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Theorem elpell1234qr 26804
Description: Membership in the set of general Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Assertion
Ref Expression
elpell1234qr  |-  ( 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 ) ) ) )
Distinct variable groups:    z, w, D    z, A, w

Proof of Theorem elpell1234qr
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 pell1234qrval 26803 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
(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 ) } )
21eleq2d 2471 . 2  |-  ( D  e.  ( NN  \NN )  -> 
( A  e.  (Pell1234QR `  D )  <->  A  e.  { a  e.  RR  |  E. z  e.  ZZ  E. w  e.  ZZ  (
a  =  ( z  +  ( ( sqr `  D )  x.  w
) )  /\  (
( z ^ 2 )  -  ( D  x.  ( w ^
2 ) ) )  =  1 ) } ) )
3 eqeq1 2410 . . . . 5  |-  ( a  =  A  ->  (
a  =  ( z  +  ( ( sqr `  D )  x.  w
) )  <->  A  =  ( z  +  ( ( sqr `  D
)  x.  w ) ) ) )
43anbi1d 686 . . . 4  |-  ( a  =  A  ->  (
( a  =  ( z  +  ( ( sqr `  D )  x.  w ) )  /\  ( ( z ^ 2 )  -  ( D  x.  (
w ^ 2 ) ) )  =  1 )  <->  ( A  =  ( z  +  ( ( sqr `  D
)  x.  w ) )  /\  ( ( z ^ 2 )  -  ( D  x.  ( w ^ 2 ) ) )  =  1 ) ) )
542rexbidv 2709 . . 3  |-  ( a  =  A  ->  ( E. z  e.  ZZ  E. w  e.  ZZ  (
a  =  ( z  +  ( ( sqr `  D )  x.  w
) )  /\  (
( z ^ 2 )  -  ( D  x.  ( w ^
2 ) ) )  =  1 )  <->  E. z  e.  ZZ  E. w  e.  ZZ  ( A  =  ( z  +  ( ( sqr `  D
)  x.  w ) )  /\  ( ( z ^ 2 )  -  ( D  x.  ( w ^ 2 ) ) )  =  1 ) ) )
65elrab 3052 . 2  |-  ( A  e.  { a  e.  RR  |  E. z  e.  ZZ  E. w  e.  ZZ  ( a  =  ( z  +  ( ( sqr `  D
)  x.  w ) )  /\  ( ( z ^ 2 )  -  ( D  x.  ( w ^ 2 ) ) )  =  1 ) }  <->  ( A  e.  RR  /\  E. z  e.  ZZ  E. w  e.  ZZ  ( A  =  ( z  +  ( ( sqr `  D
)  x.  w ) )  /\  ( ( z ^ 2 )  -  ( D  x.  ( w ^ 2 ) ) )  =  1 ) ) )
72, 6syl6bb 253 1  |-  ( 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 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2667   {crab 2670    \ cdif 3277   ` cfv 5413  (class class class)co 6040   RRcr 8945   1c1 8947    + caddc 8949    x. cmul 8951    - cmin 9247   NNcn 9956   2c2 10005   ZZcz 10238   ^cexp 11337   sqrcsqr 11993  ◻NNcsquarenn 26789  Pell1234QRcpell1234qr 26791
This theorem is referenced by:  pell1234qrre  26805  pell1234qrne0  26806  pell1234qrreccl  26807  pell1234qrmulcl  26808  pell14qrss1234  26809  pell1234qrdich  26814
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-cnex 9002  ax-resscn 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-pell1234qr 26797
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