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Theorem pell1234qrdich 30959
Description: A general Pell solution is either a positive solution, or its negation is. (Contributed by Stefan O'Rear, 18-Sep-2014.)
Assertion
Ref Expression
pell1234qrdich  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  -> 
( A  e.  (Pell14QR `  D )  \/  -u A  e.  (Pell14QR `  D )
) )

Proof of Theorem pell1234qrdich
Dummy variables  a 
b  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elpell1234qr 30949 . . 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 ) ) ) )
2 simp-4r 768 . . . . . . . . 9  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  a  e.  NN0 )  /\  E. b  e.  ZZ  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  ->  A  e.  RR )
3 oveq1 6303 . . . . . . . . . . . . . 14  |-  ( c  =  a  ->  (
c  +  ( ( sqr `  D )  x.  b ) )  =  ( a  +  ( ( sqr `  D
)  x.  b ) ) )
43eqeq2d 2471 . . . . . . . . . . . . 13  |-  ( c  =  a  ->  ( A  =  ( c  +  ( ( sqr `  D )  x.  b
) )  <->  A  =  ( a  +  ( ( sqr `  D
)  x.  b ) ) ) )
5 oveq1 6303 . . . . . . . . . . . . . . 15  |-  ( c  =  a  ->  (
c ^ 2 )  =  ( a ^
2 ) )
65oveq1d 6311 . . . . . . . . . . . . . 14  |-  ( c  =  a  ->  (
( c ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) ) )
76eqeq1d 2459 . . . . . . . . . . . . 13  |-  ( c  =  a  ->  (
( ( c ^
2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1  <->  (
( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 ) )
84, 7anbi12d 710 . . . . . . . . . . . 12  |-  ( c  =  a  ->  (
( A  =  ( c  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( c ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 )  <->  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) ) )
98rexbidv 2968 . . . . . . . . . . 11  |-  ( c  =  a  ->  ( E. b  e.  ZZ  ( A  =  (
c  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( c ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 )  <->  E. b  e.  ZZ  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) ) )
109rspcev 3210 . . . . . . . . . 10  |-  ( ( a  e.  NN0  /\  E. b  e.  ZZ  ( A  =  ( a  +  ( ( sqr `  D )  x.  b
) )  /\  (
( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 ) )  ->  E. c  e.  NN0  E. b  e.  ZZ  ( A  =  ( c  +  ( ( sqr `  D )  x.  b
) )  /\  (
( c ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 ) )
1110adantll 713 . . . . . . . . 9  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  a  e.  NN0 )  /\  E. b  e.  ZZ  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  ->  E. c  e.  NN0  E. b  e.  ZZ  ( A  =  ( c  +  ( ( sqr `  D )  x.  b
) )  /\  (
( c ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 ) )
12 elpell14qr 30947 . . . . . . . . . 10  |-  ( D  e.  ( NN  \NN )  -> 
( A  e.  (Pell14QR `  D )  <->  ( A  e.  RR  /\  E. c  e.  NN0  E. b  e.  ZZ  ( A  =  ( c  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( c ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) ) ) )
1312ad4antr 731 . . . . . . . . 9  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  a  e.  NN0 )  /\  E. b  e.  ZZ  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( A  e.  (Pell14QR `  D )  <->  ( A  e.  RR  /\  E. c  e.  NN0  E. b  e.  ZZ  ( A  =  ( c  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( c ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) ) ) )
142, 11, 13mpbir2and 922 . . . . . . . 8  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  a  e.  NN0 )  /\  E. b  e.  ZZ  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  ->  A  e.  (Pell14QR `  D
) )
1514orcd 392 . . . . . . 7  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  a  e.  NN0 )  /\  E. b  e.  ZZ  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( A  e.  (Pell14QR `  D )  \/  -u A  e.  (Pell14QR `  D )
) )
1615exp31 604 . . . . . 6  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  ->  (
a  e.  NN0  ->  ( E. b  e.  ZZ  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 )  ->  ( A  e.  (Pell14QR `  D )  \/  -u A  e.  (Pell14QR `  D ) ) ) ) )
17 simp-5r 770 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  -u a  e.  NN0 )  /\  b  e.  ZZ )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  ->  A  e.  RR )
