Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pell1234qrval Structured version   Unicode version

Theorem pell1234qrval 26951
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 5757 . . . . . . . 8  |-  ( d  =  D  ->  ( sqr `  d )  =  ( sqr `  D
) )
21oveq1d 6125 . . . . . . 7  |-  ( d  =  D  ->  (
( sqr `  d
)  x.  w )  =  ( ( sqr `  D )  x.  w
) )
32oveq2d 6126 . . . . . 6  |-  ( d  =  D  ->  (
z  +  ( ( sqr `  d )  x.  w ) )  =  ( z  +  ( ( sqr `  D
)  x.  w ) ) )
43eqeq2d 2453 . . . . 5  |-  ( d  =  D  ->  (
y  =  ( z  +  ( ( sqr `  d )  x.  w
) )  <->  y  =  ( z  +  ( ( sqr `  D
)  x.  w ) ) ) )
5 oveq1 6117 . . . . . . 7  |-  ( d  =  D  ->  (
d  x.  ( w ^ 2 ) )  =  ( D  x.  ( w ^ 2 ) ) )
65oveq2d 6126 . . . . . 6  |-  ( d  =  D  ->  (
( z ^ 2 )  -  ( d  x.  ( w ^
2 ) ) )  =  ( ( z ^ 2 )  -  ( D  x.  (
w ^ 2 ) ) ) )
76eqeq1d 2450 . . . . 5  |-  ( d  =  D  ->  (
( ( z ^
2 )  -  (
d  x.  ( w ^ 2 ) ) )  =  1  <->  (
( z ^ 2 )  -  ( D  x.  ( w ^
2 ) ) )  =  1 ) )
84, 7anbi12d 693 . . . 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 2754 . . 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 2954 . 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 26945 . 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 9112 . . 3  |-  RR  e.  _V
1312rabex 4383 . 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 5835 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 360    = wceq 1653    e. wcel 1727   E.wrex 2712   {crab 2715    \ cdif 3303   ` cfv 5483  (class class class)co 6110   RRcr 9020   1c1 9022    + caddc 9024    x. cmul 9026    - cmin 9322   NNcn 10031   2c2 10080   ZZcz 10313   ^cexp 11413   sqrcsqr 12069  ◻NNcsquarenn 26937  Pell1234QRcpell1234qr 26939
This theorem is referenced by:  elpell1234qr  26952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432  ax-cnex 9077  ax-resscn 9078
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-iota 5447  df-fun 5485  df-fv 5491  df-ov 6113  df-pell1234qr 26945
  Copyright terms: Public domain W3C validator