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Theorem pell1234qrreccl 30952
Description: General solutions of the Pell equation are closed under reciprocals. (Contributed by Stefan O'Rear, 18-Sep-2014.)
Assertion
Ref Expression
pell1234qrreccl  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  -> 
( 1  /  A
)  e.  (Pell1234QR `  D
) )

Proof of Theorem pell1234qrreccl
Dummy variables  a 
b  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elpell1234qr 30949 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
( A  e.  (Pell1234QR `  D )  <->  ( A  e.  RR  /\  E. a  e.  ZZ  E. b  e.  ZZ  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) ) ) )
21biimpa 484 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  -> 
( A  e.  RR  /\ 
E. a  e.  ZZ  E. b  e.  ZZ  ( A  =  ( a  +  ( ( sqr `  D )  x.  b
) )  /\  (
( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 ) ) )
3 pell1234qrre 30950 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  ->  A  e.  RR )
4 pell1234qrne0 30951 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  ->  A  =/=  0 )
53, 4rereccld 10392 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  -> 
( 1  /  A
)  e.  RR )
65ad2antrr 725 . . . . . . 7  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  (
1  /  A )  e.  RR )
7 simplrl 761 . . . . . . . 8  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  a  e.  ZZ )
8 simplrr 762 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  b  e.  ZZ )
98znegcld 10992 . . . . . . . 8  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  -u b  e.  ZZ )
105recnd 9639 . . . . . . . . . 10  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  -> 
( 1  /  A
)  e.  CC )
1110ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  (
1  /  A )  e.  CC )
12 zcn 10890 . . . . . . . . . . . 12  |-  ( a  e.  ZZ  ->  a  e.  CC )
1312adantr 465 . . . . . . . . . . 11  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  a  e.  CC )
1413ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  a  e.  CC )
15 eldifi 3622 . . . . . . . . . . . . . 14  |-  ( D  e.  ( NN  \NN )  ->  D  e.  NN )
1615nncnd 10572 . . . . . . . . . . . . 13  |-  ( D  e.  ( NN  \NN )  ->  D  e.  CC )
1716ad3antrrr 729 . . . . . . . . . . . 12  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  D  e.  CC )
1817sqrtcld 13279 . . . . . . . . . . 11  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  ( sqr `  D )  e.  CC )
198zcnd 10991 . . . . . . . . . . . 12  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  b  e.  CC )
2019negcld 9937 . . . . . . . . . . 11  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  -u b  e.  CC )
2118, 20mulcld 9633 . . . . . . . . . 10  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  (
( sqr `  D
)  x.  -u b
)  e.  CC )
2214, 21addcld 9632 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  (
a  +  ( ( sqr `  D )  x.  -u b ) )  e.  CC )
233recnd 9639 . . . . . . . . . 10  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  ->  A  e.  CC )
2423ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  A  e.  CC )
254ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  A  =/=  0 )
