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Theorem pell1234qrreccl 26919
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 26916 . . . 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 472 . . 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 26917 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  ->  A  e.  RR )
4 pell1234qrne0 26918 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  ->  A  =/=  0 )
53, 4rereccld 9843 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  -> 
( 1  /  A
)  e.  RR )
65ad2antrr 708 . . . . . . 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 738 . . . . . . . 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 739 . . . . . . . . 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 10379 . . . . . . . 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 9116 . . . . . . . . . 10  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  -> 
( 1  /  A
)  e.  CC )
1110ad2antrr 708 . . . . . . . . 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 10289 . . . . . . . . . . . 12  |-  ( a  e.  ZZ  ->  a  e.  CC )
1312adantr 453 . . . . . . . . . . 11  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  a  e.  CC )
1413ad2antlr 709 . . . . . . . . . 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 3471 . . . . . . . . . . . . . 14  |-  ( D  e.  ( NN  \NN )  ->  D  e.  NN )
1615nncnd 10018 . . . . . . . . . . . . 13  |-  ( D  e.  ( NN  \NN )  ->  D  e.  CC )
1716ad3antrrr 712 . . . . . . . . . . . 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 12241 . . . . . . . . . . 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 10378 . . . . . . . . . . . 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 9400 . . . . . . . . . . 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 9110 . . . . . . . . . 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 9109 . . . . . . . . 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 9116 . . . . . . . . . 10  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  ->  A  e.  CC )
2423ad2antrr 708 . . . . . . . . 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 708 . . . . . . . . 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 11537 . . . . . . . . . . . . 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 12243 . . . . . . . . . . . . . 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 6098 . . . . . . . . . . . . 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 2471 . . . . . . . . . . . 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 6099 . . . . . . . . . . 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 735 . . . . . . . . . . 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 9110 . . . . . . . . . . . 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 11490 . . . . . . . . . . . 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 644 . . . . . . . . . . 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 2478 . . . . . . . . . 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 9787 . . . . . . . . . 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 734 . . . . . . . . . . 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 9489 . . . . . . . . . . . . 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 6099 . . . . . . . . . . . 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 9419 . . . . . . . . . . . 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 2470 . . . . . . . . . . 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 6101 . . . . . . . . . 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 2480 . . . . . . . . 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 9659 . . . . . . . 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 11444 . . . . . . . . . . . 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 6099 . . . . . . . . . 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 6099 . . . . . . . . 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 2470 . . . . . . . 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 6090 . . . . . . . . . . 11  |-  ( c  =  a  ->  (
c  +  ( ( sqr `  D )  x.  d ) )  =  ( a  +  ( ( sqr `  D
)  x.  d ) ) )
5150eqeq2d 2449 . . . . . . . . . 10  |-  ( c  =  a  ->  (
( 1  /  A
)  =  ( c  +  ( ( sqr `  D )  x.  d
) )  <->  ( 1  /  A )  =  ( a  +  ( ( sqr `  D
)  x.  d ) ) ) )
52 oveq1 6090 . . . . . . . . . . . 12  |-  ( c  =  a  ->  (
c ^ 2 )  =  ( a ^
2 ) )
5352oveq1d 6098 . . . . . . . . . . 11  |-  ( c  =  a  ->  (
( c ^ 2 )  -  ( D  x.  ( d ^
2 ) ) )  =  ( ( a ^ 2 )  -  ( D  x.  (
d ^ 2 ) ) ) )
5453eqeq1d 2446 . . . . . . . . . 10  |-  ( c  =  a  ->  (
( ( c ^
2 )  -  ( D  x.  ( d ^ 2 ) ) )  =  1  <->  (
( a ^ 2 )  -  ( D  x.  ( d ^
2 ) ) )  =  1 ) )
5551, 54anbi12d 693 . . . . . . . . 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 6091 . . . . . . . . . . . 12  |-  ( d  =  -u b  ->  (
( sqr `  D
)  x.  d )  =  ( ( sqr `  D )  x.  -u b
) )
5756oveq2d 6099 . . . . . . . . . . 11  |-  ( d  =  -u b  ->  (
a  +  ( ( sqr `  D )  x.  d ) )  =  ( a  +  ( ( sqr `  D
)  x.  -u b
) ) )
5857eqeq2d 2449 . . . . . . . . . 10  |-  ( d  =  -u b  ->  (
( 1  /  A
)  =  ( a  +  ( ( sqr `  D )  x.  d
) )  <->  ( 1  /  A )  =  ( a  +  ( ( sqr `  D
)  x.  -u b
) ) ) )
59 oveq1 6090 . . . . . . . . . . . . 13  |-  ( d  =  -u b  ->  (
d ^ 2 )  =  ( -u b ^ 2 ) )
6059oveq2d 6099 . . . . . . . . . . . 12  |-  ( d  =  -u b  ->  ( D  x.  ( d ^ 2 ) )  =  ( D  x.  ( -u b ^ 2 ) ) )
6160oveq2d 6099 . . . . . . . . . . 11  |-  ( d  =  -u b  ->  (
( a ^ 2 )  -  ( D  x.  ( d ^
2 ) ) )  =  ( ( a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) ) )
6261eqeq1d 2446 . . . . . . . . . 10  |-  ( d  =  -u b  ->  (
( ( a ^
2 )  -  ( D  x.  ( d ^ 2 ) ) )  =  1  <->  (
( a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) )  =  1 ) )
6358, 62anbi12d 693 . . . . . . . . 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 3062 . . . . . . . 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 1189 . . . . . . 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 520 . . . . . 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 425 . . . . 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 2839 . . . 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 455 . . 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 26916 . . 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 453 . 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 225 1  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  -> 
( 1  /  A
)  e.  (Pell1234QR `  D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708    \ cdif 3319   ` cfv 5456  (class class class)co 6083   CCcc 8990   RRcr 8991   0cc0 8992   1c1 8993    + caddc 8995    x. cmul 8997    - cmin 9293   -ucneg 9294    / cdiv 9679   NNcn 10002   2c2 10051   ZZcz 10284   ^cexp 11384   sqrcsqr 12040  ◻NNcsquarenn 26901  Pell1234QRcpell1234qr 26903
This theorem is referenced by:  pell14qrreccl  26929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-seq 11326  df-exp 11385  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-pell1234qr 26909
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