfundval 35773 as of 17-Sep-2020. ( (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( D  e.  ( NN  \NN )  ->  (PellFund `  D )  =  sup ( { x  e.  (Pell14QR `  D )  |  1  <  x } ,  RR ,  `'  <  ) )
 
Theorempellfundre 35775 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 35776 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 35777 Weak lower bound on the Pell fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.)
 |-  ( D  e.  ( NN  \NN )  ->  1  <  (PellFund `  D ) )
 
Theorempellfundlb 35778 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 35779* 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 35780 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 35768. (Contributed by Stefan O'Rear, 18-Sep-2014.)

 |-  ( D  e.  ( NN  \NN )  ->  (PellFund `  D )  e.  (Pell1QR `  D )
 )
 
Theorempellfund14gap 35781 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 35782 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 35783 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 35784 General logarithm is a real number. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbcl 23766 instead.
 |-  (
 ( A  e.  RR+  /\  B  e.  RR+  /\  B  =/=  1 )  ->  (
 ( log `  A )  /  ( log `  B ) )  e.  RR )
 
Theoremreglogltb 35785 General logarithm preserves "less than". (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logblt 23777 instead.
 |-  (
 ( ( A  e.  RR+  /\  B  e.  RR+ )  /\  ( C  e.  RR+  /\  1  <  C ) )  ->  ( A  <  B  <->  ( ( log `  A )  /  ( log `  C ) )  <  ( ( log `  B )  /  ( log `  C ) ) ) )
 
Theoremreglogleb 35786 General logarithm preserves  <_. (Contributed by Stefan O'Rear, 19-Oct-2014.) (New usage is discouraged.) Use logbleb 23776 instead.
 |-  (
 ( ( A  e.  RR+  /\  B  e.  RR+ )  /\  ( C  e.  RR+  /\  1  <  C ) )  ->  ( A  <_  B  <->  ( ( log `  A )  /  ( log `  C ) ) 
 <_  ( ( log `  B )  /  ( log `  C ) ) ) )
 
Theoremreglogmul 35787 Multiplication law for general log. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbmul 23770 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 35788 Power law for general log. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbzexp 23769 instead.
 |-  (
 ( A  e.  RR+  /\  N  e.  ZZ  /\  ( C  e.  RR+  /\  C  =/=  1 ) )  ->  ( ( log `  ( A ^ N ) ) 
 /  ( log `  C ) )  =  ( N  x.  ( ( log `  A )  /  ( log `  C ) ) ) )
 
Theoremreglogbas 35789 General log of the base is 1. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logbid1 23761 instead.
 |-  (
 ( C  e.  RR+  /\  C  =/=  1 ) 
 ->  ( ( log `  C )  /  ( log `  C ) )  =  1
 )
 
Theoremreglog1 35790 General log of 1 is 0. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logb1 23762 instead.
 |-  (
 ( C  e.  RR+  /\  C  =/=  1 ) 
 ->  ( ( log `  1
 )  /  ( log `  C ) )  =  0 )
 
Theoremreglogexpbas 35791 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 23773 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 35792* 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 35793* 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 35794 Extend class notation to include the Robertson-Matiyasevich X sequence.
 class Xrm
 
Syntaxcrmy 35795 Extend class notation to include the Robertson-Matiyasevich Y sequence.
 class Yrm
 
Definitiondf-rmx 35796* Define the X sequence as the rational part of some solution of a special Pell equation. See frmx 35807 and rmxyval 35809 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 35797* Define the X sequence as the irrational part of some solution of a special Pell equation. See frmy 35808 and rmxyval 35809 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 35798* Value of the X sequence. Not used after rmxyval 35809 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 35799* Value of the Y sequence. Not used after rmxyval 35809 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 35800 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 ) )
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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 40901-40909
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