HomeHome Metamath Proof Explorer
Theorem List (p. 261 of 310)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21328)
  Hilbert Space Explorer  Hilbert Space Explorer
(21329-22851)
  Users' Mathboxes  Users' Mathboxes
(22852-30955)
 

Theorem List for Metamath Proof Explorer - 26001-26100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-dioph 26001* A Diophantine set is a set of natural numbers which is a projection of the zero set of some polynomial. This definition somewhat awkwardly mixes  ZZ (via mzPoly) and  NN0 (to define the zero sets); the former could be avoided by considering coincidence sets of  NN0 polynomials at the cost of requiring two, and the second is driven by consistency with our mu-recursive functions and the requirements of the Davis-Putnam-Robinson-Matiyasevich proof. Both are avoidable at a complexity cost. In particular, it is a consequence of 4sq 12885 that implicitly restricting variables to  NN0 adds no expressive power over allowing them to range over  ZZ. While this definition stipulates a specific index set for the polynomials, there is actually flexibility here, see eldioph2b 26008. (Contributed by Stefan O'Rear, 5-Oct-2014.)
 |- Dioph  =  ( n  e.  NN0  |->  ran  (  k  e.  ( ZZ>= `  n ) ,  p  e.  (mzPoly `  ( 1 ... k ) )  |->  { t  |  E. u  e.  ( NN0  ^m  (
 1 ... k ) ) ( t  =  ( u  |`  ( 1 ... n ) )  /\  ( p `  u )  =  0 ) }
 ) )
 
Theoremeldiophb 26002* Initial expression of Diophantine property of a set. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
 |-  ( D  e.  (Dioph `  N ) 
 <->  ( N  e.  NN0  /\ 
 E. k  e.  ( ZZ>=
 `  N ) E. p  e.  (mzPoly `  (
 1 ... k ) ) D  =  { t  |  E. u  e.  ( NN0  ^m  ( 1 ... k ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( p `
  u )  =  0 ) } )
 )
 
Theoremeldioph 26003* Condition for a set to be Diophantine (unpacking existential quantifier) (Contributed by Stefan O'Rear, 5-Oct-2014.)
 |-  (
 ( N  e.  NN0  /\  K  e.  ( ZZ>= `  N )  /\  P  e.  (mzPoly `  ( 1 ...
 K ) ) ) 
 ->  { t  |  E. u  e.  ( NN0  ^m  ( 1 ... K ) ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( P `
  u )  =  0 ) }  e.  (Dioph `  N ) )
 
Theoremdiophrw 26004* Renaming and adding unused witness variables does not change the Diophantine set coded by a polynomial. (Contributed by Stefan O'Rear, 7-Oct-2014.)
 |-  (
 ( S  e.  _V  /\  M : T -1-1-> S  /\  ( M  |`  O )  =  (  _I  |`  O ) )  ->  { a  |  E. b  e.  ( NN0  ^m  S ) ( a  =  ( b  |`  O )  /\  (
 ( d  e.  ( ZZ  ^m  S )  |->  ( P `  ( d  o.  M ) ) ) `  b )  =  0 ) }  =  { a  |  E. c  e.  ( NN0  ^m  T ) ( a  =  ( c  |`  O )  /\  ( P `
  c )  =  0 ) } )
 
Theoremeldioph2lem1 26005* Lemma for eldioph2 26007. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.)
 |-  (
 ( N  e.  NN0  /\  A  e.  Fin  /\  ( 1 ... N )  C_  A )  ->  E. d  e.  ( ZZ>=
 `  N ) E. e  e.  _V  (
 e : ( 1
 ... d ) -1-1-onto-> A  /\  ( e  |`  ( 1
 ... N ) )  =  (  _I  |`  ( 1
 ... N ) ) ) )
 
Theoremeldioph2lem2 26006* Lemma for eldioph2 26007. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.)
 |-  (
 ( ( N  e.  NN0  /\  -.  S  e.  Fin )  /\  ( ( 1
 ... N )  C_  S  /\  A  e.  ( ZZ>=
 `  N ) ) )  ->  E. c
 ( c : ( 1 ... A )
 -1-1-> S  /\  ( c  |`  ( 1 ... N ) )  =  (  _I  |`  ( 1 ...
 N ) ) ) )
 
