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Theorem pell14qrval 35125
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 5848 . . . . . . . 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 2416 . . . . 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 2404 . . . . 5  |-  ( a  =  D  ->  (
( ( z ^
2 )  -  (
a  x.  ( w ^ 2 ) ) )  =  1  <->  (
( z ^ 2 )  -  ( D  x.  ( w ^
2 ) ) )  =  1 ) )
84, 7anbi12d 709 . . . 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 2924 . . 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 3050 . 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 35120 . 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 9612 . . 3  |-  RR  e.  _V
1312rabex 4544 . 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 5931 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   E.wrex 2754   {crab 2757    \ cdif 3410   ` cfv 5568  (class class class)co 6277   RRcr 9520   1c1 9522    + caddc 9524    x. cmul 9526    - cmin 9840   NNcn 10575   2c2 10625   NN0cn0 10835   ZZcz 10904   ^cexp 12208   sqrcsqrt 13213  ◻NNcsquarenn 35113  Pell14QRcpell14qr 35116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629  ax-cnex 9577  ax-resscn 9578
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-iota 5532  df-fun 5570  df-fv 5576  df-ov 6280  df-pell14qr 35120
This theorem is referenced by:  elpell14qr  35126  rmxyelqirr  35187
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