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Theorem pell14qrdich 35166
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 35146 . . 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 482 . 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 763 . . . . . . . 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 10920 . . . . . . . 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 461 . . . . . 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 754 . . . . . . . . . . 11  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  A  e.  RR )
87ad2antrr 724 . . . . . . . . . 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 756 . . . . . . . . . . . 12  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  a  e.  NN0 )
109ad2antrr 724 . . . . . . . . . . 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 459 . . . . . . . . . . 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 754 . . . . . . . . . . 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 2863 . . . . . . . . . . 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 1230 . . . . . . . . . 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 530 . . . . . . . . 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 432 . . . . . . . 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 35144 . . . . . . . . 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 730 . . . . . . . 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 724 . . . . . . . . . . 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 35155 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  =/=  0 )
2221ad4antr 730 . . . . . . . . . . 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 10412 . . . . . . . . . 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 724 . . . . . . . . . . 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 459 . . . . . . . . . . . 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 35154 . . . . . . . . . . . . . . . . . 18  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR )
2726recnd 9652 . . . . . . . . . . . . . . . . 17  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  CC )
2827, 21reccld 10354 . . . . . . . . . . . . . . . 16  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( 1  /  A
)  e.  CC )
2928ad3antrrr 728 . . . . . . . . . . . . . . 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 10846 . . . . . . . . . . . . . . . . . 18  |-  ( a  e.  NN0  ->  a  e.  CC )
3130ad2antrl 726 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  a  e.  CC )
32 eldifi 3565 . . . . . . . . . . . . . . . . . . . . 21  |-  ( D  e.  ( NN  \NN )  ->  D  e.  NN )
3332nncnd 10592 . . . . . . . . . . . . . . . . . . . 20  |-  ( D  e.  ( NN  \NN )  ->  D  e.  CC )
3433ad3antrrr 728 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  D  e.  CC )
3534sqrtcld 13417 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  ( sqr `  D )  e.  CC )
36 zcn 10910 . . . . . . . . . . . . . . . . . . . 20  |-  ( b  e.  ZZ  ->  b  e.  CC )
3736ad2antll 727 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  b  e.  CC )
3837negcld 9954 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  -u b  e.  CC )
3935, 38mulcld 9646 . . . . . . . . . . . . . . . . 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 9645 . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . 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 728 . . . . . . . . . . . . . . 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 728 . . . . . . . . . . . . . . 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 10356 . . . . . . . . . . . . . . . . . 18  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  x.  (
1  /  A ) )  =  1 )
4544ad3antrrr 728 . . . . . . . . . . . . . . . . 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 758 . . . . . . . . . . . . . . . . 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 2446 . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . . . 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 9646 . . . . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . . . 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 12320 . . . . . . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . . . . . . 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 12366 . . . . . . . . . . . . . . . . . . . . 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 ) ) )
5434sqsqrtd 13419 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  ( ( sqr `  D ) ^
2 )  =  D )
5554oveq1d 6293 . . . . . . . . . . . . . . . . . . . . 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 2444 . . . . . . . . . . . . . . . . . . . 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 6294 . . . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . . . . . . . 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 10051 . . . . . . . . . . . . . . . . . . . . . 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 6294 . . . . . . . . . . . . . . . . . . . . 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 9903 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( a  e.  CC  /\  ( ( sqr `  D
)  x.  b )  e.  CC )  -> 
( a  +  -u ( ( sqr `  D
)  x.  b ) )  =  ( a  -  ( ( sqr `  D )  x.  b
) ) )
6362eqcomd 2410 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( a  e.  CC  /\  ( ( sqr `  D
)  x.  b )  e.  CC )  -> 
( a  -  (
( sqr `  D
)  x.  b ) )  =  ( a  +  -u ( ( sqr `  D )  x.  b
) ) )
6431, 49, 63syl2anc 659 . . . . . . . . . . . . . . . . . . . . 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 2446 . . . . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . . . 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 6296 . . . . . . . . . . . . . . . . . 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 2453 . . . . . . . . . . . . . . . . 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 715 . . . . . . . . . . . . . . . 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 2443 . . . . . . . . . . . . . . 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 10225 . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . 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 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 ) )  /\  -u b  e.  NN0 )  ->  b  e.  CC )
74 sqneg 12273 . . . . . . . . . . . . . . . . 17  |-  ( b  e.  CC  ->  ( -u b ^ 2 )  =  ( b ^
2 ) )
7573, 74syl 17 . . . . . . . . . . . . . . . 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 6294 . . . . . . . . . . . . . . 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 6294 . . . . . . . . . . . . . 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 763 . . . . . . . . . . . . . 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 2443 . . . . . . . . . . . . 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 530 . . . . . . . . . . . 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 6286 . . . . . . . . . . . . . . . 16  |-  ( c  =  -u b  ->  (
( sqr `  D
)  x.  c )  =  ( ( sqr `  D )  x.  -u b
) )
8281oveq2d 6294 . . . . . . . . . . . . . . 15  |-  ( c  =  -u b  ->  (
a  +  ( ( sqr `  D )  x.  c ) )  =  ( a  +  ( ( sqr `  D
)  x.  -u b
) ) )
8382eqeq2d 2416 . . . . . . . . . . . . . 14  |-  ( c  =  -u b  ->  (
( 1  /  A
)  =  ( a  +  ( ( sqr `  D )  x.  c
) )  <->  ( 1  /  A )  =  ( a  +  ( ( sqr `  D
)  x.  -u b
) ) ) )
84 oveq1 6285 . . . . . . . . . . . . . . . . 17  |-  ( c  =  -u b  ->  (
c ^ 2 )  =  ( -u b ^ 2 ) )
8584oveq2d 6294 . . . . . . . . . . . . . . . 16  |-  ( c  =  -u b  ->  ( D  x.  ( c ^ 2 ) )  =  ( D  x.  ( -u b ^ 2 ) ) )
8685oveq2d 6294 . . . . . . . . . . . . . . 15  |-  ( c  =  -u b  ->  (
( a ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  ( ( a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) ) )
8786eqeq1d 2404 . . . . . . . . . . . . . 14  |-  ( c  =  -u b  ->  (
( ( a ^
2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  1  <->  (
( a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) )  =  1 ) )
8883, 87anbi12d 709 . . . . . . . . . . . . 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 3160 . . . . . . . . . . . 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 659 . . . . . . . . . . 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 2862 . . . . . . . . . . 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 659 . . . . . . . . . 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 530 . . . . . . . . 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 432 . . . . . . . 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 35144 . . . . . . . . 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 730 . . . . . . . 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 839 . . . . . 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 432 . . . 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 2903 . . 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 601 . 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 366    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2755    \ cdif 3411   ` cfv 5569  (class class class)co 6278   CCcc 9520   RRcr 9521   0cc0 9522   1c1 9523    + caddc 9525    x. cmul 9527    - cmin 9841   -ucneg 9842    / cdiv 10247   NNcn 10576   2c2 10626   NN0cn0 10836   ZZcz 10905   ^cexp 12210   sqrcsqrt 13215  ◻NNcsquarenn 35133  Pell1QRcpell1qr 35134  Pell14QRcpell14qr 35136
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-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  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-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-sup 7935  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-seq 12152  df-exp 12211  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-pell1qr 35139  df-pell14qr 35140  df-pell1234qr 35141
This theorem is referenced by:  elpell1qr2  35169
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