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Theorem elpell14qr 26912
Description: Membership in the set of positive Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Assertion
Ref Expression
elpell14qr  |-  ( 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 ) ) ) )
Distinct variable groups:    z, w, D    z, A, w

Proof of Theorem elpell14qr
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 pell14qrval 26911 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
(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 ) } )
21eleq2d 2503 . 2  |-  ( D  e.  ( NN  \NN )  -> 
( A  e.  (Pell14QR `  D )  <->  A  e.  { a  e.  RR  |  E. z  e.  NN0  E. w  e.  ZZ  (
a  =  ( z  +  ( ( sqr `  D )  x.  w
) )  /\  (
( z ^ 2 )  -  ( D  x.  ( w ^
2 ) ) )  =  1 ) } ) )
3 eqeq1 2442 . . . . 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 2748 . . 3  |-  ( a  =  A  ->  ( E. z  e.  NN0  E. w  e.  ZZ  (
a  =  ( z  +  ( ( sqr `  D )  x.  w
) )  /\  (
( z ^ 2 )  -  ( D  x.  ( w ^
2 ) ) )  =  1 )  <->  E. z  e.  NN0  E. w  e.  ZZ  ( A  =  ( z  +  ( ( sqr `  D
)  x.  w ) )  /\  ( ( z ^ 2 )  -  ( D  x.  ( w ^ 2 ) ) )  =  1 ) ) )
65elrab 3092 . 2  |-  ( A  e.  { a  e.  RR  |  E. z  e.  NN0  E. w  e.  ZZ  ( a  =  ( z  +  ( ( sqr `  D
)  x.  w ) )  /\  ( ( z ^ 2 )  -  ( D  x.  ( w ^ 2 ) ) )  =  1 ) }  <->  ( A  e.  RR  /\  E. z  e.  NN0  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.  (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 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2706   {crab 2709    \ cdif 3317   ` cfv 5454  (class class class)co 6081   RRcr 8989   1c1 8991    + caddc 8993    x. cmul 8995    - cmin 9291   NNcn 10000   2c2 10049   NN0cn0 10221   ZZcz 10282   ^cexp 11382   sqrcsqr 12038  ◻NNcsquarenn 26899  Pell14QRcpell14qr 26902
This theorem is referenced by:  pell14qrss1234  26919  pell14qrgt0  26922  pell1234qrdich  26924  pell1qrss14  26931  pell14qrdich  26932  rmxycomplete  26980
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-cnex 9046  ax-resscn 9047
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-pell14qr 26906
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