Theoremeldioph2 26007* Construct a Diophantine set from a polynomial with witness variables drawn from any set whatsoever, via mzpcompact2 25996. (Contributed by Stefan O'Rear, 8-Oct-2014.) (Revised by Stefan O'Rear, 5-Jun-2015.)
 |-  (
 ( N  e.  NN0  /\  ( S  e.  _V  /\  ( 1 ... N )  C_  S )  /\  P  e.  (mzPoly `  S ) )  ->  { t  |  E. u  e.  ( NN0  ^m  S ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( P `
  u )  =  0 ) }  e.  (Dioph `  N ) )
 
Theoremeldioph2b 26008* While Diophantine sets were defined to have a finite number of witness variables consequtively following the observable variables, this is not necessary; they can equivalently be taken to use any witness set  ( S  \ 
( 1 ... N
) ). For instance, in diophin 26018 we use this to take the two input sets to have disjoint witness sets. (Contributed by Stefan O'Rear, 8-Oct-2014.)
 |-  (
 ( ( N  e.  NN0  /\  S  e.  _V )  /\  ( -.  S  e.  Fin  /\  ( 1 ... N )  C_  S ) ) 
 ->  ( A  e.  (Dioph `  N )  <->  E. p  e.  (mzPoly `  S ) A  =  { t  |  E. u  e.  ( NN0  ^m  S ) ( t  =  ( u  |`  ( 1
 ... N ) ) 
 /\  ( p `  u )  =  0
 ) } ) )
 
Theoremeldiophelnn0 26009 Remove antecedent on  B from Diophantine set constructors. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  ( A  e.  (Dioph `  B )  ->  B  e.  NN0 )
 
Theoremeldioph3b 26010* Define Diophantine sets in terms of polynomials with variables indexed by  NN. This avoids a quantifier over the number of witness variables and will be easier to use than eldiophb 26002 in most cases. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  ( A  e.  (Dioph `  N ) 
 <->  ( N  e.  NN0  /\ 
 E. p  e.  (mzPoly `  NN ) A  =  { t  |  E. u  e.  ( NN0  ^m  NN ) ( t  =  ( u  |`  ( 1
 ... N ) ) 
 /\  ( p `  u )  =  0
 ) } ) )
 
Theoremeldioph3 26011* Inference version of eldioph3b 26010 with quantifier expanded. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  (
 ( N  e.  NN0  /\  P  e.  (mzPoly `  NN ) )  ->  { t  |  E. u  e.  ( NN0  ^m  NN ) ( t  =  ( u  |`  ( 1 ... N ) )  /\  ( P `
  u )  =  0 ) }  e.  (Dioph `  N ) )
 
16.15.12  Diophantine sets 2 miscellanea
 
Theoremellz1 26012 Membership in a set of lower integers. (Contributed by Stefan O'Rear, 9-Oct-2014.)
 |-  ( B  e.  ZZ  ->  ( A  e.  ( ZZ  \  ( ZZ>= `  ( B  +  1 ) ) )  <->  ( A  e.  ZZ  /\  A  <_  B ) ) )
 
Theoremlzunuz 26013 A set of lower integers and upper integers which abut or overlap is all of the integers. (Contributed by Stefan O'Rear, 9-Oct-2014.)
 |-  (
 ( A  e.  ZZ  /\  B  e.  ZZ  /\  B  <_  ( A  +  1 ) )  ->  ( ( ZZ  \  ( ZZ>= `  ( A  +  1 ) ) )  u.  ( ZZ>= `  B ) )  =  ZZ )
 
Theoremfz1eqin 26014 Express a one-based finite range as the intersection of lower integers with  NN. (Contributed by Stefan O'Rear, 9-Oct-2014.)
 |-  ( N  e.  NN0  ->  (
 1 ... N )  =  ( ( ZZ  \  ( ZZ>= `  ( N  +  1 ) ) )  i^i  NN )
 )
 
Theoremlzenom 26015 Lower integers are countably infinite. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  ( N  e.  ZZ  ->  ( ZZ  \  ( ZZ>= `  ( N  +  1
 ) ) )  ~~  om )
 
