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Theorem pell1234qrval 30990
Description: Value of the set of general Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Assertion
Ref Expression
pell1234qrval  |-  ( D  e.  ( NN  \NN )  -> 
(Pell1234QR `  D )  =  { y  e.  RR  |  E. z  e.  ZZ  E. w  e.  ZZ  (
y  =  ( z  +  ( ( sqr `  D )  x.  w
) )  /\  (
( z ^ 2 )  -  ( D  x.  ( w ^
2 ) ) )  =  1 ) } )
Distinct variable group:    y, z, w, D

Proof of Theorem pell1234qrval
Dummy variable  d is distinct from all other variables.
StepHypRef Expression
1 fveq2 5872 . . . . . . . 8  |-  ( d  =  D  ->  ( sqr `  d )  =  ( sqr `  D
) )
21oveq1d 6311 . . . . . . 7  |-  ( d  =  D  ->  (
( sqr `  d
)  x.  w )  =  ( ( sqr `  D )  x.  w
) )
32oveq2d 6312 . . . . . 6  |-  ( d  =  D  ->  (
z  +  ( ( sqr `  d )  x.  w ) )  =  ( z  +  ( ( sqr `  D
)  x.  w ) ) )
43eqeq2d 2471 . . . . 5  |-  ( d  =  D  ->  (
y  =  ( z  +  ( ( sqr `  d )  x.  w
) )  <->  y  =  ( z  +  ( ( sqr `  D
)  x.  w ) ) ) )
5 oveq1 6303 . . . . . . 7  |-  ( d  =  D  ->  (
d  x.  ( w ^ 2 ) )  =  ( D  x.  ( w ^ 2 ) ) )
65oveq2d 6312 . . . . . 6  |-  ( d  =  D  ->  (
( z ^ 2 )  -  ( d  x.  ( w ^
2 ) ) )  =  ( ( z ^ 2 )  -  ( D  x.  (
w ^ 2 ) ) ) )
76eqeq1d 2459 . . . . 5  |-  ( d  =  D  ->  (
( ( z ^
2 )  -  (
d  x.  ( w ^ 2 ) ) )  =  1  <->  (
( z ^ 2 )  -  ( D  x.  ( w ^
2 ) ) )  =  1 ) )
84, 7anbi12d 710 . . . 4  |-  ( d  =  D  ->  (
( y  =  ( z  +  ( ( sqr `  d )  x.  w ) )  /\  ( ( z ^ 2 )  -  ( d  x.  (
w ^ 2 ) ) )  =  1 )  <->  ( y  =  ( z  +  ( ( sqr `  D
)  x.  w ) )  /\  ( ( z ^ 2 )  -  ( D  x.  ( w ^ 2 ) ) )  =  1 ) ) )
982rexbidv 2975 . . 3  |-  ( d  =  D  ->  ( E. z  e.  ZZ  E. w  e.  ZZ  (
y  =  ( z  +  ( ( sqr `  d )  x.  w
) )  /\  (
( z ^ 2 )  -  ( d  x.  ( w ^
2 ) ) )  =  1 )  <->  E. z  e.  ZZ  E. w  e.  ZZ  ( y  =  ( z  +  ( ( sqr `  D
)  x.  w ) )  /\  ( ( z ^ 2 )  -  ( D  x.  ( w ^ 2 ) ) )  =  1 ) ) )
109rabbidv 3101 . 2  |-  ( d  =  D  ->  { y  e.  RR  |  E. z  e.  ZZ  E. w  e.  ZZ  ( y  =  ( z  +  ( ( sqr `  d
)  x.  w ) )  /\  ( ( z ^ 2 )  -  ( d  x.  ( w ^ 2 ) ) )  =  1 ) }  =  { y  e.  RR  |  E. z  e.  ZZ  E. w  e.  ZZ  (
y  =  ( z  +  ( ( sqr `  D )  x.  w
) )  /\  (
( z ^ 2 )  -  ( D  x.  ( w ^
2 ) ) )  =  1 ) } )
11 df-pell1234qr 30984 . 2  |- Pell1234QR  =  ( d  e.  ( NN 
\NN )  |->  { y  e.  RR  |  E. z  e.  ZZ  E. w  e.  ZZ  ( y  =  ( z  +  ( ( sqr `  d
)  x.  w ) )  /\  ( ( z ^ 2 )  -  ( d  x.  ( w ^ 2 ) ) )  =  1 ) } )
12 reex 9600 . . 3  |-  RR  e.  _V
1312rabex 4607 . 2  |-  { y  e.  RR  |  E. z  e.  ZZ  E. w  e.  ZZ  ( y  =  ( z  +  ( ( sqr `  D
)  x.  w ) )  /\  ( ( z ^ 2 )  -  ( D  x.  ( w ^ 2 ) ) )  =  1 ) }  e.  _V
1410, 11, 13fvmpt 5956 1  |-  ( D  e.  ( NN  \NN )  -> 
(Pell1234QR `  D )  =  { y  e.  RR  |  E. z  e.  ZZ  E. w  e.  ZZ  (
y  =  ( z  +  ( ( sqr `  D )  x.  w
) )  /\  (
( z ^ 2 )  -  ( D  x.  ( w ^
2 ) ) )  =  1 ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   E.wrex 2808   {crab 2811    \ cdif 3468   ` cfv 5594  (class class class)co 6296   RRcr 9508   1c1 9510    + caddc 9512    x. cmul 9514    - cmin 9824   NNcn 10556   2c2 10606   ZZcz 10885   ^cexp 12169   sqrcsqrt 13078  ◻NNcsquarenn 30976  Pell1234QRcpell1234qr 30978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-cnex 9565  ax-resscn 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-pell1234qr 30984
This theorem is referenced by:  elpell1234qr  30991
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