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Theorem pell1234qrreccl 27042
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 27039 . . . 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 470 . . 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 27040 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  ->  A  e.  RR )
4 pell1234qrne0 27041 . . . . . . . . 9  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  ->  A  =/=  0 )
53, 4rereccld 9603 . . . . . . . 8  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  -> 
( 1  /  A
)  e.  RR )
65ad2antrr 706 . . . . . . 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 736 . . . . . . . 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 737 . . . . . . . . 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 10135 . . . . . . . 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 )
10 eldifi 3311 . . . . . . . . . . . . . . . . 17  |-  ( D  e.  ( NN  \NN )  ->  D  e.  NN )
1110nncnd 9778 . . . . . . . . . . . . . . . 16  |-  ( D  e.  ( NN  \NN )  ->  D  e.  CC )
1211ad3antrrr 710 . . . . . . . . . . . . . . 15  |-  ( ( ( ( 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 )
1312sqrcld 11935 . . . . . . . . . . . . . 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 )  e.  CC )
148zcnd 10134 . . . . . . . . . . . . . 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 ) )  ->  b  e.  CC )
1513, 14sqmuld 11273 . . . . . . . . . . . . 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 ) ) )
1612sqsqrd 11937 . . . . . . . . . . . . . 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 )
1716oveq1d 5889 . . . . . . . . . . . . 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 ) ) )
1815, 17eqtr2d 2329 . . . . . . . . . . . 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 ) )
1918oveq2d 5890 . . . . . . . . . . 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 ) ) )
20 simprr 733 . . . . . . . . . . 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 )
21 zcn 10045 . . . . . . . . . . . . . 14  |-  ( a  e.  ZZ  ->  a  e.  CC )
2221adantr 451 . . . . . . . . . . . . 13  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  a  e.  CC )
2322ad2antlr 707 . . . . . . . . . . . 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  e.  CC )
2413, 14mulcld 8871 . . . . . . . . . . . 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 )
25 subsq 11226 . . . . . . . . . . . 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 ) ) ) )
2623, 24, 25syl2anc 642 . . . . . . . . . . 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 ) ) ) )
2719, 20, 263eqtr3d 2336 . . . . . . . . . 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 ) ) ) )
283recnd 8877 . . . . . . . . . . . 12  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  ->  A  e.  CC )
2928ad2antrr 706 . . . . . . . . . . 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  e.  CC )
304ad2antrr 706 . . . . . . . . . . 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  =/=  0 )
3129, 30recidd 9547 . . . . . . . . . 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 )
32 simprl 732 . . . . . . . . . . 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 ) ) )
3313, 14mulneg2d 9249 . . . . . . . . . . . . 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 ) )
3433oveq2d 5890 . . . . . . . . . . . 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
) ) )
3523, 24negsubd 9179 . . . . . . . . . . . 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
) ) )
3634, 35eqtrd 2328 . . . . . . . . . . 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 ) ) )
3732, 36oveq12d 5892 . . . . . . . . . 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
) ) ) )
3827, 31, 373eqtr4d 2338 . . . . . . . . 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
) ) ) )
395recnd 8877 . . . . . . . . . . 11  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  -> 
( 1  /  A
)  e.  CC )
4039ad2antrr 706 . . . . . . . . . 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 )  e.  CC )
4114negcld 9160 . . . . . . . . . . . 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 ) )  ->  -u b  e.  CC )