Theoremelmapresaun 26016 fresaun 5269 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  (
 ( F  e.  ( C  ^m  A )  /\  G  e.  ( C  ^m  B )  /\  ( F  |`  ( A  i^i  B ) )  =  ( G  |`  ( A  i^i  B ) ) ) 
 ->  ( F  u.  G )  e.  ( C  ^m  ( A  u.  B ) ) )
 
Theoremelmapresaunres2 26017 fresaunres2 5270 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014.)
 |-  (
 ( F  e.  ( C  ^m  A )  /\  G  e.  ( C  ^m  B )  /\  ( F  |`  ( A  i^i  B ) )  =  ( G  |`  ( A  i^i  B ) ) ) 
 ->  ( ( F  u.  G )  |`  B )  =  G )
 
16.15.13  Diophantine sets 2: union and intersection. Monotone Boolean algebra
 
Theoremdiophin 26018 If two sets are Diophantine, so is their intersection. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  (
 ( A  e.  (Dioph `  N )  /\  B  e.  (Dioph `  N )
 )  ->  ( A  i^i  B )  e.  (Dioph `  N ) )
 
Theoremdiophun 26019 If two sets are Diophantine, so is their union. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  (
 ( A  e.  (Dioph `  N )  /\  B  e.  (Dioph `  N )
 )  ->  ( A  u.  B )  e.  (Dioph `  N ) )
 
Theoremeldiophss 26020 Diophantine sets are sets of tuples of natural numbers. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  ( A  e.  (Dioph `  B )  ->  A  C_  ( NN0  ^m  ( 1 ...
 B ) ) )
 
16.15.14  Diophantine sets 3: construction
 
Theoremdiophrex 26021* Projecting a Diophantine set by removing a coordinate results in a Diophantine set. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  (
 ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M ) ) 
 ->  { t  |  E. u  e.  S  t  =  ( u  |`  ( 1
 ... N ) ) }  e.  (Dioph `  N ) )
 
Theoremeq0rabdioph 26022* This is the first of a number of theorems which allow sets to be proven Diophantine by syntactic induction, and models the correspondence between Diophantine sets and monotone existential first order logic. This first theorem shows that the zero set of an implicit polynomial is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  (
 ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ...
 N ) )  |->  A )  e.  (mzPoly `  (
 1 ... N ) ) )  ->  { t  e.  ( NN0  ^m  (
 1 ... N ) )  |  A  =  0 }  e.  (Dioph `  N ) )
 
Theoremeqrabdioph 26023* Diophantine set builder for equality of polynomial expressions. Note that the two expressions need not be non-negative; only variables are so constrained. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  (
 ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ...
 N ) )  |->  A )  e.  (mzPoly `  (
 1 ... N ) ) 
 /\  ( t  e.  ( ZZ  ^m  (
 1 ... N ) ) 
 |->  B )  e.  (mzPoly `  ( 1 ... N ) ) )  ->  { t  e.  ( NN0  ^m  ( 1 ...
 N ) )  |  A  =  B }  e.  (Dioph `  N )
 )
 
Theorem0dioph 26024 The null set is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  ( A  e.  NN0  ->  (/)  e.  (Dioph `  A ) )
 
Theoremvdioph 26025 The "universal" set (as large as possible given eldiophss 26020) is Diophantine. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  ( A  e.  NN0  ->  ( NN0  ^m  ( 1 ...
 A ) )  e.  (Dioph `  A )
 )
 
Theoremanrabdioph 26026* Diophantine set builder for conjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  (
 ( { t  e.  ( NN0  ^m  (
 1 ... N ) )  |  ph }  e.  (Dioph `  N )  /\  { t  e.  ( NN0  ^m  ( 1 ... N ) )  |  ps }  e.  (Dioph `  N ) )  ->  { t  e.  ( NN0  ^m  (
 1 ... N ) )  |  ( ph  /\  ps ) }  e.  (Dioph `  N ) )
 
Theoremorrabdioph 26027* Diophantine set builder for disjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  (
 ( { t  e.  ( NN0  ^m  (
 1 ... N ) )  |  ph }  e.  (Dioph `  N )  /\  { t  e.  ( NN0  ^m  ( 1 ... N ) )  |  ps }  e.  (Dioph `  N ) )  ->  { t  e.  ( NN0  ^m  (
 1 ... N ) )  |  ( ph  \/  ps ) }  e.  (Dioph `  N ) )
 