2618, 19sqmuld 12324 . . . . . . . . . . . . 13  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  (
( ( sqr `  D
)  x.  b ) ^ 2 )  =  ( ( ( sqr `  D ) ^ 2 )  x.  ( b ^ 2 ) ) )
2717sqsqrtd 13281 . . . . . . . . . . . . . 14  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  (
( sqr `  D
) ^ 2 )  =  D )
2827oveq1d 6311 . . . . . . . . . . . . 13  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  (
( ( sqr `  D
) ^ 2 )  x.  ( b ^
2 ) )  =  ( D  x.  (
b ^ 2 ) ) )
2926, 28eqtr2d 2499 . . . . . . . . . . . 12  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  ( D  x.  ( b ^ 2 ) )  =  ( ( ( sqr `  D )  x.  b ) ^
2 ) )
3029oveq2d 6312 . . . . . . . . . . 11  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  (
( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  ( ( a ^ 2 )  -  ( ( ( sqr `  D )  x.  b
) ^ 2 ) ) )
31 simprr 757 . . . . . . . . . . 11  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  (
( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 )
3218, 19mulcld 9633 . . . . . . . . . . . 12  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  (
( sqr `  D
)  x.  b )  e.  CC )
33 subsq 12277 . . . . . . . . . . . 12  |-  ( ( 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 ) ) ) )
3414, 32, 33syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  (
( a ^ 2 )  -  ( ( ( sqr `  D
)  x.  b ) ^ 2 ) )  =  ( ( a  +  ( ( sqr `  D )  x.  b
) )  x.  (
a  -  ( ( sqr `  D )  x.  b ) ) ) )
3530, 31, 343eqtr3d 2506 . . . . . . . . . 10  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  1  =  ( ( a  +  ( ( sqr `  D )  x.  b
) )  x.  (
a  -  ( ( sqr `  D )  x.  b ) ) ) )
3624, 25recidd 10336 . . . . . . . . . 10  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  ( A  x.  ( 1  /  A ) )  =  1 )
37 simprl 756 . . . . . . . . . . 11  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  A  =  ( a  +  ( ( sqr `  D
)  x.  b ) ) )
3818, 19mulneg2d 10031 . . . . . . . . . . . . 13  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  (
( sqr `  D
)  x.  -u b
)  =  -u (
( sqr `  D
)  x.  b ) )
3938oveq2d 6312 . . . . . . . . . . . 12  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  (
a  +  ( ( sqr `  D )  x.  -u b ) )  =  ( a  + 
-u ( ( sqr `  D )  x.  b
) ) )
4014, 32negsubd 9956 . . . . . . . . . . . 12  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  (
a  +  -u (
( sqr `  D
)  x.  b ) )  =  ( a  -  ( ( sqr `  D )  x.  b
) ) )
4139, 40eqtrd 2498 . . . . . . . . . . 11  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  (
a  +  ( ( sqr `  D )  x.  -u b ) )  =  ( a  -  ( ( sqr `  D
)  x.  b ) ) )
4237, 41oveq12d 6314 . . . . . . . . . 10  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  ( A  x.  ( a  +  ( ( sqr `  D )  x.  -u b
) ) )  =  ( ( a  +  ( ( sqr `  D
)  x.  b ) )  x.  ( a  -  ( ( sqr `  D )  x.  b
) ) ) )
4335, 36, 423eqtr4d 2508 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  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
) ) ) )
4411, 22, 24, 25, 43mulcanad 10205 . . . . . . . 8  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  (
1  /  A )  =  ( a  +  ( ( sqr `  D
)  x.  -u b
) ) )
45 sqneg 12230 . . . . . . . . . . . 12  |-  ( b  e.  CC  ->  ( -u b ^ 2 )  =  ( b ^
