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Theorem pell14qrdich 29057
Description: A positive Pell solution is either in the first quadrant, or its reciprocal is. (Contributed by Stefan O'Rear, 18-Sep-2014.)
Assertion
Ref Expression
pell14qrdich  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  e.  (Pell1QR `  D )  \/  (
1  /  A )  e.  (Pell1QR `  D
) ) )

Proof of Theorem pell14qrdich
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elpell14qr 29037 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
( A  e.  (Pell14QR `  D )  <->  ( A  e.  RR  /\  E. a  e.  NN0  E. b  e.  ZZ  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) ) ) )
21biimpa 481 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  e.  RR  /\ 
E. a  e.  NN0  E. b  e.  ZZ  ( A  =  ( a  +  ( ( sqr `  D )  x.  b
) )  /\  (
( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 ) ) )
3 simplrr 755 . . . . . . . 8  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
b  e.  ZZ )
4 elznn0 10651 . . . . . . . 8  |-  ( b  e.  ZZ  <->  ( b  e.  RR  /\  ( b  e.  NN0  \/  -u b  e.  NN0 ) ) )
53, 4sylib 196 . . . . . . 7  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( b  e.  RR  /\  ( b  e.  NN0  \/  -u b  e.  NN0 ) ) )
65simprd 460 . . . . . 6  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( b  e.  NN0  \/  -u b  e.  NN0 ) )
7 simplr 749 . . . . . . . . . . 11  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  A  e.  RR )
87ad2antrr 720 . . . . . . . . . 10  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  ZZ ) )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  /\  b  e.  NN0 )  ->  A  e.  RR )
9 simprl 750 . . . . . . . . . . . 12  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  a  e.  NN0 )
109ad2antrr 720 . . . . . . . . . . 11  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  ZZ ) )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  /\  b  e.  NN0 )  -> 
a  e.  NN0 )
11 simpr 458 . . . . . . . . . . 11  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  ZZ ) )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  /\  b  e.  NN0 )  -> 
b  e.  NN0 )
12 simplr 749 . . . . . . . . . . 11  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  ZZ ) )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  /\  b  e.  NN0 )  -> 
( A  =  ( a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )
13 rsp2e 2771 . . . . . . . . . . 11  |-  ( ( a  e.  NN0  /\  b  e.  NN0  /\  ( A  =  ( a  +  ( ( sqr `  D )  x.  b
) )  /\  (
( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 ) )  ->  E. a  e.  NN0  E. b  e.  NN0  ( A  =  ( a  +  ( ( sqr `  D )  x.  b
) )  /\  (
( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 ) )
1410, 11, 12, 13syl3anc 1213 . . . . . . . . . 10  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  ZZ ) )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  /\  b  e.  NN0 )  ->  E. a  e.  NN0  E. b  e.  NN0  ( A  =  ( a  +  ( ( sqr `  D )  x.  b
) )  /\  (
( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 ) )
158, 14jca 529 . . . . . . . . 9  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  ZZ ) )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  /\  b  e.  NN0 )  -> 
( A  e.  RR  /\ 
E. a  e.  NN0  E. b  e.  NN0  ( A  =  ( a  +  ( ( sqr `  D )  x.  b
) )  /\  (
( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 ) ) )
1615ex 434 . . . . . . . 8  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( b  e.  NN0  ->  ( A  e.  RR  /\ 
E. a  e.  NN0  E. b  e.  NN0  ( A  =  ( a  +  ( ( sqr `  D )  x.  b
) )  /\  (
( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 ) ) ) )
17 elpell1qr 29035 . . . . . . . . 9  |-  ( D  e.  ( NN  \NN )  -> 
( A  e.  (Pell1QR `  D )  <->  ( A  e.  RR  /\  E. a  e.  NN0  E. b  e. 
NN0  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) ) ) )
1817ad4antr 726 . . . . . . . 8  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( A  e.  (Pell1QR `  D )  <->  ( A  e.  RR  /\  E. a  e.  NN0  E. b  e. 