Theorem3anrabdioph 26028* Diophantine set builder for ternary conjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  (
 ( { t  e.  ( NN0  ^m  (
 1 ... N ) )  |  ph }  e.  (Dioph `  N )  /\  { t  e.  ( NN0  ^m  ( 1 ... N ) )  |  ps }  e.  (Dioph `  N )  /\  { t  e.  ( NN0  ^m  (
 1 ... N ) )  |  ch }  e.  (Dioph `  N ) ) 
 ->  { t  e.  ( NN0  ^m  ( 1 ...
 N ) )  |  ( ph  /\  ps  /\ 
 ch ) }  e.  (Dioph `  N ) )
 
Theorem3orrabdioph 26029* Diophantine set builder for ternary disjunctions. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  (
 ( { t  e.  ( NN0  ^m  (
 1 ... N ) )  |  ph }  e.  (Dioph `  N )  /\  { t  e.  ( NN0  ^m  ( 1 ... N ) )  |  ps }  e.  (Dioph `  N )  /\  { t  e.  ( NN0  ^m  (
 1 ... N ) )  |  ch }  e.  (Dioph `  N ) ) 
 ->  { t  e.  ( NN0  ^m  ( 1 ...
 N ) )  |  ( ph  \/  ps  \/  ch ) }  e.  (Dioph `  N ) )
 
16.15.15  Diophantine sets 4 miscellanea
 
Theorem2sbcrex 26030* Exchange an existential quantifier with two substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( [. A  /  a ]. [. B  /  b ]. E. c  e.  C  ph  <->  E. c  e.  C  [. A  /  a ]. [. B  /  b ]. ph )
 
Theoremsbc2rexg 26031* Exchange a substitution with two existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  ( A  e.  V  ->  (
 [. A  /  a ]. E. b  e.  B  E. c  e.  C  ph  <->  E. b  e.  B  E. c  e.  C  [. A  /  a ]. ph )
 )
 
Theoremsbc4rexg 26032* Exchange a substitution with 4 existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  ( A  e.  V  ->  (
 [. A  /  a ]. E. b  e.  B  E. c  e.  C  E. d  e.  D  E. e  e.  E  ph  <->  E. b  e.  B  E. c  e.  C  E. d  e.  D  E. e  e.  E  [. A  /  a ]. ph ) )
 
TheoremsbcbiiiOLD 26033 Fully inferenced rewriting under an explicit substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  A  e.  _V   &    |-  ( ph  <->  ps )   =>    |-  ( [. A  /  a ]. ph  <->  [. A  /  a ]. ps )
 
Theoremsbcrot3 26034* Rotate a sequence of three explicit substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  ( [. A  /  a ]. [. B  /  b ]. [. C  /  c ]. ph  <->  [. B  /  b ]. [. C  /  c ]. [. A  /  a ]. ph )
 
Theoremsbcrot5 26035* Rotate a sequence of five explicit substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  ( [. A  /  a ]. [. B  /  b ]. [. C  /  c ]. [. D  /  d ]. [. E  /  e ]. ph  <->  [. B  /  b ]. [. C  /  c ]. [. D  /  d ]. [. E  /  e ]. [. A  /  a ]. ph )
 
Theoremsbccomieg 26036* Commute two explicit substitutions, using an implicit substitution to rewrite the exiting substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  (
 a  =  A  ->  B  =  C )   =>    |-  ( [. A  /  a ]. [. B  /  b ]. ph  <->  [. C  /  b ]. [. A  /  a ]. ph )
 
Theoremsbcrot3gOLD 26037* Rotate a sequence of three explicit substitutions, closed theorem. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  (
 ( A  e.  D  /\  B  e.  E  /\  A. b  C  e.  F )  ->  ( [. A  /  a ]. [. B  /  b ]. [. C  /  c ]. ph  <->  [. B  /  b ]. [. C  /  c ]. [. A  /  a ]. ph ) )
 
Theoremsbcrot3OLD 26038* Rotate a sequence of three explicit substitutions. Substituted values must be manifest sets. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( [. A  /  a ]. [. B  /  b ]. [. C  /  c ]. ph  <->  [. B  /  b ]. [. C  /  c ]. [. A  /  a ]. ph )
 
