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Theorem pell1qrval 30375
Description: Value of the set of first-quadrant Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Assertion
Ref Expression
pell1qrval  |-  ( D  e.  ( NN  \NN )  -> 
(Pell1QR `  D )  =  { y  e.  RR  |  E. z  e.  NN0  E. w  e.  NN0  (
y  =  ( z  +  ( ( sqr `  D )  x.  w
) )  /\  (
( z ^ 2 )  -  ( D  x.  ( w ^
2 ) ) )  =  1 ) } )
Distinct variable group:    y, z, w, D

Proof of Theorem pell1qrval
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 fveq2 5859 . . . . . . . 8  |-  ( a  =  D  ->  ( sqr `  a )  =  ( sqr `  D
) )
21oveq1d 6292 . . . . . . 7  |-  ( a  =  D  ->  (
( sqr `  a
)  x.  w )  =  ( ( sqr `  D )  x.  w
) )
32oveq2d 6293 . . . . . 6  |-  ( a  =  D  ->  (
z  +  ( ( sqr `  a )  x.  w ) )  =  ( z  +  ( ( sqr `  D
)  x.  w ) ) )
43eqeq2d 2476 . . . . 5  |-  ( a  =  D  ->  (
y  =  ( z  +  ( ( sqr `  a )  x.  w
) )  <->  y  =  ( z  +  ( ( sqr `  D
)  x.  w ) ) ) )
5 oveq1 6284 . . . . . . 7  |-  ( a  =  D  ->  (
a  x.  ( w ^ 2 ) )  =  ( D  x.  ( w ^ 2 ) ) )
65oveq2d 6293 . . . . . 6  |-  ( a  =  D  ->  (
( z ^ 2 )  -  ( a  x.  ( w ^
2 ) ) )  =  ( ( z ^ 2 )  -  ( D  x.  (
w ^ 2 ) ) ) )
76eqeq1d 2464 . . . . 5  |-  ( a  =  D  ->  (
( ( z ^
2 )  -  (
a  x.  ( w ^ 2 ) ) )  =  1  <->  (
( z ^ 2 )  -  ( D  x.  ( w ^
2 ) ) )  =  1 ) )
84, 7anbi12d 710 . . . 4  |-  ( a  =  D  ->  (
( y  =  ( z  +  ( ( sqr `  a )  x.  w ) )  /\  ( ( z ^ 2 )  -  ( a  x.  (
w ^ 2 ) ) )  =  1 )  <->  ( y  =  ( z  +  ( ( sqr `  D
)  x.  w ) )  /\  ( ( z ^ 2 )  -  ( D  x.  ( w ^ 2 ) ) )  =  1 ) ) )
982rexbidv 2975 . . 3  |-  ( a  =  D  ->  ( E. z  e.  NN0  E. w  e.  NN0  (
y  =  ( z  +  ( ( sqr `  a )  x.  w
) )  /\  (
( z ^ 2 )  -  ( a  x.  ( w ^
2 ) ) )  =  1 )  <->  E. z  e.  NN0  E. w  e. 
NN0  ( y  =  ( z  +  ( ( sqr `  D
)  x.  w ) )  /\  ( ( z ^ 2 )  -  ( D  x.  ( w ^ 2 ) ) )  =  1 ) ) )
109rabbidv 3100 . 2  |-  ( a  =  D  ->  { y  e.  RR  |  E. z  e.  NN0  E. w  e.  NN0  ( y  =  ( z  +  ( ( sqr `  a
)  x.  w ) )  /\  ( ( z ^ 2 )  -  ( a  x.  ( w ^ 2 ) ) )  =  1 ) }  =  { y  e.  RR  |  E. z  e.  NN0  E. w  e.  NN0  (
y  =  ( z  +  ( ( sqr `  D )  x.  w
) )  /\  (
( z ^ 2 )  -  ( D  x.  ( w ^
2 ) ) )  =  1 ) } )
11 df-pell1qr 30371 . 2  |- Pell1QR  =  ( a  e.  ( NN 
\NN )  |->  { y  e.  RR  |  E. z  e.  NN0  E. w  e. 
NN0  ( y  =  ( z  +  ( ( sqr `  a
)  x.  w ) )  /\  ( ( z ^ 2 )  -  ( a  x.  ( w ^ 2 ) ) )  =  1 ) } )
12 reex 9574 . . 3  |-  RR  e.  _V
1312rabex 4593 . 2  |-  { y  e.  RR  |  E. z  e.  NN0  E. w  e.  NN0  ( y  =  ( z  +  ( ( sqr `  D
)  x.  w ) )  /\  ( ( z ^ 2 )  -  ( D  x.  ( w ^ 2 ) ) )  =  1 ) }  e.  _V
1410, 11, 13fvmpt 5943 1  |-  ( D  e.  ( NN  \NN )  -> 
(Pell1QR `  D )  =  { y  e.  RR  |  E. z  e.  NN0  E. w  e.  NN0  (
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 1374    e. wcel 1762   E.wrex 2810   {crab 2813    \ cdif 3468   ` cfv 5581  (class class class)co 6277   RRcr 9482   1c1 9484    + caddc 9486    x. cmul 9488    - cmin 9796   NNcn 10527   2c2 10576   NN0cn0 10786   ^cexp 12124   sqrcsqr 13018  ◻NNcsquarenn 30365  Pell1QRcpell1qr 30366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681  ax-cnex 9539  ax-resscn 9540
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-iota 5544  df-fun 5583  df-fv 5589  df-ov 6280  df-pell1qr 30371
This theorem is referenced by:  elpell1qr  30376
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