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Theorem pell14qrdich 26277
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 26257 . . 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 470 . 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 737 . . . . . . . 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 10127 . . . . . . . 8  |-  ( b  e.  ZZ  <->  ( b  e.  RR  /\  ( b  e.  NN0  \/  -u b  e.  NN0 ) ) )
53, 4sylib 188 . . . . . . 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 449 . . . . . 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 731 . . . . . . . . . . 11  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  A  e.  RR )
87ad2antrr 706 . . . . . . . . . 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 732 . . . . . . . . . . . 12  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  a  e.  NN0 )
109ad2antrr 706 . . . . . . . . . . 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 447 . . . . . . . . . . 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 731 . . . . . . . . . . 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 2682 . . . . . . . . . . 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 1182 . . . . . . . . . 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 518 . . . . . . . . 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 423 . . . . . . . 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 26255 . . . . . . . . . 10  |-  ( 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 ) ) ) )
1817adantr 451 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( 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 ) ) ) )
1918ad3antrrr 710 . . . . . . . 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 ) ) ) )
2016, 19sylibrd 225 . . . . . . 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
) ) )
217ad2antrr 706 . . . . . . . . . . 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 )
22 simpll 730 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  ->  D  e.  ( NN  \NN ) )
2322ad3antrrr 710 . . . . . . . . . . . 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 )  ->  D  e.  ( NN 
\NN ) )
24 simplr 731 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  ->  A  e.  (Pell14QR `  D )
)
2524ad3antrrr 710 . . . . . . . . . . . 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 )  ->  A  e.  (Pell14QR `  D
) )
26 pell14qrne0 26266 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  =/=  0 )
2723, 25, 26syl2anc 642 . . . . . . . . . . 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 )
2821, 27rereccld 9674 . . . . . . . . . 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 )
299ad2antrr 706 . . . . . . . . . . 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 )
30 simpr 447 . . . . . . . . . . . 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 )
31 pell14qrre 26265 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR )
3231recnd 8948 . . . . . . . . . . . . . . . . . . 19  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  CC )
3332, 26recidd 9618 . . . . . . . . . . . . . . . . . 18  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  x.  (
1  /  A ) )  =  1 )
3433ad3antrrr 710 . . . . . . . . . . . . . . . . 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 )
35 simprr 733 . . . . . . . . . . . . . . . . 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 )
3634, 35eqtr4d 2393 . . . . . . . . . . . . . . . 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 ) ) ) )
37 nn0cn 10064 . . . . . . . . . . . . . . . . . . . . 21  |-  ( a  e.  NN0  ->  a  e.  CC )
3837ad2antrl 708 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  a  e.  CC )
3938adantr 451 . . . . . . . . . . . . . . . . . . 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 )
40 eldifi 3374 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( D  e.  ( NN  \NN )  ->  D  e.  NN )
4140nncnd 9849 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( D  e.  ( NN  \NN )  ->  D  e.  CC )
4241ad3antrrr 710 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  D  e.  CC )
4342sqrcld 12009 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  ( sqr `  D )  e.  CC )
44 zcn 10118 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( b  e.  ZZ  ->  b  e.  CC )
4544ad2antll 709 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  b  e.  CC )
4643, 45mulcld 8942 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  ( ( sqr `  D )  x.  b )  e.  CC )
4746adantr 451 . . . . . . . . . . . . . . . . . . 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 )
48 subsq 11300 . . . . . . . . . . . . . . . . . . 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 ) ) ) )
4939, 47, 48syl2anc 642 . . . . . . . . . . . . . . . . . 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 ) ) ) )
5043, 45sqmuld 11347 . . . . . . . . . . . . . . . . . . . . 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 ) ) )
5142sqsqrd 12011 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  ( ( sqr `  D ) ^
2 )  =  D )
5251oveq1d 5957 . . . . . . . . . . . . . . . . . . . . 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 ) ) )
5350, 52eqtr2d 2391 . . . . . . . . . . . . . . . . . . . 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 ) )
5453oveq2d 5958 . . . . . . . . . . . . . . . . . . 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 ) ) )
5554adantr 451 . . . . . . . . . . . . . . . . . 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 ) ) )
56 simpr 447 . . . . . . . . . . . . . . . . . . 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 ) ) )
5743, 45mulneg2d 9320 . . . . . . . . . . . . . . . . . . . . . 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 ) )
5857oveq2d 5958 . . . . . . . . . . . . . . . . . . . . 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 ) ) )
59 negsub 9182 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( a  e.  CC  /\  ( ( sqr `  D
)  x.  b )  e.  CC )  -> 
( a  +  -u ( ( sqr `  D
)  x.  b ) )  =  ( a  -  ( ( sqr `  D )  x.  b
) ) )
6059eqcomd 2363 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( a  e.  CC  /\  ( ( sqr `  D
)  x.  b )  e.  CC )  -> 
( a  -  (
( sqr `  D
)  x.  b ) )  =  ( a  +  -u ( ( sqr `  D )  x.  b
) ) )
6138, 46, 60syl2anc 642 . . . . . . . . . . . . . . . . . . . . 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 ) ) )