Theoremsbcrot5OLD 26039* Rotate a sequence of five explicit substitutions. Substituted values must be manifest sets. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   &    |-  E  e.  _V   =>    |-  ( [. A  /  a ]. [. B  /  b ]. [. C  /  c ]. [. D  /  d ]. [. E  /  e ]. ph  <->  [. B  /  b ]. [. C  /  c ]. [. D  /  d ]. [. E  /  e ]. [. A  /  a ]. ph )
 
TheoremsbccomiegOLD 26040* Commute two explicit substitutions, using an implicit substitution to rewrite the exiting substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  (
 a  =  A  ->  B  =  C )   =>    |-  ( ( A  e.  V  /\  C  e.  W )  ->  ( [. A  /  a ]. [. B  /  b ]. ph  <->  [. C  /  b ]. [. A  /  a ]. ph ) )
 
16.15.16  Diophantine sets 4: Quantification
 
Theoremrexrabdioph 26041* Diophantine set builder for existential quantification. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  M  =  ( N  +  1 )   &    |-  ( v  =  ( t `  M )  ->  ( ps  <->  ch ) )   &    |-  ( u  =  ( t  |`  ( 1 ... N ) )  ->  ( ch  <->  ph ) )   =>    |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0  ^m  ( 1 ...
 M ) )  | 
 ph }  e.  (Dioph `  M ) )  ->  { u  e.  ( NN0  ^m  ( 1 ...
 N ) )  | 
 E. v  e.  NN0  ps
 }  e.  (Dioph `  N ) )
 
Theoremrexfrabdioph 26042* Diophantine set builder for existential quantifier, explicit substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  M  =  ( N  +  1 )   =>    |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0  ^m  ( 1 ...
 M ) )  | 
 [. ( t  |`  ( 1 ... N ) )  /  u ].
 [. ( t `  M )  /  v ]. ph }  e.  (Dioph `  M ) )  ->  { u  e.  ( NN0  ^m  ( 1 ...
 N ) )  | 
 E. v  e.  NN0  ph
 }  e.  (Dioph `  N ) )
 
Theorem2rexfrabdioph 26043* Diophantine set builder for existential quantifier, explicit substitution, two variables. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  M  =  ( N  +  1 )   &    |-  L  =  ( M  +  1 )   =>    |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0  ^m  ( 1 ...
 L ) )  | 
 [. ( t  |`  ( 1 ... N ) )  /  u ].
 [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ].
 ph }  e.  (Dioph `  L ) )  ->  { u  e.  ( NN0  ^m  ( 1 ...
 N ) )  | 
 E. v  e.  NN0  E. w  e.  NN0  ph }  e.  (Dioph `  N ) )
 
Theorem3rexfrabdioph 26044* Diophantine set builder for existential quantifier, explicit substitution, two variables. (Contributed by Stefan O'Rear, 17-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  M  =  ( N  +  1 )   &    |-  L  =  ( M  +  1 )   &    |-  K  =  ( L  +  1 )   =>    |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0  ^m  (
 1 ... K ) )  |  [. ( t  |`  ( 1 ... N ) )  /  u ].
 [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ].
 [. ( t `  K )  /  x ].
 ph }  e.  (Dioph `  K ) )  ->  { u  e.  ( NN0  ^m  ( 1 ...
 N ) )  | 
 E. v  e.  NN0  E. w  e.  NN0  E. x  e.  NN0  ph }  e.  (Dioph `  N ) )
 
Theorem4rexfrabdioph 26045* Diophantine set builder for existential quantifier, explicit substitution, four variables. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  M  =  ( N  +  1 )   &    |-  L  =  ( M  +  1 )   &    |-  K  =  ( L  +  1 )   &    |-  J  =  ( K  +  1 )   =>    |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0  ^m  ( 1 ...
 J ) )  | 
 [. ( t  |`  ( 1 ... N ) )  /  u ].
 [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ].
 [. ( t `  K )  /  x ].
 [. ( t `  J )  /  y ]. ph }  e.  (Dioph `  J ) )  ->  { u  e.  ( NN0  ^m  ( 1 ...
 N ) )  | 
 E. v  e.  NN0  E. w  e.  NN0  E. x  e.  NN0  E. y  e. 
 NN0  ph }  e.  (Dioph `  N ) )
 
