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Theorem elpell14qr 27037
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 27036 . . 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 2363 . 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 2302 . . . . 5  |-  ( a  =  A  ->  (
a  =  ( z  +  ( ( sqr `  D )  x.  w
) )  <->  A  =  ( z  +  ( ( sqr `  D
)  x.  w ) ) ) )
43anbi1d 685 . . . 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 2599 . . 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 2936 . 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 252 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   {crab 2560    \ cdif 3162   ` cfv 5271  (class class class)co 5874   RRcr 8752   1c1 8754    + caddc 8756    x. cmul 8758    - cmin 9053   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   ^cexp 11120   sqrcsqr 11734  ◻NNcsquarenn 27024  Pell14QRcpell14qr 27027
This theorem is referenced by:  pell14qrss1234  27044  pell14qrgt0  27047  pell1234qrdich  27049  pell1qrss14  27056  pell14qrdich  27057  rmxycomplete  27105
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-cnex 8809  ax-resscn 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-pell14qr 27031
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