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Theorem pell1234qrval 26258
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 5608 . . . . . . . 8  |-  ( d  =  D  ->  ( sqr `  d )  =  ( sqr `  D
) )
21oveq1d 5960 . . . . . . 7  |-  ( d  =  D  ->  (
( sqr `  d
)  x.  w )  =  ( ( sqr `  D )  x.  w
) )
32oveq2d 5961 . . . . . 6  |-  ( d  =  D  ->  (
z  +  ( ( sqr `  d )  x.  w ) )  =  ( z  +  ( ( sqr `  D
)  x.  w ) ) )
43eqeq2d 2369 . . . . 5  |-  ( d  =  D  ->  (
y  =  ( z  +  ( ( sqr `  d )  x.  w
) )  <->  y  =  ( z  +  ( ( sqr `  D
)  x.  w ) ) ) )
5 oveq1 5952 . . . . . . 7  |-  ( d  =  D  ->  (
d  x.  ( w ^ 2 ) )  =  ( D  x.  ( w ^ 2 ) ) )
65oveq2d 5961 . . . . . 6  |-  ( d  =  D  ->  (
( z ^ 2 )  -  ( d  x.  ( w ^
2 ) ) )  =  ( ( z ^ 2 )  -  ( D  x.  (
w ^ 2 ) ) ) )
76eqeq1d 2366 . . . . 5  |-  ( d  =  D  ->  (
( ( z ^
2 )  -  (
d  x.  ( w ^ 2 ) ) )  =  1  <->  (
( z ^ 2 )  -  ( D  x.  ( w ^
2 ) ) )  =  1 ) )
84, 7anbi12d 691 . . . 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 2662 . . 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 2856 . 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 26252 . 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 8918 . . 3  |-  RR  e.  _V
1312rabex 4246 . 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 5685 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 358    = wceq 1642    e. wcel 1710   E.wrex 2620   {crab 2623    \ cdif 3225   ` cfv 5337  (class class class)co 5945   RRcr 8826   1c1 8828    + caddc 8830    x. cmul 8832    - cmin 9127   NNcn 9836   2c2 9885   ZZcz 10116   ^cexp 11197   sqrcsqr 11814  ◻NNcsquarenn 26244  Pell1234QRcpell1234qr 26246
This theorem is referenced by:  elpell1234qr  26259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295  ax-cnex 8883  ax-resscn 8884
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fv 5345  df-ov 5948  df-pell1234qr 26252
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