NN0  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) ) ) )
1916, 18sylibrd 234 . . . . . . 7  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( b  e.  NN0  ->  A  e.  (Pell1QR `  D
) ) )
207ad2antrr 720 . . . . . . . . . . 11  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  ZZ ) )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  /\  -u b  e.  NN0 )  ->  A  e.  RR )
21 pell14qrne0 29046 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  =/=  0 )
2221ad4antr 726 . . . . . . . . . . 11  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  ZZ ) )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  /\  -u b  e.  NN0 )  ->  A  =/=  0 )
2320, 22rereccld 10148 . . . . . . . . . 10  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  ZZ ) )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  /\  -u b  e.  NN0 )  ->  ( 1  /  A
)  e.  RR )
249ad2antrr 720 . . . . . . . . . . 11  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  ZZ ) )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  /\  -u b  e.  NN0 )  ->  a  e.  NN0 )
25 simpr 458 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  ZZ ) )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  /\  -u b  e.  NN0 )  -> 
-u b  e.  NN0 )
26 pell14qrre 29045 . . . . . . . . . . . . . . . . . 18  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR )
2726recnd 9402 . . . . . . . . . . . . . . . . 17  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  CC )
2827, 21reccld 10090 . . . . . . . . . . . . . . . 16  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( 1  /  A
)  e.  CC )
2928ad3antrrr 724 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( 1  /  A
)  e.  CC )
30 nn0cn 10579 . . . . . . . . . . . . . . . . . 18  |-  ( a  e.  NN0  ->  a  e.  CC )
3130ad2antrl 722 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  a  e.  CC )
32 eldifi 3468 . . . . . . . . . . . . . . . . . . . . 21  |-  ( D  e.  ( NN  \NN )  ->  D  e.  NN )
3332nncnd 10328 . . . . . . . . . . . . . . . . . . . 20  |-  ( D  e.  ( NN  \NN )  ->  D  e.  CC )
3433ad3antrrr 724 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  D  e.  CC )
3534sqrcld 12909 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  ( sqr `  D )  e.  CC )
36 zcn 10641 . . . . . . . . . . . . . . . . . . . 20  |-  ( b  e.  ZZ  ->  b  e.  CC )
3736ad2antll 723 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  b  e.  CC )
3837negcld 9696 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  -u b  e.  CC )
3935, 38mulcld 9396 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  ( ( sqr `  D )  x.  -u b )  e.  CC )
4031, 39addcld 9395 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  ( a  +  ( ( sqr `  D )  x.  -u b
) )  e.  CC )
4140adantr 462 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( a  +  ( ( sqr `  D
)  x.  -u b
) )  e.  CC )
4227ad3antrrr 724 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  ->  A  e.  CC )
4321ad3antrrr 724 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  ->  A  =/=  0 )
4427, 21recidd 10092 . . . . . . . . . . . . . . . . . 18  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  x.  (
1  /  A ) )  =  1 )
4544ad3antrrr 724 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( A  x.  (
1  /  A ) )  =  1 )
46 simprr 751 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( ( a ^
2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )
4745, 46eqtr4d 2470 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( A  x.  (
