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Theorem pell1234qrval 29191
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 6106 . . . . . . 7  |-  ( d  =  D  ->  (
( sqr `  d
)  x.  w )  =  ( ( sqr `  D )  x.  w
) )
32oveq2d 6107 . . . . . 6  |-  ( d  =  D  ->  (
z  +  ( ( sqr `  d )  x.  w ) )  =  ( z  +  ( ( sqr `  D
)  x.  w ) ) )
43eqeq2d 2454 . . . . 5  |-  ( d  =  D  ->  (
y  =  ( z  +  ( ( sqr `  d )  x.  w
) )  <->  y  =  ( z  +  ( ( sqr `  D
)  x.  w ) ) ) )
5 oveq1 6098 . . . . . . 7  |-  ( d  =  D  ->  (
d  x.  ( w ^ 2 ) )  =  ( D  x.  ( w ^ 2 ) ) )
65oveq2d 6107 . . . . . 6  |-  ( d  =  D  ->  (
( z ^ 2 )  -  ( d  x.  ( w ^
2 ) ) )  =  ( ( z ^ 2 )  -  ( D  x.  (
w ^ 2 ) ) ) )
76eqeq1d 2451 . . . . 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 2758 . . 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 2964 . 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 29185 . 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 9373 . . 3  |-  RR  e.  _V
1312rabex 4443 . 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 5774 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 1369    e. wcel 1756   E.wrex 2716   {crab 2719    \ cdif 3325   ` cfv 5418  (class class class)co 6091   RRcr 9281   1c1 9283    + caddc 9285    x. cmul 9287    - cmin 9595   NNcn 10322   2c2 10371   ZZcz 10646   ^cexp 11865   sqrcsqr 12722  ◻NNcsquarenn 29177  Pell1234QRcpell1234qr 29179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531  ax-cnex 9338  ax-resscn 9339
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-pell1234qr 29185
This theorem is referenced by:  elpell1234qr  29192
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