1817renegcld 10007 . . . . . . . . . . 11  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  -u a  e.  NN0 )  /\  b  e.  ZZ )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  ->  -u A  e.  RR )
19 simpllr 760 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  -u a  e.  NN0 )  /\  b  e.  ZZ )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  ->  -u a  e.  NN0 )
20 znegcl 10920 . . . . . . . . . . . . 13  |-  ( b  e.  ZZ  ->  -u b  e.  ZZ )
2120ad2antlr 726 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  -u a  e.  NN0 )  /\  b  e.  ZZ )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  ->  -u b  e.  ZZ )
22 simprl 756 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  -u a  e.  NN0 )  /\  b  e.  ZZ )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  ->  A  =  ( a  +  ( ( sqr `  D )  x.  b
) ) )
2322negeqd 9833 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  -u a  e.  NN0 )  /\  b  e.  ZZ )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  ->  -u A  =  -u (
a  +  ( ( sqr `  D )  x.  b ) ) )
24 zcn 10890 . . . . . . . . . . . . . . 15  |-  ( a  e.  ZZ  ->  a  e.  CC )
2524ad4antlr 732 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  -u a  e.  NN0 )  /\  b  e.  ZZ )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
a  e.  CC )
26 eldifi 3622 . . . . . . . . . . . . . . . . . 18  |-  ( D  e.  ( NN  \NN )  ->  D  e.  NN )
2726nncnd 10572 . . . . . . . . . . . . . . . . 17  |-  ( D  e.  ( NN  \NN )  ->  D  e.  CC )
2827ad5antr 733 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  -u a  e.  NN0 )  /\  b  e.  ZZ )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  ->  D  e.  CC )
2928sqrtcld 13279 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  -u a  e.  NN0 )  /\  b  e.  ZZ )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( sqr `  D
)  e.  CC )
30 zcn 10890 . . . . . . . . . . . . . . . 16  |-  ( b  e.  ZZ  ->  b  e.  CC )
3130ad2antlr 726 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  -u a  e.  NN0 )  /\  b  e.  ZZ )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
b  e.  CC )
3229, 31mulcld 9633 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  -u a  e.  NN0 )  /\  b  e.  ZZ )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( ( sqr `  D
)  x.  b )  e.  CC )
3325, 32negdid 9963 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  -u a  e.  NN0 )  /\  b  e.  ZZ )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  ->  -u ( a  +  ( ( sqr `  D
)  x.  b ) )  =  ( -u a  +  -u ( ( sqr `  D )  x.  b ) ) )
34 mulneg2 10015 . . . . . . . . . . . . . . . 16  |-  ( ( ( sqr `  D
)  e.  CC  /\  b  e.  CC )  ->  ( ( sqr `  D
)  x.  -u b
)  =  -u (
( sqr `  D
)  x.  b ) )
3534eqcomd 2465 . . . . . . . . . . . . . . 15  |-  ( ( ( sqr `  D
)  e.  CC  /\  b  e.  CC )  -> 
-u ( ( sqr `  D )  x.  b
)  =  ( ( sqr `  D )  x.  -u b ) )
3629, 31, 35syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  -u a  e.  NN0 )  /\  b  e.  ZZ )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  ->  -u ( ( sqr `  D
)  x.  b )  =  ( ( sqr `  D )  x.  -u b
) )
3736oveq2d 6312 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  -u a  e.  NN0 )  /\  b  e.  ZZ )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( -u a  +  -u ( ( sqr `  D
)  x.  b ) )  =  ( -u a  +  ( ( sqr `  D )  x.  -u b ) ) )
3823, 33, 373eqtrd 2502 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  -u a  e.  NN0 )  /\  b  e.  ZZ )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  ->  -u A  =  ( -u a  +  ( ( sqr `  D )  x.  -u b ) ) )
39 sqneg 12230 . . . . . . . . . . . . . . 15  |-  ( a  e.  CC  ->  ( -u a ^ 2 )  =  ( a ^
2 ) )
4025, 39syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  -u a  e.  NN0 )  /\  b  e.  ZZ )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( -u a ^ 2 )  =  ( a ^ 2 ) )
41 sqneg 12230 . . . . . . . . . . . . . . . 16  |-  ( b  e.  CC  ->  ( -u b ^ 2 )  =  ( b ^
2 ) )
4231, 41syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  -u a  e.  NN0 )  /\  b  e.  ZZ )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( -u b ^ 2 )  =  ( b ^ 2 ) )
4342oveq2d 6312 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  -u a  e.  NN0 )  /\  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 ) ) )
