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Theorem pell1234qrreccl 29200
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 29197 . . . 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 29198 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  ->  A  e.  RR )
4 pell1234qrne0 29199 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  ->  A  =/=  0 )
53, 4rereccld 10163 . . . . . . . 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 759 . . . . . . . 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 760 . . . . . . . . 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 10754 . . . . . . . 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 9417 . . . . . . . . . 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 10656 . . . . . . . . . . . 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 3483 . . . . . . . . . . . . . 14  |-  ( D  e.  ( NN  \NN )  ->  D  e.  NN )
1615nncnd 10343 . . . . . . . . . . . . 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 )
1817sqrcld 12928 . . . . . . . . . . 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 10753 . . . . . . . . . . . 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 9711 . . . . . . . . . . 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 9411 . . . . . . . . . 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 9410 . . . . . . . . 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 9417 . . . . . . . . . 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 12025 . . . . . . . . . . . . 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 ) ) )
2717sqsqrd 12930 . . . . . . . . . . . . . 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 6111 . . . . . . . . . . . . 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 2476 . . . . . . . . . . . 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 6112 . . . . . . . . . . 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 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 ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 )
3218, 19mulcld 9411 . . . . . . . . . . . 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 11978 . . . . . . . . . . . 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 2483 . . . . . . . . . 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 10107 . . . . . . . . . 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 755 . . . . . . . . . . 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 9803 . . . . . . . . . . . . 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 6112 . . . . . . . . . . . 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 9730 . . . . . . . . . . . 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 2475 . . . . . . . . . . 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 6114 . . . . . . . . . 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 2485 . . . . . . . . 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 9976 . . . . . . . 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 11931 . . . . . . . . . . . 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 6112 . . . . . . . . . 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 6112 . . . . . . . . 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 2475 . . . . . . . 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 6103 . . . . . . . . . . 11  |-  ( c  =  a  ->  (
c  +  ( ( sqr `  D )  x.  d ) )  =  ( a  +  ( ( sqr `  D
)  x.  d ) ) )
5150eqeq2d 2454 . . . . . . . . . 10  |-  ( c  =  a  ->  (
( 1  /  A
)  =  ( c  +  ( ( sqr `  D )  x.  d
) )  <->  ( 1  /  A )  =  ( a  +  ( ( sqr `  D
)  x.  d ) ) ) )
52 oveq1 6103 . . . . . . . . . . . 12  |-  ( c  =  a  ->  (
c ^ 2 )  =  ( a ^
2 ) )
5352oveq1d 6111 . . . . . . . . . . 11  |-  ( c  =  a  ->  (
( c ^ 2 )  -  ( D  x.  ( d ^
2 ) ) )  =  ( ( a ^ 2 )  -  ( D  x.  (
d ^ 2 ) ) ) )
5453eqeq1d 2451 . . . . . . . . . 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 6104 . . . . . . . . . . . 12  |-  ( d  =  -u b  ->  (
( sqr `  D
)  x.  d )  =  ( ( sqr `  D )  x.  -u b
) )
5756oveq2d 6112 . . . . . . . . . . 11  |-  ( d  =  -u b  ->  (
a  +  ( ( sqr `  D )  x.  d ) )  =  ( a  +  ( ( sqr `  D
)  x.  -u b
) ) )
5857eqeq2d 2454 . . . . . . . . . 10  |-  ( d  =  -u b  ->  (
( 1  /  A
)  =  ( a  +  ( ( sqr `  D )  x.  d
) )  <->  ( 1  /  A )  =  ( a  +  ( ( sqr `  D
)  x.  -u b
) ) ) )
59 oveq1 6103 . . . . . . . . . . . . 13  |-  ( d  =  -u b  ->  (
d ^ 2 )  =  ( -u b ^ 2 ) )
6059oveq2d 6112 . . . . . . . . . . . 12  |-  ( d  =  -u b  ->  ( D  x.  ( d ^ 2 ) )  =  ( D  x.  ( -u b ^ 2 ) ) )
6160oveq2d 6112 . . . . . . . . . . 11  |-  ( d  =  -u b  ->  (
( a ^ 2 )  -  ( D  x.  ( d ^
2 ) ) )  =  ( ( a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) ) )
6261eqeq1d 2451 . . . . . . . . . 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 3086 . . . . . . . 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 1222 . . . . . . 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 2853 . . . 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 29197 . . 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 1369    e. wcel 1756    =/= wne 2611   E.wrex 2721    \ cdif 3330   ` cfv 5423  (class class class)co 6096   CCcc 9285   RRcr 9286   0cc0 9287   1c1 9288    + caddc 9290    x. cmul 9292    - cmin 9600   -ucneg 9601    / cdiv 9998   NNcn 10327   2c2 10376   ZZcz 10651   ^cexp 11870   sqrcsqr 12727  ◻NNcsquarenn 29182  Pell1234QRcpell1234qr 29184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-seq 11812  df-exp 11871  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-pell1234qr 29190
This theorem is referenced by:  pell14qrreccl  29210
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