4213, 41mulcld 8871 . . . . . . . . . . 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
)  x.  -u b
)  e.  CC )
4323, 42addcld 8870 . . . . . . . . . 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  +  ( ( sqr `  D )  x.  -u b ) )  e.  CC )
4440, 43, 29, 30mulcand 9417 . . . . . . . . 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
) ) )  <->  ( 1  /  A )  =  ( a  +  ( ( sqr `  D
)  x.  -u b
) ) ) )
4538, 44mpbid 201 . . . . . . . 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
) ) )
46 sqneg 11180 . . . . . . . . . . . 12  |-  ( b  e.  CC  ->  ( -u b ^ 2 )  =  ( b ^
2 ) )
4714, 46syl 15 . . . . . . . . . . 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 ) )
4847oveq2d 5890 . . . . . . . . . 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 ) ) )
4948oveq2d 5890 . . . . . . . . 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 ) ) ) )
5049, 20eqtrd 2328 . . . . . . . 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 )
51 oveq1 5881 . . . . . . . . . . 11  |-  ( c  =  a  ->  (
c  +  ( ( sqr `  D )  x.  d ) )  =  ( a  +  ( ( sqr `  D
)  x.  d ) ) )
5251eqeq2d 2307 . . . . . . . . . 10  |-  ( c  =  a  ->  (
( 1  /  A
)  =  ( c  +  ( ( sqr `  D )  x.  d
) )  <->  ( 1  /  A )  =  ( a  +  ( ( sqr `  D
)  x.  d ) ) ) )
53 oveq1 5881 . . . . . . . . . . . 12  |-  ( c  =  a  ->  (
c ^ 2 )  =  ( a ^
2 ) )
5453oveq1d 5889 . . . . . . . . . . 11  |-  ( c  =  a  ->  (
( c ^ 2 )  -  ( D  x.  ( d ^
2 ) ) )  =  ( ( a ^ 2 )  -  ( D  x.  (
d ^ 2 ) ) ) )
5554eqeq1d 2304 . . . . . . . . . 10  |-  ( c  =  a  ->  (
( ( c ^
2 )  -  ( D  x.  ( d ^ 2 ) ) )  =  1  <->  (
( a ^ 2 )  -  ( D  x.  ( d ^
2 ) ) )  =  1 ) )
5652, 55anbi12d 691 . . . . . . . . 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 ) ) )
57 oveq2 5882 . . . . . . . . . . . 12  |-  ( d  =  -u b  ->  (
( sqr `  D
)  x.  d )  =  ( ( sqr `  D )  x.  -u b
) )
5857oveq2d 5890 . . . . . . . . . . 11  |-  ( d  =  -u b  ->  (
a  +  ( ( sqr `  D )  x.  d ) )  =  ( a  +  ( ( sqr `  D
)  x.  -u b
) ) )
5958eqeq2d 2307 . . . . . . . . . 10  |-  ( d  =  -u b  ->  (
( 1  /  A
)  =  ( a  +  ( ( sqr `  D )  x.  d
) )  <->  ( 1  /  A )  =  ( a  +  ( ( sqr `  D
)  x.  -u b
) ) ) )
60 oveq1 5881 . . . . . . . . . . . . 13  |-  ( d  =  -u b  ->  (
d ^ 2 )  =  ( -u b ^ 2 ) )
6160oveq2d 5890 . . . . . . . . . . . 12  |-  ( d  =  -u b  ->  ( D  x.  ( d ^ 2 ) )  =  ( D  x.  ( -u b ^ 2 ) ) )
6261oveq2d 5890 . . . . . . . . . . 11  |-  ( d  =  -u b  ->  (
( a ^ 2 )  -  ( D  x.  ( d ^
2 ) ) )  =  ( ( a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) ) )
6362eqeq1d 2304 . . . . . . . . . 10  |-  ( d  =  -u b  ->  (
( ( a ^
2 )  -  ( D  x.  ( d ^ 2 ) ) )  =  1  <->  (
( a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) )  =  1 ) )
6459, 63anbi12d 691 . . . . . . . . 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 ) ) )
6556, 64rspc2ev 2905 . . . . . . . 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 ) )
667, 9, 45, 50, 65syl112anc 1186 . . . . . . 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 ) )
676, 66jca 518 . . . . . 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 ) ) )
6867ex 423 . . . . 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 ) ) ) )
6968rexlimdvva 2687 . . . 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 ) ) ) )
7069adantld 453 . . 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 ) ) ) )
712, 70mpd 14 . 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 ) ) )
72 elpell1234qr 27039 . . 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 ) ) ) )
7372adantr 451 . 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 ) ) ) )
7471, 73mpbird 223 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 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557    \ cdif 3162   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    - cmin 9053   -ucneg 9054    / cdiv 9439   NNcn 9762   2c2 9811   ZZcz 10040   ^cexp 11120   sqrcsqr 11734  ◻NNcsquarenn 27024  Pell1234QRcpell1234qr 27026
This theorem is referenced by:  pell14qrreccl  27052
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-pell1234qr 27032
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