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Theorem pell1qrval 26079
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 5563 . . . . . . . 8  |-  ( a  =  D  ->  ( sqr `  a )  =  ( sqr `  D
) )
21oveq1d 5915 . . . . . . 7  |-  ( a  =  D  ->  (
( sqr `  a
)  x.  w )  =  ( ( sqr `  D )  x.  w
) )
32oveq2d 5916 . . . . . 6  |-  ( a  =  D  ->  (
z  +  ( ( sqr `  a )  x.  w ) )  =  ( z  +  ( ( sqr `  D
)  x.  w ) ) )
43eqeq2d 2327 . . . . 5  |-  ( a  =  D  ->  (
y  =  ( z  +  ( ( sqr `  a )  x.  w
) )  <->  y  =  ( z  +  ( ( sqr `  D
)  x.  w ) ) ) )
5 oveq1 5907 . . . . . . 7  |-  ( a  =  D  ->  (
a  x.  ( w ^ 2 ) )  =  ( D  x.  ( w ^ 2 ) ) )
65oveq2d 5916 . . . . . 6  |-  ( a  =  D  ->  (
( z ^ 2 )  -  ( a  x.  ( w ^
2 ) ) )  =  ( ( z ^ 2 )  -  ( D  x.  (
w ^ 2 ) ) ) )
76eqeq1d 2324 . . . . 5  |-  ( a  =  D  ->  (
( ( z ^
2 )  -  (
a  x.  ( w ^ 2 ) ) )  =  1  <->  (
( z ^ 2 )  -  ( D  x.  ( w ^
2 ) ) )  =  1 ) )
84, 7anbi12d 691 . . . 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 2620 . . 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 2814 . 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 26075 . 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 8873 . . 3  |-  RR  e.  _V
1312rabex 4202 . 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 5640 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 set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701   E.wrex 2578   {crab 2581    \ cdif 3183   ` cfv 5292  (class class class)co 5900   RRcr 8781   1c1 8783    + caddc 8785    x. cmul 8787    - cmin 9082   NNcn 9791   2c2 9840   NN0cn0 10012   ^cexp 11151   sqrcsqr 11765  ◻NNcsquarenn 26069  Pell1QRcpell1qr 26070
This theorem is referenced by:  elpell1qr  26080
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251  ax-cnex 8838  ax-resscn 8839
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-iota 5256  df-fun 5294  df-fv 5300  df-ov 5903  df-pell1qr 26075
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