Theorem6rexfrabdioph 26046* Diophantine set builder for existential quantifier, explicit substitution, six variables. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  M  =  ( N  +  1 )   &    |-  L  =  ( M  +  1 )   &    |-  K  =  ( L  +  1 )   &    |-  J  =  ( K  +  1 )   &    |-  I  =  ( J  +  1 )   &    |-  H  =  ( I  +  1 )   =>    |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0  ^m  (
 1 ... H ) )  |  [. ( t  |`  ( 1 ... N ) )  /  u ].
 [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ].
 [. ( t `  K )  /  x ].
 [. ( t `  J )  /  y ]. [. ( t `  I )  /  z ]. [. ( t `  H )  /  p ].
 ph }  e.  (Dioph `  H ) )  ->  { u  e.  ( NN0  ^m  ( 1 ...
 N ) )  | 
 E. v  e.  NN0  E. w  e.  NN0  E. x  e.  NN0  E. y  e. 
 NN0  E. z  e.  NN0  E. p  e.  NN0  ph }  e.  (Dioph `  N ) )
 
Theorem7rexfrabdioph 26047* Diophantine set builder for existential quantifier, explicit substitution, seven variables. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  M  =  ( N  +  1 )   &    |-  L  =  ( M  +  1 )   &    |-  K  =  ( L  +  1 )   &    |-  J  =  ( K  +  1 )   &    |-  I  =  ( J  +  1 )   &    |-  H  =  ( I  +  1 )   &    |-  G  =  ( H  +  1 )   =>    |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0  ^m  ( 1 ...
 G ) )  | 
 [. ( t  |`  ( 1 ... N ) )  /  u ].
 [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ].
 [. ( t `  K )  /  x ].
 [. ( t `  J )  /  y ]. [. ( t `  I )  /  z ]. [. ( t `  H )  /  p ].
 [. ( t `  G )  /  q ]. ph }  e.  (Dioph `  G ) )  ->  { u  e.  ( NN0  ^m  ( 1 ...
 N ) )  | 
 E. v  e.  NN0  E. w  e.  NN0  E. x  e.  NN0  E. y  e. 
 NN0  E. z  e.  NN0  E. p  e.  NN0  E. q  e.  NN0  ph }  e.  (Dioph `  N ) )
 
16.15.17  Diophantine sets 5: Arithmetic sets
 
Theoremrabdiophlem1 26048* Lemma for arithmetic diophantine sets. Convert polynomial-ness of an expression into a constraint suitable for ralimi 2580. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  (
 ( t  e.  ( ZZ  ^m  ( 1 ...
 N ) )  |->  A )  e.  (mzPoly `  (
 1 ... N ) ) 
 ->  A. t  e.  ( NN0  ^m  ( 1 ...
 N ) ) A  e.  ZZ )
 
Theoremrabdiophlem2 26049* Lemma for arithmetic diophantine sets. Reuse a polynomial expression under a new quantifier. (Contributed by Stefan O'Rear, 10-Oct-2014.)
 |-  M  =  ( N  +  1 )   =>    |-  ( ( N  e.  NN0  /\  ( u  e.  ( ZZ  ^m  ( 1 ...
 N ) )  |->  A )  e.  (mzPoly `  (
 1 ... N ) ) )  ->  ( t  e.  ( ZZ  ^m  (
 1 ... M ) ) 
 |->  [_ ( t  |`  ( 1 ... N ) )  /  u ]_ A )  e.  (mzPoly `  ( 1 ... M ) ) )
 
Theoremelnn0rabdioph 26050* Diophantine set builder for nonnegativity constraints. The first builder which uses a witness variable internally; an expression is nonnegative if there is a nonnegative integer equal to it. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  (
 ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ...
 N ) )  |->  A )  e.  (mzPoly `  (
 1 ... N ) ) )  ->  { t  e.  ( NN0  ^m  (
 1 ... N ) )  |  A  e.  NN0 }  e.  (Dioph `  N ) )
 
Theoremrexzrexnn0 26051* Rewrite a quantification over integers into a quantification over naturals. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ( x  =  -u y  ->  ( ph  <->  ch ) )   =>    |-  ( E. x  e.  ZZ  ph  <->  E. y  e.  NN0  ( ps  \/  ch )
 )
 