1  /  A ) )  =  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) ) )
4831adantr 462 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  /\  A  =  ( a  +  ( ( sqr `  D
)  x.  b ) ) )  ->  a  e.  CC )
4935, 37mulcld 9396 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  ( ( sqr `  D )  x.  b )  e.  CC )
5049adantr 462 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  /\  A  =  ( a  +  ( ( sqr `  D
)  x.  b ) ) )  ->  (
( sqr `  D
)  x.  b )  e.  CC )
51 subsq 11959 . . . . . . . . . . . . . . . . . . 19  |-  ( ( a  e.  CC  /\  ( ( sqr `  D
)  x.  b )  e.  CC )  -> 
( ( a ^
2 )  -  (
( ( sqr `  D
)  x.  b ) ^ 2 ) )  =  ( ( a  +  ( ( sqr `  D )  x.  b
) )  x.  (
a  -  ( ( sqr `  D )  x.  b ) ) ) )
5248, 50, 51syl2anc 656 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  /\  A  =  ( a  +  ( ( sqr `  D
)  x.  b ) ) )  ->  (
( a ^ 2 )  -  ( ( ( sqr `  D
)  x.  b ) ^ 2 ) )  =  ( ( a  +  ( ( sqr `  D )  x.  b
) )  x.  (
a  -  ( ( sqr `  D )  x.  b ) ) ) )
5335, 37sqmuld 12006 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  ( (
( sqr `  D
)  x.  b ) ^ 2 )  =  ( ( ( sqr `  D ) ^ 2 )  x.  ( b ^ 2 ) ) )
5434sqsqrd 12911 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  ( ( sqr `  D ) ^
2 )  =  D )
5554oveq1d 6097 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  ( (
( sqr `  D
) ^ 2 )  x.  ( b ^
2 ) )  =  ( D  x.  (
b ^ 2 ) ) )
5653, 55eqtr2d 2468 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  ( D  x.  ( b ^ 2 ) )  =  ( ( ( sqr `  D
)  x.  b ) ^ 2 ) )
5756oveq2d 6098 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  ( (
a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  ( ( a ^
2 )  -  (
( ( sqr `  D
)  x.  b ) ^ 2 ) ) )
5857adantr 462 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  /\  A  =  ( a  +  ( ( sqr `  D
)  x.  b ) ) )  ->  (
( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  ( ( a ^ 2 )  -  ( ( ( sqr `  D )  x.  b
) ^ 2 ) ) )
59 simpr 458 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  /\  A  =  ( a  +  ( ( sqr `  D
)  x.  b ) ) )  ->  A  =  ( a  +  ( ( sqr `  D
)  x.  b ) ) )
6035, 37mulneg2d 9788 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  ( ( sqr `  D )  x.  -u b )  =  -u ( ( sqr `  D
)  x.  b ) )
6160oveq2d 6098 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  ( a  +  ( ( sqr `  D )  x.  -u b
) )  =  ( a  +  -u (
( sqr `  D
)  x.  b ) ) )
62 negsub 9647 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( a  e.  CC  /\  ( ( sqr `  D
)  x.  b )  e.  CC )  -> 
( a  +  -u ( ( sqr `  D
)  x.  b ) )  =  ( a  -  ( ( sqr `  D )  x.  b
) ) )
6362eqcomd 2440 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( a  e.  CC  /\  ( ( sqr `  D
)  x.  b )  e.  CC )  -> 
( a  -  (
( sqr `  D
)  x.  b ) )  =  ( a  +  -u ( ( sqr `  D )  x.  b
) ) )
6431, 49, 63syl2anc 656 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  ( a  -  ( ( sqr `  D )  x.  b
) )  =  ( a  +  -u (
( sqr `  D
)  x.  b ) ) )
6561, 64eqtr4d 2470 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  ( a  +  ( ( sqr `  D )  x.  -u b
) )  =  ( a  -  ( ( sqr `  D )  x.  b ) ) )
6665adantr 462 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  /\  A  =  ( a  +  ( ( sqr `  D
)  x.  b ) ) )  ->  (
a  +  ( ( sqr `  D )  x.  -u b ) )  =  ( a  -  ( ( sqr `  D
)  x.  b ) ) )
6759, 66oveq12d 6100 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  /\  A  =  ( a  +  ( ( sqr `  D
)  x.  b ) ) )  ->  ( A  x.  ( a  +  ( ( sqr `  D )  x.  -u b
