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Theorem List for Metamath Proof Explorer - 26801-26900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsbcrot3 26801* 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 26802* 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 26803* 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 26804* Rotate a sequence of three explicit substitutions, closed theorem. (Contributed by Stefan O'Rear, 11-Oct-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  (
 ( 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 26805* Rotate a sequence of three explicit substitutions. Substituted values must be manifest sets. (Contributed by Stefan O'Rear, 11-Oct-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   =>    |-  ( [. A  /  a ]. [. B  /  b ]. [. C  /  c ]. ph  <->  [. B  /  b ]. [. C  /  c ]. [. A  /  a ]. ph )
 
Theoremsbcrot5OLD 26806* Rotate a sequence of five explicit substitutions. Substituted values must be manifest sets. (Contributed by Stefan O'Rear, 11-Oct-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  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 26807* Commute two explicit substitutions, using an implicit substitution to rewrite the exiting substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  (
 a  =  A  ->  B  =  C )   =>    |-  ( ( A  e.  V  /\  C  e.  W )  ->  ( [. A  /  a ]. [. B  /  b ]. ph  <->  [. C  /  b ]. [. A  /  a ]. ph ) )
 
19.16.16  Diophantine sets 4: Quantification
 
Theoremrexrabdioph 26808* 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 26809* 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 26810* 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 26811* 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 26812* 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 26813* 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 26814* 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 ) )
 
19.16.17  Diophantine sets 5: Arithmetic sets
 
Theoremrabdiophlem1 26815* Lemma for arithmetic diophantine sets. Convert polynomial-ness of an expression into a constraint suitable for ralimi 2773. (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 26816* 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 26817* 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 26818* 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 26819* 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 26820* 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 26821* 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 26822* 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 26823* 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 26824* 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 )
 )
 
19.16.18  Diophantine sets 6 miscellanea
 
Theoremfz1ssnn 26825 A finite set of positive integers is a set of positive integers. (Contributed by Stefan O'Rear, 16-Oct-2014.)
 |-  (
 1 ... A )  C_  NN
 
19.16.19  Diophantine sets 6: reusability. renumbering of variables
 
Theoremeldioph4b 26826* 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 26827* Forward-only version of eldioph4b 26826. (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 26828* 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 26829* 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 26830* 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 ) )
 
19.16.20  Pigeonhole Principle and cardinality helpers
 
Theoremfphpd 26831* 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 26832* 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 26833 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 26834* 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 )
 
19.16.21  A non-closed set of reals is infinite
 
Theoremrencldnfilem 26835* Lemma for rencldnfi 26836. (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 26836* A set of real numbers which comes arbitrarily close to some target yet excludes it is infinite. The work is done in rencldnfilem 26835 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 )
 
19.16.22  Miscellanea for Lagrange's theorem
 
Theoremicodiamlt 26837 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 26838 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 ) )
 
19.16.23  Lagrange's rational approximation theorem
 
Theoremirrapxlem1 26839* Lemma for irrapx1 26845. Divides the unit interval into  B half-open sections and using the pigeonhole principle fphpdo 26832 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 26840* Lemma for irrapx1 26845. 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 26841* Lemma for irrapx1 26845. 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 26842* Lemma for irrapx1 26845. 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 26843* Lemma for irrapx1 26845. 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 26844* Lemma for irrapx1 26845. 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 26845* 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 )
 
19.16.24  Pell equations 1: A nontrivial solution always exists
 
Theorempellexlem1 26846 Lemma for pellex 26852. 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 26847 Lemma for pellex 26852. 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 26848* Lemma for pellex 26852. 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 26849* Lemma for pellex 26852. Invoking irrapx1 26845, 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 26850* Lemma for pellex 26852. Invoking fiphp3d 26834, 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 26851* Lemma for pellex 26852. 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 26852* 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 )
 
19.16.25  Pell equations 2: Algebraic number theory of the solution set
 
Syntaxcsquarenn 26853 Extend class notation to include the set of square natural numbers.
 classNN
 
Syntaxcpell1qr 26854 Extend class notation to include the class of quadrant-1 Pell solutions.
 class Pell1QR
 
Syntaxcpell1234qr 26855 Extend class notation to include the class of any-quadrant Pell solutions.
 class Pell1234QR
 
Syntaxcpell14qr 26856 Extend class notation to include the class of positive Pell solutions.
 class Pell14QR
 
Syntaxcpellfund 26857 Extend class notation to include the Pell-equation fundamental solution function.
 class PellFund
 
Definitiondf-squarenn 26858 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 26859* 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 26860* 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 26861* 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 26862* 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 26863* 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 26864* 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 26865* 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 26866* 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 26867* 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 26868* 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 26869 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 26870 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 26871 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 26872 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 26873 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 26874 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 26875 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 26876 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 26877 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 26878 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 26879 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 26880 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 26881 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 26882 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 26883 Lemma for pell14qrexpcl 26884. (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 26884 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 26885 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 26886 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 26887 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 26888 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 26889 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 26890 Lemma for pell1qrgap 26891. (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 26891 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 26892 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 26893 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 26894 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 )
 )
 
19.16.26  Pell equations 3: characterizing fundamental solution
 
Theoreminfmrgelbi 26895* 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 26896* 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 26897* 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 26898 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 26899 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 26900 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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