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Theorem pell14qrval 29186
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 5689 . . . . . . . 8  |-  ( a  =  D  ->  ( sqr `  a )  =  ( sqr `  D
) )
21oveq1d 6104 . . . . . . 7  |-  ( a  =  D  ->  (
( sqr `  a
)  x.  w )  =  ( ( sqr `  D )  x.  w
) )
32oveq2d 6105 . . . . . 6  |-  ( a  =  D  ->  (
z  +  ( ( sqr `  a )  x.  w ) )  =  ( z  +  ( ( sqr `  D
)  x.  w ) ) )
43eqeq2d 2452 . . . . 5  |-  ( a  =  D  ->  (
y  =  ( z  +  ( ( sqr `  a )  x.  w
) )  <->  y  =  ( z  +  ( ( sqr `  D
)  x.  w ) ) ) )
5 oveq1 6096 . . . . . . 7  |-  ( a  =  D  ->  (
a  x.  ( w ^ 2 ) )  =  ( D  x.  ( w ^ 2 ) ) )
65oveq2d 6105 . . . . . 6  |-  ( a  =  D  ->  (
( z ^ 2 )  -  ( a  x.  ( w ^
2 ) ) )  =  ( ( z ^ 2 )  -  ( D  x.  (
w ^ 2 ) ) ) )
76eqeq1d 2449 . . . . 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 2756 . . 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 2962 . 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 29181 . 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 9371 . . 3  |-  RR  e.  _V
1312rabex 4441 . 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 5772 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 369    = wceq 1369    e. wcel 1756   E.wrex 2714   {crab 2717    \ cdif 3323   ` cfv 5416  (class class class)co 6089   RRcr 9279   1c1 9281    + caddc 9283    x. cmul 9285    - cmin 9593   NNcn 10320   2c2 10369   NN0cn0 10577   ZZcz 10644   ^cexp 11863   sqrcsqr 12720  ◻NNcsquarenn 29174  Pell14QRcpell14qr 29177
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 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529  ax-cnex 9336  ax-resscn 9337
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-iota 5379  df-fun 5418  df-fv 5424  df-ov 6092  df-pell14qr 29181
This theorem is referenced by:  elpell14qr  29187  rmxyelqirr  29248
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