2 ) )
4619, 45syl 16 . . . . . . . . . . 11  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  ( -u b ^ 2 )  =  ( b ^
2 ) )
4746oveq2d 6312 . . . . . . . . . 10  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  ( D  x.  ( -u b ^ 2 ) )  =  ( D  x.  ( b ^ 2 ) ) )
4847oveq2d 6312 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  (
( a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) )  =  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) ) )
4948, 31eqtrd 2498 . . . . . . . 8  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  (
( a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) )  =  1 )
50 oveq1 6303 . . . . . . . . . . 11  |-  ( c  =  a  ->  (
c  +  ( ( sqr `  D )  x.  d ) )  =  ( a  +  ( ( sqr `  D
)  x.  d ) ) )
5150eqeq2d 2471 . . . . . . . . . 10  |-  ( c  =  a  ->  (
( 1  /  A
)  =  ( c  +  ( ( sqr `  D )  x.  d
) )  <->  ( 1  /  A )  =  ( a  +  ( ( sqr `  D
)  x.  d ) ) ) )
52 oveq1 6303 . . . . . . . . . . . 12  |-  ( c  =  a  ->  (
c ^ 2 )  =  ( a ^
2 ) )
5352oveq1d 6311 . . . . . . . . . . 11  |-  ( c  =  a  ->  (
( c ^ 2 )  -  ( D  x.  ( d ^
2 ) ) )  =  ( ( a ^ 2 )  -  ( D  x.  (
d ^ 2 ) ) ) )
5453eqeq1d 2459 . . . . . . . . . 10  |-  ( c  =  a  ->  (
( ( c ^
2 )  -  ( D  x.  ( d ^ 2 ) ) )  =  1  <->  (
( a ^ 2 )  -  ( D  x.  ( d ^
2 ) ) )  =  1 ) )
5551, 54anbi12d 710 . . . . . . . . 9  |-  ( c  =  a  ->  (
( ( 1  /  A )  =  ( c  +  ( ( sqr `  D )  x.  d ) )  /\  ( ( c ^ 2 )  -  ( D  x.  (
d ^ 2 ) ) )  =  1 )  <->  ( ( 1  /  A )  =  ( a  +  ( ( sqr `  D
)  x.  d ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( d ^ 2 ) ) )  =  1 ) ) )
56 oveq2 6304 . . . . . . . . . . . 12  |-  ( d  =  -u b  ->  (
( sqr `  D
)  x.  d )  =  ( ( sqr `  D )  x.  -u b
) )
5756oveq2d 6312 . . . . . . . . . . 11  |-  ( d  =  -u b  ->  (
a  +  ( ( sqr `  D )  x.  d ) )  =  ( a  +  ( ( sqr `  D
)  x.  -u b
) ) )
5857eqeq2d 2471 . . . . . . . . . 10  |-  ( d  =  -u b  ->  (
( 1  /  A
)  =  ( a  +  ( ( sqr `  D )  x.  d
) )  <->  ( 1  /  A )  =  ( a  +  ( ( sqr `  D
)  x.  -u b
) ) ) )
59 oveq1 6303 . . . . . . . . . . . . 13  |-  ( d  =  -u b  ->  (
d ^ 2 )  =  ( -u b ^ 2 ) )
6059oveq2d 6312 . . . . . . . . . . . 12  |-  ( d  =  -u b  ->  ( D  x.  ( d ^ 2 ) )  =  ( D  x.  ( -u b ^ 2 ) ) )
6160oveq2d 6312 . . . . . . . . . . 11  |-  ( d  =  -u b  ->  (
( a ^ 2 )  -  ( D  x.  ( d ^
2 ) ) )  =  ( ( a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) ) )
6261eqeq1d 2459 . . . . . . . . . 10  |-  ( d  =  -u b  ->  (
( ( a ^
2 )  -  ( D  x.  ( d ^ 2 ) ) )  =  1  <->  (
( a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) )  =  1 ) )
6358, 62anbi12d 710 . . . . . . . . 9  |-  ( d  =  -u b  ->  (
( ( 1  /  A )  =  ( a  +  ( ( sqr `  D )  x.  d ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
d ^ 2 ) ) )  =  1 )  <->  ( ( 1  /  A )  =  ( a  +  ( ( sqr `  D
)  x.  -u b
) )  /\  (
( a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) )  =  1 ) ) )
6455, 63rspc2ev 3221 . . . . . . . 8  |-  ( ( a  e.  ZZ  /\  -u b  e.  ZZ  /\  ( ( 1  /  A )  =  ( a  +  ( ( sqr `  D )  x.  -u b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) )  =  1 ) )  ->  E. c  e.  ZZ  E. d  e.  ZZ  ( ( 1  /  A )  =  ( c  +  ( ( sqr `  D
