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Theorem pell1234qrval 26807
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 5691 . . . . . . . 8  |-  ( d  =  D  ->  ( sqr `  d )  =  ( sqr `  D
) )
21oveq1d 6059 . . . . . . 7  |-  ( d  =  D  ->  (
( sqr `  d
)  x.  w )  =  ( ( sqr `  D )  x.  w
) )
32oveq2d 6060 . . . . . 6  |-  ( d  =  D  ->  (
z  +  ( ( sqr `  d )  x.  w ) )  =  ( z  +  ( ( sqr `  D
)  x.  w ) ) )
43eqeq2d 2419 . . . . 5  |-  ( d  =  D  ->  (
y  =  ( z  +  ( ( sqr `  d )  x.  w
) )  <->  y  =  ( z  +  ( ( sqr `  D
)  x.  w ) ) ) )
5 oveq1 6051 . . . . . . 7  |-  ( d  =  D  ->  (
d  x.  ( w ^ 2 ) )  =  ( D  x.  ( w ^ 2 ) ) )
65oveq2d 6060 . . . . . 6  |-  ( d  =  D  ->  (
( z ^ 2 )  -  ( d  x.  ( w ^
2 ) ) )  =  ( ( z ^ 2 )  -  ( D  x.  (
w ^ 2 ) ) ) )
76eqeq1d 2416 . . . . 5  |-  ( d  =  D  ->  (
( ( z ^
2 )  -  (
d  x.  ( w ^ 2 ) ) )  =  1  <->  (
( z ^ 2 )  -  ( D  x.  ( w ^
2 ) ) )  =  1 ) )
84, 7anbi12d 692 . . . 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 2713 . . 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 2912 . 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 26801 . 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 9041 . . 3  |-  RR  e.  _V
1312rabex 4318 . 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 5769 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 set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2671   {crab 2674    \ cdif 3281   ` cfv 5417  (class class class)co 6044   RRcr 8949   1c1 8951    + caddc 8953    x. cmul 8955    - cmin 9251   NNcn 9960   2c2 10009   ZZcz 10242   ^cexp 11341   sqrcsqr 11997  ◻NNcsquarenn 26793  Pell1234QRcpell1234qr 26795
This theorem is referenced by:  elpell1234qr  26808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367  ax-cnex 9006  ax-resscn 9007
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-iota 5381  df-fun 5419  df-fv 5425  df-ov 6047  df-pell1234qr 26801
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