Theoremlerabdioph 26052* Diophantine set builder for the less or equals relation. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  (
 ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ...
 N ) )  |->  A )  e.  (mzPoly `  (
 1 ... N ) ) 
 /\  ( t  e.  ( ZZ  ^m  (
 1 ... N ) ) 
 |->  B )  e.  (mzPoly `  ( 1 ... N ) ) )  ->  { t  e.  ( NN0  ^m  ( 1 ...
 N ) )  |  A  <_  B }  e.  (Dioph `  N )
 )
 
Theoremeluzrabdioph 26053* Diophantine set builder for membership in a fixed set of upper integers. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  (
 ( N  e.  NN0  /\  M  e.  ZZ  /\  ( t  e.  ( ZZ  ^m  ( 1 ...
 N ) )  |->  A )  e.  (mzPoly `  (
 1 ... N ) ) )  ->  { t  e.  ( NN0  ^m  (
 1 ... N ) )  |  A  e.  ( ZZ>=
 `  M ) }  e.  (Dioph `  N )
 )
 
Theoremelnnrabdioph 26054* Diophantine set builder for positivity. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  (
 ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ...
 N ) )  |->  A )  e.  (mzPoly `  (
 1 ... N ) ) )  ->  { t  e.  ( NN0  ^m  (
 1 ... N ) )  |  A  e.  NN }  e.  (Dioph `  N ) )
 
Theoremltrabdioph 26055* Diophantine set builder for the strict less than relation. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  (
 ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ...
 N ) )  |->  A )  e.  (mzPoly `  (
 1 ... N ) ) 
 /\  ( t  e.  ( ZZ  ^m  (
 1 ... N ) ) 
 |->  B )  e.  (mzPoly `  ( 1 ... N ) ) )  ->  { t  e.  ( NN0  ^m  ( 1 ...
 N ) )  |  A  <  B }  e.  (Dioph `  N )
 )
 
Theoremnerabdioph 26056* Diophantine set builder for inequality. This not quite trivial theorem touches on something important; Diophantine sets are not closed under negation, but they contain an important subclass that is, namely the recursive sets. With this theorem and De Morgan's laws, all quantifier-free formulae can be negated. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  (
 ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ...
 N ) )  |->  A )  e.  (mzPoly `  (
 1 ... N ) ) 
 /\  ( t  e.  ( ZZ  ^m  (
 1 ... N ) ) 
 |->  B )  e.  (mzPoly `  ( 1 ... N ) ) )  ->  { t  e.  ( NN0  ^m  ( 1 ...
 N ) )  |  A  =/=  B }  e.  (Dioph `  N )
 )
 
Theoremdvdsrabdioph 26057* Divisibility is a Diophantine relation. (Contributed by Stefan O'Rear, 11-Oct-2014.)
 |-  (
 ( N  e.  NN0  /\  ( t  e.  ( ZZ  ^m  ( 1 ...
 N ) )  |->  A )  e.  (mzPoly `  (
 1 ... N ) ) 
 /\  ( t  e.  ( ZZ  ^m  (
 1 ... N ) ) 
 |->  B )  e.  (mzPoly `  ( 1 ... N ) ) )  ->  { t  e.  ( NN0  ^m  ( 1 ...
 N ) )  |  A  ||  B }  e.  (Dioph `  N )
 )
 
16.15.18  Diophantine sets 6 miscellanea
 
Theoremfz1ssnn 26058 A finite set of positive integers is a set of positive integers. (Contributed by Stefan O'Rear, 16-Oct-2014.)
 |-  (
 1 ... A )  C_  NN
 
Theoremftp 26059 A function with a domain of three elements. (Contributed by Stefan O'Rear, 17-Oct-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  X  e.  _V   &    |-  Y  e.  _V   &    |-  Z  e.  _V   &    |-  A  =/=  B   &    |-  A  =/=  C   &    |-  B  =/=  C   =>    |- 
 { <. A ,  X >. ,  <. B ,  Y >. ,  <. C ,  Z >. } : { A ,  B ,  C } --> { X ,  Y ,  Z }
 
