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Theorem elpell1234qr 30618
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 30617 . . 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 2537 . 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 2471 . . . . 5  |-  ( a  =  A  ->  (
a  =  ( z  +  ( ( sqr `  D )  x.  w
) )  <->  A  =  ( z  +  ( ( sqr `  D
)  x.  w ) ) ) )
43anbi1d 704 . . . 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 2980 . . 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 3261 . 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 261 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815   {crab 2818    \ cdif 3473   ` cfv 5588  (class class class)co 6285   RRcr 9492   1c1 9494    + caddc 9496    x. cmul 9498    - cmin 9806   NNcn 10537   2c2 10586   ZZcz 10865   ^cexp 12135   sqrcsqrt 13032  ◻NNcsquarenn 30603  Pell1234QRcpell1234qr 30605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-cnex 9549  ax-resscn 9550
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6288  df-pell1234qr 30611
This theorem is referenced by:  pell1234qrre  30619  pell1234qrne0  30620  pell1234qrreccl  30621  pell1234qrmulcl  30622  pell14qrss1234  30623  pell1234qrdich  30628
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