) ) )  =  ( ( a  +  ( ( sqr `  D
)  x.  b ) )  x.  ( a  -  ( ( sqr `  D )  x.  b
) ) ) )
6852, 58, 673eqtr4d 2477 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  /\  A  =  ( a  +  ( ( sqr `  D
)  x.  b ) ) )  ->  (
( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  ( A  x.  ( a  +  ( ( sqr `  D
)  x.  -u b
) ) ) )
6968adantrr 711 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( ( a ^
2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  ( A  x.  ( a  +  ( ( sqr `  D
)  x.  -u b
) ) ) )
7047, 69eqtrd 2467 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( A  x.  (
1  /  A ) )  =  ( A  x.  ( a  +  ( ( sqr `  D
)  x.  -u b
) ) ) )
7129, 41, 42, 43, 70mulcanad 9961 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( 1  /  A
)  =  ( a  +  ( ( sqr `  D )  x.  -u b
) ) )
7271adantr 462 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  ZZ ) )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  /\  -u b  e.  NN0 )  ->  ( 1  /  A
)  =  ( a  +  ( ( sqr `  D )  x.  -u b
) ) )
7337ad2antrr 720 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  ZZ ) )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  /\  -u b  e.  NN0 )  ->  b  e.  CC )
74 sqneg 11912 . . . . . . . . . . . . . . . . 17  |-  ( b  e.  CC  ->  ( -u b ^ 2 )  =  ( b ^
2 ) )
7573, 74syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  ZZ ) )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  /\  -u b  e.  NN0 )  ->  ( -u b ^
2 )  =  ( b ^ 2 ) )
7675oveq2d 6098 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  ZZ ) )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  /\  -u b  e.  NN0 )  ->  ( D  x.  ( -u b ^ 2 ) )  =  ( D  x.  ( b ^
2 ) ) )
7776oveq2d 6098 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  ZZ ) )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  /\  -u b  e.  NN0 )  ->  ( ( a ^
2 )  -  ( D  x.  ( -u b ^ 2 ) ) )  =  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) ) )
78 simplrr 755 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  ZZ ) )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  /\  -u b  e.  NN0 )  ->  ( ( a ^
2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )
7977, 78eqtrd 2467 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  ZZ ) )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  /\  -u b  e.  NN0 )  ->  ( ( a ^
2 )  -  ( D  x.  ( -u b ^ 2 ) ) )  =  1 )
8072, 79jca 529 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  ZZ ) )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  /\  -u b  e.  NN0 )  ->  ( ( 1  /  A )  =  ( a  +  ( ( sqr `  D )  x.  -u b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) )  =  1 ) )
81 oveq2 6090 . . . . . . . . . . . . . . . 16  |-  ( c  =  -u b  ->  (
( sqr `  D
)  x.  c )  =  ( ( sqr `  D )  x.  -u b
) )
8281oveq2d 6098 . . . . . . . . . . . . . . 15  |-  ( c  =  -u b  ->  (
a  +  ( ( sqr `  D )  x.  c ) )  =  ( a  +  ( ( sqr `  D
)  x.  -u b
) ) )
8382eqeq2d 2446 . . . . . . . . . . . . . 14  |-  ( c  =  -u b  ->  (
( 1  /  A
)  =  ( a  +  ( ( sqr `  D )  x.  c
) )  <->  ( 1  /  A )  =  ( a  +  ( ( sqr `  D
)  x.  -u b
) ) ) )
84 oveq1 6089 . . . . . . . . . . . . . . . . 17  |-  ( c  =  -u b  ->  (
c ^ 2 )  =  ( -u b ^ 2 ) )
8584oveq2d 6098 . . . . . . . . . . . . . . . 16  |-  ( c  =  -u b  ->  ( D  x.  ( c ^ 2 ) )  =  ( D  x.  ( -u b ^ 2 ) ) )
8685oveq2d 6098 . . . . . . . . . . . . . . 15  |-  ( c  =  -u b  ->  (
( a ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  ( ( a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) ) )