16.15.19  Diophantine sets 6: reusability. renumbering of variables
 
Theoremeldioph4b 26060* Membership in Dioph expressed using a quantified union to add witness variables instead of a restriction to remove them. (Contributed by Stefan O'Rear, 16-Oct-2014.)
 |-  W  e.  _V   &    |-  -.  W  e.  Fin   &    |-  ( W  i^i  NN )  =  (/)   =>    |-  ( S  e.  (Dioph `  N )  <->  ( N  e.  NN0  /\  E. p  e.  (mzPoly `  ( W  u.  (
 1 ... N ) ) ) S  =  {
 t  e.  ( NN0  ^m  ( 1 ... N ) )  |  E. w  e.  ( NN0  ^m  W ) ( p `  ( t  u.  w ) )  =  0 } ) )
 
Theoremeldioph4i 26061* Forward-only version of eldioph4b 26060. (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 26062* 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 26063* 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 26064* 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 ) )
 
16.15.20  Pigeonhole Principle and cardinality helpers
 
Theoremfphpd 26065* 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 26066* 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 26067 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 26068* 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 )
 
16.15.21  A non-closed set of reals is infinite
 
Theoremrencldnfilem 26069* Lemma for rencldnfi 26070. (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 26070* A set of real numbers which comes arbitrarily close to some target yet excludes it is infinite. The work is done in rencldnfilem 26069 using infima; this theorem removes the requirement that A be non-empty. (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 )
 
16.15.22  Miscellanea for Lagrange's theorem
 
Theoremicodiamlt 26071 Two elements in a half-open interval have separation strictly less than the difference between the endpoints. (Contributed by Stefan O'Rear, 12-Sep-2014.)
 |-  (
 ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,) B ) 
 /\  D  e.  ( A [,) B ) ) )  ->  ( abs `  ( C  -  D ) )  <  ( B  -  A ) )
 
Theoremmodelico 26072 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 ) )
 
16.15.23  Lagrange's rational approximation theorem
 
Theoremirrapxlem1 26073* Lemma for irrapx1 26079. Divides the unit interval into  B half-open sections and using the pigeonhole principle fphpdo 26066 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 26074* Lemma for irrapx1 26079. 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 26075* Lemma for irrapx1 26079. 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 26076* Lemma for irrapx1 26079. 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 26077* Lemma for irrapx1 26079. 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 26078* Lemma for irrapx1 26079. 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 26079* 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 )
 
16.15.24  Pell equations 1: A nontrivial solution always exists
 
Theorempellexlem1 26080 Lemma for pellex 26086. 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 26081 Lemma for pellex 26086. 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 26082* Lemma for pellex 26086. 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 26083* Lemma for pellex 26086. Invoking irrapx1 26079, 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 26084* Lemma for pellex 26086. Invoking fiphp3d 26068, 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 26085* Lemma for pellex 26086. 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 26086* 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 )
 
16.15.25  Pell equations 2: Algebraic number theory of the solution set
 
Syntaxcsquarenn 26087 Extend class notation to include the set of square natural numbers.
 classNN
 
Syntaxcpell1qr 26088 Extend class notation to include the class of quadrant-1 Pell solutions.
 class Pell1QR
 
Syntaxcpell1234qr 26089 Extend class notation to include the class of any-quadrant Pell solutions.
 class Pell1234QR
 
Syntaxcpell14qr 26090 Extend class notation to include the class of positive Pell solutions.
 class Pell14QR
 
Syntaxcpellfund 26091 Extend class notation to include the Pell-equation fundamental solution function.
 class PellFund
 
Definitiondf-squarenn 26092 Define the set of square natural numbers. (Contributed by Stefan O'Rear, 18-Sep-2014.)
 |-NN  =  { x  e. 
 NN  |  ( sqr `  x )  e.  QQ }
 
Definitiondf-pell1qr 26093* 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 26094* 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 26095* 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 26096* A function mapping Pell discriminants to the corresponding fundamental solution. (Contributed by Stefan O'Rear, 18-Sep-2014.)
 |- PellFund  =  ( x  e.  ( NN  \NN ) 
 |->  sup ( { z  e.  (Pell14QR `  x )  |  1  <  z } ,  RR ,  `'  <  ) )
 
Theorempell1qrval 26097* 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 26098* 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 26099* 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 26100* 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 ) ) ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-30955
  Copyright terms: Public domain < Previous  Next >