)  x.  d ) )  /\  ( ( c ^ 2 )  -  ( D  x.  ( d ^ 2 ) ) )  =  1 ) )
657, 9, 44, 49, 64syl112anc 1232 . . . . . . 7  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  E. c  e.  ZZ  E. d  e.  ZZ  ( ( 1  /  A )  =  ( c  +  ( ( sqr `  D
)  x.  d ) )  /\  ( ( c ^ 2 )  -  ( D  x.  ( d ^ 2 ) ) )  =  1 ) )
666, 65jca 532 . . . . . 6  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) )  ->  (
( 1  /  A
)  e.  RR  /\  E. c  e.  ZZ  E. d  e.  ZZ  (
( 1  /  A
)  =  ( c  +  ( ( sqr `  D )  x.  d
) )  /\  (
( c ^ 2 )  -  ( D  x.  ( d ^
2 ) ) )  =  1 ) ) )
6766ex 434 . . . . 5  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  (Pell1234QR `  D ) )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  -> 
( ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  ->  (
( 1  /  A
)  e.  RR  /\  E. c  e.  ZZ  E. d  e.  ZZ  (
( 1  /  A
)  =  ( c  +  ( ( sqr `  D )  x.  d
) )  /\  (
( c ^ 2 )  -  ( D  x.  ( d ^
2 ) ) )  =  1 ) ) ) )
6867rexlimdvva 2956 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  -> 
( E. a  e.  ZZ  E. b  e.  ZZ  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  ->  (
( 1  /  A
)  e.  RR  /\  E. c  e.  ZZ  E. d  e.  ZZ  (
( 1  /  A
)  =  ( c  +  ( ( sqr `  D )  x.  d
) )  /\  (
( c ^ 2 )  -  ( D  x.  ( d ^
2 ) ) )  =  1 ) ) ) )
6968adantld 467 . . 3  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  -> 
( ( A  e.  RR  /\  E. a  e.  ZZ  E. b  e.  ZZ  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( ( 1  /  A )  e.  RR  /\ 
E. c  e.  ZZ  E. d  e.  ZZ  (
( 1  /  A
)  =  ( c  +  ( ( sqr `  D )  x.  d
) )  /\  (
( c ^ 2 )  -  ( D  x.  ( d ^
2 ) ) )  =  1 ) ) ) )
702, 69mpd 15 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  -> 
( ( 1  /  A )  e.  RR  /\ 
E. c  e.  ZZ  E. d  e.  ZZ  (
( 1  /  A
)  =  ( c  +  ( ( sqr `  D )  x.  d
) )  /\  (
( c ^ 2 )  -  ( D  x.  ( d ^
2 ) ) )  =  1 ) ) )
71 elpell1234qr 30949 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
( ( 1  /  A )  e.  (Pell1234QR `  D )  <->  ( (
1  /  A )  e.  RR  /\  E. c  e.  ZZ  E. d  e.  ZZ  ( ( 1  /  A )  =  ( c  +  ( ( sqr `  D
)  x.  d ) )  /\  ( ( c ^ 2 )  -  ( D  x.  ( d ^ 2 ) ) )  =  1 ) ) ) )
7271adantr 465 . 2  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  -> 
( ( 1  /  A )  e.  (Pell1234QR `  D )  <->  ( (
1  /  A )  e.  RR  /\  E. c  e.  ZZ  E. d  e.  ZZ  ( ( 1  /  A )  =  ( c  +  ( ( sqr `  D
)  x.  d ) )  /\  ( ( c ^ 2 )  -  ( D  x.  ( d ^ 2 ) ) )  =  1 ) ) ) )
7370, 72mpbird 232 1  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  -> 
( 1  /  A
)  e.  (Pell1234QR `  D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808    \ cdif 3468   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514    - cmin 9824   -ucneg 9825    / cdiv 10227   NNcn 10556   2c2 10606   ZZcz 10885   ^cexp 12168   sqrcsqrt 13077  ◻NNcsquarenn 30934  Pell1234QRcpell1234qr 30936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-seq 12110  df-exp 12169  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-pell1234qr 30942
This theorem is referenced by:  pell14qrreccl  30962
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