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Theorem pell14qrval 26913
Description: Value of the set of positive Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Assertion
Ref Expression
pell14qrval  |-  ( D  e.  ( NN  \NN )  -> 
(Pell14QR `  D )  =  { y  e.  RR  |  E. z  e.  NN0  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 pell14qrval
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 fveq2 5730 . . . . . . . 8  |-  ( a  =  D  ->  ( sqr `  a )  =  ( sqr `  D
) )
21oveq1d 6098 . . . . . . 7  |-  ( a  =  D  ->  (
( sqr `  a
)  x.  w )  =  ( ( sqr `  D )  x.  w
) )
32oveq2d 6099 . . . . . 6  |-  ( a  =  D  ->  (
z  +  ( ( sqr `  a )  x.  w ) )  =  ( z  +  ( ( sqr `  D
)  x.  w ) ) )
43eqeq2d 2449 . . . . 5  |-  ( a  =  D  ->  (
y  =  ( z  +  ( ( sqr `  a )  x.  w
) )  <->  y  =  ( z  +  ( ( sqr `  D
)  x.  w ) ) ) )
5 oveq1 6090 . . . . . . 7  |-  ( a  =  D  ->  (
a  x.  ( w ^ 2 ) )  =  ( D  x.  ( w ^ 2 ) ) )
65oveq2d 6099 . . . . . 6  |-  ( a  =  D  ->  (
( z ^ 2 )  -  ( a  x.  ( w ^
2 ) ) )  =  ( ( z ^ 2 )  -  ( D  x.  (
w ^ 2 ) ) ) )
76eqeq1d 2446 . . . . 5  |-  ( a  =  D  ->  (
( ( z ^
2 )  -  (
a  x.  ( w ^ 2 ) ) )  =  1  <->  (
( z ^ 2 )  -  ( D  x.  ( w ^
2 ) ) )  =  1 ) )
84, 7anbi12d 693 . . . 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 2750 . . 3  |-  ( a  =  D  ->  ( E. z  e.  NN0  E. w  e.  ZZ  (
y  =  ( z  +  ( ( sqr `  a )  x.  w
) )  /\  (
( z ^ 2 )  -  ( a  x.  ( w ^
2 ) ) )  =  1 )  <->  E. z  e.  NN0  E. w  e.  ZZ  ( y  =  ( z  +  ( ( sqr `  D
)  x.  w ) )  /\  ( ( z ^ 2 )  -  ( D  x.  ( w ^ 2 ) ) )  =  1 ) ) )
109rabbidv 2950 . 2  |-  ( a  =  D  ->  { y  e.  RR  |  E. z  e.  NN0  E. w  e.  ZZ  ( y  =  ( z  +  ( ( sqr `  a
)  x.  w ) )  /\  ( ( z ^ 2 )  -  ( a  x.  ( w ^ 2 ) ) )  =  1 ) }  =  { y  e.  RR  |  E. z  e.  NN0  E. w  e.  ZZ  (
y  =  ( z  +  ( ( sqr `  D )  x.  w
) )  /\  (
( z ^ 2 )  -  ( D  x.  ( w ^
2 ) ) )  =  1 ) } )
11 df-pell14qr 26908 . 2  |- Pell14QR  =  ( a  e.  ( NN 
\NN )  |->  { y  e.  RR  |  E. z  e.  NN0  E. w  e.  ZZ  ( y  =  ( z  +  ( ( sqr `  a
)  x.  w ) )  /\  ( ( z ^ 2 )  -  ( a  x.  ( w ^ 2 ) ) )  =  1 ) } )
12 reex 9083 . . 3  |-  RR  e.  _V
1312rabex 4356 . 2  |-  { y  e.  RR  |  E. z  e.  NN0  E. w  e.  ZZ  ( y  =  ( z  +  ( ( sqr `  D
)  x.  w ) )  /\  ( ( z ^ 2 )  -  ( D  x.  ( w ^ 2 ) ) )  =  1 ) }  e.  _V
1410, 11, 13fvmpt 5808 1  |-  ( D  e.  ( NN  \NN )  -> 
(Pell14QR `  D )  =  { y  e.  RR  |  E. z  e.  NN0  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 1726   E.wrex 2708   {crab 2711    \ cdif 3319   ` cfv 5456  (class class class)co 6083   RRcr 8991   1c1 8993    + caddc 8995    x. cmul 8997    - cmin 9293   NNcn 10002   2c2 10051   NN0cn0 10223   ZZcz 10284   ^cexp 11384   sqrcsqr 12040  ◻NNcsquarenn 26901  Pell14QRcpell14qr 26904
This theorem is referenced by:  elpell14qr  26914  rmxyelqirr  26975
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-cnex 9048  ax-resscn 9049
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-pell14qr 26908
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