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Theorem List for Metamath Proof Explorer - 26201-26300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsbcrot5 26201* 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 26202* 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 26203* 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 26204* 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 26205* 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 26206* 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 ) )
 
16.16.16  Diophantine sets 4: Quantification
 
Theoremrexrabdioph 26207* 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 26208* 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 26209* 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 26210* 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 26211* 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 26212* 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 26213* 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 ) )
 
16.16.17  Diophantine sets 5: Arithmetic sets
 
Theoremrabdiophlem1 26214* Lemma for arithmetic diophantine sets. Convert polynomial-ness of an expression into a constraint suitable for ralimi 2589. (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 26215* 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 26216* 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 26217* 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 26218* 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 26219* 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 26220* 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 26221* 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 26222* 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 26223* 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 )
 )
 
16.16.18  Diophantine sets 6 miscellanea
 
Theoremfz1ssnn 26224 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 26225 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 }
 
16.16.19  Diophantine sets 6: reusability. renumbering of variables
 
Theoremeldioph4b 26226* 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 26227* Forward-only version of eldioph4b 26226. (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 26228* 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 26229* 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 26230* 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 ) )
 
16.16.20  Pigeonhole Principle and cardinality helpers
 
Theoremfphpd 26231* 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 26232* 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 26233 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 26234* 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 )
 
16.16.21  A non-closed set of reals is infinite
 
Theoremrencldnfilem 26235* Lemma for rencldnfi 26236. (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 26236* A set of real numbers which comes arbitrarily close to some target yet excludes it is infinite. The work is done in rencldnfilem 26235 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 )
 
16.16.22  Miscellanea for Lagrange's theorem
 
Theoremicodiamlt 26237 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 26238 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 ) )
 
16.16.23  Lagrange's rational approximation theorem
 
Theoremirrapxlem1 26239* Lemma for irrapx1 26245. Divides the unit interval into  B half-open sections and using the pigeonhole principle fphpdo 26232 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 26240* Lemma for irrapx1 26245. 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 26241* Lemma for irrapx1 26245. 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 26242* Lemma for irrapx1 26245. 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 26243* Lemma for irrapx1 26245. 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 26244* Lemma for irrapx1 26245. 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 26245* 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 )
 
16.16.24  Pell equations 1: A nontrivial solution always exists
 
Theorempellexlem1 26246 Lemma for pellex 26252. 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 26247 Lemma for pellex 26252. 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 26248* Lemma for pellex 26252. 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 26249* Lemma for pellex 26252. Invoking irrapx1 26245, 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 26250* Lemma for pellex 26252. Invoking fiphp3d 26234, 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 26251* Lemma for pellex 26252. 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 26252* 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 )
 
16.16.25  Pell equations 2: Algebraic number theory of the solution set
 
Syntaxcsquarenn 26253 Extend class notation to include the set of square natural numbers.
 classNN
 
Syntaxcpell1qr 26254 Extend class notation to include the class of quadrant-1 Pell solutions.
 class Pell1QR
 
Syntaxcpell1234qr 26255 Extend class notation to include the class of any-quadrant Pell solutions.
 class Pell1234QR
 
Syntaxcpell14qr 26256 Extend class notation to include the class of positive Pell solutions.
 class Pell14QR
 
Syntaxcpellfund 26257 Extend class notation to include the Pell-equation fundamental solution function.
 class PellFund
 
Definitiondf-squarenn 26258 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 26259* 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 26260* 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 26261* 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 26262* 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 26263* 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 26264* 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 26265* 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 26266* 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 26267* 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 26268* 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 26269 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 26270 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 26271 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 26272 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 26273 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 26274 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 26275 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 26276 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 26277 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 26278 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 26279 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 26280 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 26281 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 26282 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 26283 Lemma for pell14qrexpcl 26284. (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 26284 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 26285 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 26286 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 26287 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 26288 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 26289 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 26290 Lemma for pell1qrgap 26291. (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 26291 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 26292 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 26293 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 26294 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 26295* 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 26296* 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 26297* 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 26298 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 26299 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 26300 Weak lower bound on the Pell fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.)
 |-  ( D  e.  ( NN  \NN )  ->  1  <  (PellFund `  D ) )
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