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Theorem elpell1234qr 26928
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 26927 . . 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 2505 . 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 2444 . . . . 5  |-  ( a  =  A  ->  (
a  =  ( z  +  ( ( sqr `  D )  x.  w
) )  <->  A  =  ( z  +  ( ( sqr `  D
)  x.  w ) ) ) )
43anbi1d 687 . . . 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 2750 . . 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 3094 . 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 254 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708   {crab 2711    \ cdif 3319   ` cfv 5457  (class class class)co 6084   RRcr 8994   1c1 8996    + caddc 8998    x. cmul 9000    - cmin 9296   NNcn 10005   2c2 10054   ZZcz 10287   ^cexp 11387   sqrcsqr 12043  ◻NNcsquarenn 26913  Pell1234QRcpell1234qr 26915
This theorem is referenced by:  pell1234qrre  26929  pell1234qrne0  26930  pell1234qrreccl  26931  pell1234qrmulcl  26932  pell14qrss1234  26933  pell1234qrdich  26938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-cnex 9051  ax-resscn 9052
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-pell1234qr 26921
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