8786eqeq1d 2443 . . . . . . . . . . . . . 14  |-  ( c  =  -u b  ->  (
( ( a ^
2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  1  <->  (
( a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) )  =  1 ) )
8883, 87anbi12d 705 . . . . . . . . . . . . 13  |-  ( c  =  -u b  ->  (
( ( 1  /  A )  =  ( a  +  ( ( sqr `  D )  x.  c ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
c ^ 2 ) ) )  =  1 )  <->  ( ( 1  /  A )  =  ( a  +  ( ( sqr `  D
)  x.  -u b
) )  /\  (
( a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) )  =  1 ) ) )
8988rspcev 3064 . . . . . . . . . . . 12  |-  ( (
-u b  e.  NN0  /\  ( ( 1  /  A )  =  ( a  +  ( ( sqr `  D )  x.  -u b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) )  =  1 ) )  ->  E. c  e.  NN0  ( ( 1  /  A )  =  ( a  +  ( ( sqr `  D
)  x.  c ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  1 ) )
9025, 80, 89syl2anc 656 . . . . . . . . . . 11  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  ZZ ) )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  /\  -u b  e.  NN0 )  ->  E. c  e.  NN0  ( ( 1  /  A )  =  ( a  +  ( ( sqr `  D )  x.  c ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
c ^ 2 ) ) )  =  1 ) )
91 rspe 2769 . . . . . . . . . . 11  |-  ( ( a  e.  NN0  /\  E. c  e.  NN0  (
( 1  /  A
)  =  ( a  +  ( ( sqr `  D )  x.  c
) )  /\  (
( a ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  1 ) )  ->  E. a  e.  NN0  E. c  e.  NN0  (
( 1  /  A
)  =  ( a  +  ( ( sqr `  D )  x.  c
) )  /\  (
( a ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  1 ) )
9224, 90, 91syl2anc 656 . . . . . . . . . 10  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  ZZ ) )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  /\  -u b  e.  NN0 )  ->  E. a  e.  NN0  E. c  e.  NN0  (
( 1  /  A
)  =  ( a  +  ( ( sqr `  D )  x.  c
) )  /\  (
( a ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  1 ) )
9323, 92jca 529 . . . . . . . . 9  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  /\  A  e.  RR )  /\  ( a  e.  NN0  /\  b  e.  ZZ ) )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  /\  -u b  e.  NN0 )  ->  ( ( 1  /  A )  e.  RR  /\ 
E. a  e.  NN0  E. c  e.  NN0  (
( 1  /  A
)  =  ( a  +  ( ( sqr `  D )  x.  c
) )  /\  (
( a ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  1 ) ) )
9493ex 434 . . . . . . . 8  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( -u b  e.  NN0  ->  ( ( 1  /  A )  e.  RR  /\ 
E. a  e.  NN0  E. c  e.  NN0  (
( 1  /  A
)  =  ( a  +  ( ( sqr `  D )  x.  c
) )  /\  (
( a ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  1 ) ) ) )
95 elpell1qr 29035 . . . . . . . . 9  |-  ( D  e.  ( NN  \NN )  -> 
( ( 1  /  A )  e.  (Pell1QR `  D )  <->  ( (
1  /  A )  e.  RR  /\  E. a  e.  NN0  E. c  e.  NN0  ( ( 1  /  A )  =  ( a  +  ( ( sqr `  D
)  x.  c ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  1 ) ) ) )
9695ad4antr 726 . . . . . . . 8  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( ( 1  /  A )  e.  (Pell1QR `  D )  <->  ( (
1  /  A )  e.  RR  /\  E. a  e.  NN0  E. c  e.  NN0  ( ( 1  /  A )  =  ( a  +  ( ( sqr `  D
)  x.  c ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  1 ) ) ) )
9794, 96sylibrd 234 . . . . . . 7  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( -u b  e.  NN0  ->  ( 1  /  A
)  e.  (Pell1QR `  D
) ) )
9819, 97orim12d 829 . . . . . 6  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( ( b  e. 
NN0  \/  -u b  e. 
