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Theorem pell14qrdich 26822
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 26802 . . 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 471 . 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 738 . . . . . . . 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 10252 . . . . . . . 8  |-  ( b  e.  ZZ  <->  ( b  e.  RR  /\  ( b  e.  NN0  \/  -u b  e.  NN0 ) ) )
53, 4sylib 189 . . . . . . 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 450 . . . . . 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 732 . . . . . . . . . . 11  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  A  e.  RR )
87ad2antrr 707 . . . . . . . . . 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 733 . . . . . . . . . . . 12  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  a  e.  NN0 )
109ad2antrr 707 . . . . . . . . . . 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 448 . . . . . . . . . . 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 732 . . . . . . . . . . 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 2729 . . . . . . . . . . 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 1184 . . . . . . . . . 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 519 . . . . . . . . 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 424 . . . . . . . 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 26800 . . . . . . . . 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 713 . . . . . . . 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 226 . . . . . . 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 707 . . . . . . . . . . 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 26811 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  =/=  0 )
2221ad4antr 713 . . . . . . . . . . 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 9797 . . . . . . . . . 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 707 . . . . . . . . . . 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 448 . . . . . . . . . . . 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 26810 . . . . . . . . . . . . . . . . . 18  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  RR )
2726recnd 9070 . . . . . . . . . . . . . . . . 17  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  ->  A  e.  CC )
2827, 21reccld 9739 . . . . . . . . . . . . . . . 16  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( 1  /  A
)  e.  CC )
2928ad3antrrr 711 . . . . . . . . . . . . . . 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 10187 . . . . . . . . . . . . . . . . . 18  |-  ( a  e.  NN0  ->  a  e.  CC )
3130ad2antrl 709 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  a  e.  CC )
32 eldifi 3429 . . . . . . . . . . . . . . . . . . . . 21  |-  ( D  e.  ( NN  \NN )  ->  D  e.  NN )
3332nncnd 9972 . . . . . . . . . . . . . . . . . . . 20  |-  ( D  e.  ( NN  \NN )  ->  D  e.  CC )
3433ad3antrrr 711 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  D  e.  CC )
3534sqrcld 12194 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  ( sqr `  D )  e.  CC )
36 zcn 10243 . . . . . . . . . . . . . . . . . . . 20  |-  ( b  e.  ZZ  ->  b  e.  CC )
3736ad2antll 710 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  b  e.  CC )
3837negcld 9354 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  -u b  e.  CC )
3935, 38mulcld 9064 . . . . . . . . . . . . . . . . 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 9063 . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . 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 711 . . . . . . . . . . . . . . 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 711 . . . . . . . . . . . . . . 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 9741 . . . . . . . . . . . . . . . . . 18  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell14QR `  D ) )  -> 
( A  x.  (
1  /  A ) )  =  1 )
4544ad3antrrr 711 . . . . . . . . . . . . . . . . 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 734 . . . . . . . . . . . . . . . . 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 2439 . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . . . . . 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 9064 . . . . . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . . . . . 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 11443 . . . . . . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . . . . . . 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 11490 . . . . . . . . . . . . . . . . . . . . 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 12196 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell14QR `  D )
)  /\  A  e.  RR )  /\  (
a  e.  NN0  /\  b  e.  ZZ )
)  ->  ( ( sqr `  D ) ^
2 )  =  D )
5554oveq1d 6055 . . . . . . . . . . . . . . . . . . . . 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 2437 . . . . . . . . . . . . . . . . . . . 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 6056 . . . . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . . . . 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 448 . . . . . . . . . . . . . . . . . . 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 9443 . . . . . . . . . . . . . . . . . . . . . 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 6056 . . . . . . . . . . . . . . . . . . . . 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 9305 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( a  e.  CC  /\  ( ( sqr `  D
)  x.  b )  e.  CC )  -> 
( a  +  -u ( ( sqr `  D
)  x.  b ) )  =  ( a  -  ( ( sqr `  D )  x.  b
) ) )
6362eqcomd 2409 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( a  e.  CC  /\  ( ( sqr `  D
)  x.  b )  e.  CC )  -> 
( a  -  (
( sqr `  D
)  x.  b ) )  =  ( a  +  -u ( ( sqr `  D )  x.  b
) ) )
6431, 49, 63syl2anc 643 . . . . . . . . . . . . . . . . . . . . 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 2439 . . . . . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . . . . . 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 6058 . . . . . . . . . . . . . . . . . 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 2446 . . . . . . . . . . . . . . . . 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 698 . . . . . . . . . . . . . . . 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 2436 . . . . . . . . . . . . . . 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 9613 . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . 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 707 . . . . . . . . . . . . . . . . 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 11397 . . . . . . . . . . . . . . . . 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 6056 . . . . . . . . . . . . . . 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 6056 . . . . . . . . . . . . . 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 738 . . . . . . . . . . . . . 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 2436 . . . . . . . . . . . . 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 519 . . . . . . . . . . . 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 6048 . . . . . . . . . . . . . . . 16  |-  ( c  =  -u b  ->  (
( sqr `  D
)  x.  c )  =  ( ( sqr `  D )  x.  -u b
) )
8281oveq2d 6056 . . . . . . . . . . . . . . 15  |-  ( c  =  -u b  ->  (
a  +  ( ( sqr `  D )  x.  c ) )  =  ( a  +  ( ( sqr `  D
)  x.  -u b
) ) )
8382eqeq2d 2415 . . . . . . . . . . . . . 14  |-  ( c  =  -u b  ->  (
( 1  /  A
)  =  ( a  +  ( ( sqr `  D )  x.  c
) )  <->  ( 1  /  A )  =  ( a  +  ( ( sqr `  D
)  x.  -u b
) ) ) )
84 oveq1 6047 . . . . . . . . . . . . . . . . 17  |-  ( c  =  -u b  ->  (
c ^ 2 )  =  ( -u b ^ 2 ) )
8584oveq2d 6056 . . . . . . . . . . . . . . . 16  |-  ( c  =  -u b  ->  ( D  x.  ( c ^ 2 ) )  =  ( D  x.  ( -u b ^ 2 ) ) )
8685oveq2d 6056 . . . . . . . . . . . . . . 15  |-  ( c  =  -u b  ->  (
( a ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  ( ( a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) ) )
8786eqeq1d 2412 . . . . . . . . . . . . . 14  |-  ( c  =  -u b  ->  (
( ( a ^
2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  1  <->  (
( a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) )  =  1 ) )
8883, 87anbi12d 692 . . . . . . . . . . . . 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 3012 . . . . . . . . . . . 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 643 . . . . . . . . . . 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 2727 . . . . . . . . . . 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 643 . . . . . . . . . 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 519 . . . . . . . . 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 424 . . . . . . . 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 26800 . . . . . . . . 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 713 . . . . . . . 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 226 . . . . . . 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 812 . . . . . 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 424 . . . 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 2797 . . 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 587 . 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 set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667    \ cdif 3277   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    - cmin 9247   -ucneg 9248    / cdiv 9633   NNcn 9956   2c2 10005   NN0cn0 10177   ZZcz 10238   ^cexp 11337   sqrcsqr 11993  ◻NNcsquarenn 26789  Pell1QRcpell1qr 26790  Pell14QRcpell14qr 26792
This theorem is referenced by:  elpell1qr2  26825
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-pell1qr 26795  df-pell14qr 26796  df-pell1234qr 26797
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