6258, 61eqtr4d 2393 . . . . . . . . . . . . . . . . . . . 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 ) ) )
6362adantr 451 . . . . . . . . . . . . . . . . . . 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 ) ) )
6456, 63oveq12d 5960 . . . . . . . . . . . . . . . . . 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
) ) ) )
6549, 55, 643eqtr4d 2400 . . . . . . . . . . . . . . . . 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
) ) ) )
6665adantrr 697 . . . . . . . . . . . . . . . 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
) ) ) )
6736, 66eqtrd 2390 . . . . . . . . . . . . . . 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
) ) ) )
6832, 26reccld 9616 . . . . . . . . . . . . . . . . 17  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( 1  /  A
)  e.  CC )
6968ad3antrrr 710 . . . . . . . . . . . . . . . 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 ) )  -> 
( 1  /  A
)  e.  CC )
7045negcld 9231 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  -u b  e.  CC )
7143, 70mulcld 8942 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  ( ( sqr `  D )  x.  -u b )  e.  CC )
7238, 71addcld 8941 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( 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 )
7372adantr 451 . . . . . . . . . . . . . . . 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  +  ( ( sqr `  D
)  x.  -u b
) )  e.  CC )
7432ad3antrrr 710 . . . . . . . . . . . . . . . 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  e.  CC )
7526ad3antrrr 710 . . . . . . . . . . . . . . . 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  =/=  0 )
7669, 73, 74, 75mulcand 9488 . . . . . . . . . . . . . . 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
) ) )  <->  ( 1  /  A )  =  ( a  +  ( ( sqr `  D
)  x.  -u b
) ) ) )
7767, 76mpbid 201 . . . . . . . . . . . . . 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
) ) )
7877adantr 451 . . . . . . . . . . . . 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
) ) )
7945ad2antrr 706 . . . . . . . . . . . . . . . . 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 )
80 sqneg 11254 . . . . . . . . . . . . . . . . 17  |-  ( b  e.  CC  ->  ( -u b ^ 2 )  =  ( b ^
2 ) )
8179, 80syl 15 . . . . . . . . . . . . . . . 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 ) )
8281oveq2d 5958 . . . . . . . . . . . . . . 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 ) ) )
8382oveq2d 5958 . . . . . . . . . . . . . 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 ) ) ) )
84 simplrr 737 . . . . . . . . . . . . . 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 )
8583, 84eqtrd 2390 . . . . . . . . . . . . 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 )
8678, 85jca 518 . . . . . . . . . . . 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 ) )
87 oveq2 5950 . . . . . . . . . . . . . . . 16  |-  ( c  =  -u b  ->  (
( sqr `  D
)  x.  c )  =  ( ( sqr `  D )  x.  -u b
) )
8887oveq2d 5958 . . . . . . . . . . . . . . 15  |-  ( c  =  -u b  ->  (
a  +  ( ( sqr `  D )  x.  c ) )  =  ( a  +  ( ( sqr `  D
)  x.  -u b
) ) )
8988eqeq2d 2369 . . . . . . . . . . . . . 14  |-  ( c  =  -u b  ->  (
( 1  /  A
)  =  ( a  +  ( ( sqr `  D )  x.  c
) )  <->  ( 1  /  A )  =  ( a  +  ( ( sqr `  D
)  x.  -u b
) ) ) )
90 oveq1 5949 . . . . . . . . . . . . . . . . 17  |-  ( c  =  -u b  ->  (
c ^ 2 )  =  ( -u b ^ 2 ) )
9190oveq2d 5958 . . . . . . . . . . . . . . . 16  |-  ( c  =  -u b  ->  ( D  x.  ( c ^ 2 ) )  =  ( D  x.  ( -u b ^ 2 ) ) )
9291oveq2d 5958 . . . . . . . . . . . . . . 15  |-  ( c  =  -u b  ->  (
( a ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  ( ( a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) ) )
9392eqeq1d 2366 . . . . . . . . . . . . . 14  |-  ( c  =  -u b  ->  (
( ( a ^
2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  1  <->  (
( a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) )  =  1 ) )
9489, 93anbi12d 691 . . . . . . . . . . . . 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 ) ) )
9594rspcev 2960 . . . . . . . . . . . 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 ) )
9630, 86, 95syl2anc 642 . . . . . . . . . . 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 ) )
97 rspe 2680 . . . . . . . . . . 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 ) )
9829, 96, 97syl2anc 642 . . . . . . . . . 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 ) )
9928, 98jca 518 . . . . . . . . 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 ) ) )
10099ex 423 . . . . . . . 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 ) ) ) )
101 elpell1qr 26255 . . . . . . . . . 10  |-  ( 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 ) ) ) )
102101adantr 451 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( ( 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 ) ) ) )
103102ad3antrrr 710 . . . . . . . 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 ) ) ) )
104100, 103sylibrd 225 . . . . . . 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
) ) )
10520, 104orim12d 811 . . . . . 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 )
) ) )
1066, 105mpd 14 . . . . 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
) ) )
107106ex 423 . . . 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
) ) ) )
108107rexlimdvva 2750 . . 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 )
) ) )
109108expimpd 586 . 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
) ) ) )
1102, 109mpd 14 1  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  e.  (Pell1QR `  D )  \/  (
1  /  A )  e.  (Pell1QR `  D
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   E.wrex 2620    \ cdif 3225   ` cfv 5334  (class class class)co 5942   CCcc 8822   RRcr 8823   0cc0 8824   1c1 8825    + caddc 8827    x. cmul 8829    - cmin 9124   -ucneg 9125    / cdiv 9510   NNcn 9833   2c2 9882   NN0cn0 10054   ZZcz 10113   ^cexp 11194   sqrcsqr 11808  ◻NNcsquarenn 26244  Pell1QRcpell1qr 26245  Pell14QRcpell14qr 26247
This theorem is referenced by:  elpell1qr2  26280
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-sup 7281  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-n0 10055  df-z 10114  df-uz 10320  df-rp 10444  df-seq 11136  df-exp 11195  df-cj 11674  df-re 11675  df-im 11676  df-sqr 11810  df-abs 11811  df-pell1qr 26250  df-pell14qr 26251  df-pell1234qr 26252
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