NN0 )  ->  ( A  e.  (Pell1QR `  D
)  \/  ( 1  /  A )  e.  (Pell1QR `  D )
) ) )
996, 98mpd 15 . . . . 5  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D
) )  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( A  e.  (Pell1QR `  D )  \/  (
1  /  A )  e.  (Pell1QR `  D
) ) )
10099ex 434 . . . 4  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  ( ( A  =  ( a  +  ( ( sqr `  D )  x.  b
) )  /\  (
( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 )  -> 
( A  e.  (Pell1QR `  D )  \/  (
1  /  A )  e.  (Pell1QR `  D
) ) ) )
101100rexlimdvva 2840 . . 3  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  ->  ( E. a  e.  NN0  E. b  e.  ZZ  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  ->  ( A  e.  (Pell1QR `  D
)  \/  ( 1  /  A )  e.  (Pell1QR `  D )
) ) )
102101expimpd 600 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( A  e.  RR  /\  E. a  e.  NN0  E. b  e.  ZZ  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( A  e.  (Pell1QR `  D )  \/  (
1  /  A )  e.  (Pell1QR `  D
) ) ) )
1032, 102mpd 15 1  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  e.  (Pell1QR `  D )  \/  (
1  /  A )  e.  (Pell1QR `  D
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1364    e. wcel 1757    =/= wne 2598   E.wrex 2708    \ cdif 3315   ` cfv 5408  (class class class)co 6082   CCcc 9270   RRcr 9271   0cc0 9272   1c1 9273    + caddc 9275    x. cmul 9277    - cmin 9585   -ucneg 9586    / cdiv 9983   NNcn 10312   2c2 10361   NN0cn0 10569   ZZcz 10636   ^cexp 11851   sqrcsqr 12708  ◻NNcsquarenn 29024  Pell1QRcpell1qr 29025  Pell14QRcpell14qr 29027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2416  ax-sep 4403  ax-nul 4411  ax-pow 4460  ax-pr 4521  ax-un 6363  ax-cnex 9328  ax-resscn 9329  ax-1cn 9330  ax-icn 9331  ax-addcl 9332  ax-addrcl 9333  ax-mulcl 9334  ax-mulrcl 9335  ax-mulcom 9336  ax-addass 9337  ax-mulass 9338  ax-distr 9339  ax-i2m1 9340  ax-1ne0 9341  ax-1rid 9342  ax-rnegex 9343  ax-rrecex 9344  ax-cnre 9345  ax-pre-lttri 9346  ax-pre-lttrn 9347  ax-pre-ltadd 9348  ax-pre-mulgt0 9349  ax-pre-sup 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1702  df-eu 2260  df-mo 2261  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2966  df-sbc 3178  df-csb 3279  df-dif 3321  df-un 3323  df-in 3325  df-ss 3332  df-pss 3334  df-nul 3628  df-if 3782  df-pw 3852  df-sn 3868  df-pr 3870  df-tp 3872  df-op 3874  df-uni 4082  df-iun 4163  df-br 4283  df-opab 4341  df-mpt 4342  df-tr 4376  df-eprel 4621  df-id 4625  df-po 4630  df-so 4631  df-fr 4668  df-we 4670  df-ord 4711  df-on 4712  df-lim 4713  df-suc 4714  df-xp 4835  df-rel 4836  df-cnv 4837  df-co 4838  df-dm 4839  df-rn 4840  df-res 4841  df-ima 4842  df-iota 5371  df-fun 5410  df-fn 5411  df-f 5412  df-f1 5413  df-fo 5414  df-f1o 5415  df-fv 5416  df-riota 6041  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-om 6468  df-2nd 6569  df-recs 6820  df-rdg 6854  df-er 7091  df-en 7301  df-dom 7302  df-sdom 7303  df-sup 7681  df-pnf 9410  df-mnf 9411  df-xr 9412  df-ltxr 9413  df-le 9414  df-sub 9587  df-neg 9588  df-div 9984  df-nn 10313  df-2 10370  df-3 10371  df-n0 10570  df-z 10637  df-uz 10852  df-rp 10982  df-seq 11793  df-exp 11852  df-cj 12574  df-re 12575  df-im 12576  df-sqr 12710  df-abs 12711  df-pell1qr 29030  df-pell14qr 29031  df-pell1234qr 29032
This theorem is referenced by:  elpell1qr2  29060
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