4440, 43oveq12d 6314 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  -u a  e.  NN0 )  /\  b  e.  ZZ )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( ( -u a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) )  =  ( ( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) ) )
45 simprr 757 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  -u 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 )
4644, 45eqtrd 2498 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  -u a  e.  NN0 )  /\  b  e.  ZZ )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( ( -u a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) )  =  1 )
47 oveq1 6303 . . . . . . . . . . . . . . 15  |-  ( c  =  -u a  ->  (
c  +  ( ( sqr `  D )  x.  d ) )  =  ( -u a  +  ( ( sqr `  D )  x.  d
) ) )
4847eqeq2d 2471 . . . . . . . . . . . . . 14  |-  ( c  =  -u a  ->  ( -u A  =  ( c  +  ( ( sqr `  D )  x.  d
) )  <->  -u A  =  ( -u a  +  ( ( sqr `  D
)  x.  d ) ) ) )
49 oveq1 6303 . . . . . . . . . . . . . . . 16  |-  ( c  =  -u a  ->  (
c ^ 2 )  =  ( -u a ^ 2 ) )
5049oveq1d 6311 . . . . . . . . . . . . . . 15  |-  ( c  =  -u a  ->  (
( c ^ 2 )  -  ( D  x.  ( d ^
2 ) ) )  =  ( ( -u a ^ 2 )  -  ( D  x.  (
d ^ 2 ) ) ) )
5150eqeq1d 2459 . . . . . . . . . . . . . 14  |-  ( c  =  -u a  ->  (
( ( c ^
2 )  -  ( D  x.  ( d ^ 2 ) ) )  =  1  <->  (
( -u a ^ 2 )  -  ( D  x.  ( d ^
2 ) ) )  =  1 ) )
5248, 51anbi12d 710 . . . . . . . . . . . . 13  |-  ( c  =  -u a  ->  (
( -u A  =  ( c  +  ( ( sqr `  D )  x.  d ) )  /\  ( ( c ^ 2 )  -  ( D  x.  (
d ^ 2 ) ) )  =  1 )  <->  ( -u A  =  ( -u a  +  ( ( sqr `  D )  x.  d
) )  /\  (
( -u a ^ 2 )  -  ( D  x.  ( d ^
2 ) ) )  =  1 ) ) )
53 oveq2 6304 . . . . . . . . . . . . . . . 16  |-  ( d  =  -u b  ->  (
( sqr `  D
)  x.  d )  =  ( ( sqr `  D )  x.  -u b
) )
5453oveq2d 6312 . . . . . . . . . . . . . . 15  |-  ( d  =  -u b  ->  ( -u a  +  ( ( sqr `  D )  x.  d ) )  =  ( -u a  +  ( ( sqr `  D )  x.  -u b
) ) )
5554eqeq2d 2471 . . . . . . . . . . . . . 14  |-  ( d  =  -u b  ->  ( -u A  =  ( -u a  +  ( ( sqr `  D )  x.  d ) )  <->  -u A  =  ( -u a  +  ( ( sqr `  D
)  x.  -u b
) ) ) )
56 oveq1 6303 . . . . . . . . . . . . . . . . 17  |-  ( d  =  -u b  ->  (
d ^ 2 )  =  ( -u b ^ 2 ) )
5756oveq2d 6312 . . . . . . . . . . . . . . . 16  |-  ( d  =  -u b  ->  ( D  x.  ( d ^ 2 ) )  =  ( D  x.  ( -u b ^ 2 ) ) )
5857oveq2d 6312 . . . . . . . . . . . . . . 15  |-  ( d  =  -u b  ->  (
( -u a ^ 2 )  -  ( D  x.  ( d ^
2 ) ) )  =  ( ( -u a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) ) )
5958eqeq1d 2459 . . . . . . . . . . . . . 14  |-  ( d  =  -u b  ->  (
( ( -u a ^ 2 )  -  ( D  x.  (
d ^ 2 ) ) )  =  1  <-> 
( ( -u a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) )  =  1 ) )
6055, 59anbi12d 710 . . . . . . . . . . . . 13  |-  ( d  =  -u b  ->  (
( -u A  =  (
-u a  +  ( ( sqr `  D
)  x.  d ) )  /\  ( (
-u a ^ 2 )  -  ( D  x.  ( d ^
2 ) ) )  =  1 )  <->  ( -u A  =  ( -u a  +  ( ( sqr `  D )  x.  -u b
) )  /\  (
( -u a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) )  =  1 ) ) )
6152, 60rspc2ev 3221 . . . . . . . . . . . 12  |-  ( (
-u a  e.  NN0  /\  -u b  e.  ZZ  /\  ( -u A  =  ( -u a  +  ( ( sqr `  D
)  x.  -u b
) )  /\  (
( -u a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) )  =  1 ) )  ->  E. c  e.  NN0  E. d  e.  ZZ  ( -u A  =  ( c  +  ( ( sqr `  D
)  x.  d ) )  /\  ( ( c ^ 2 )  -  ( D  x.  ( d ^ 2 ) ) )  =  1 ) )
6219, 21, 38, 46, 61syl112anc 1232 . . . . . . . . . . 11  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  -u a  e.  NN0 )  /\  b  e.  ZZ )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  ->  E. c  e.  NN0  E. d  e.  ZZ  ( -u A  =  ( c  +  ( ( sqr `  D )  x.  d
) )  /\  (
( c ^ 2 )  -  ( D  x.  ( d ^
2 ) ) )  =  1 ) )
63 elpell14qr 30947 . . . . . . . . . . . 12  |-  ( D  e.  ( NN  \NN )  -> 
( -u A  e.  (Pell14QR `  D )  <->  ( -u A  e.  RR  /\  E. c  e.  NN0  E. d  e.  ZZ  ( -u A  =  ( c  +  ( ( sqr `  D
)  x.  d ) )  /\  ( ( c ^ 2 )  -  ( D  x.  ( d ^ 2 ) ) )  =  1 ) ) ) )
6463ad5antr 733 . . . . . . . . . . 11  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  -u a  e.  NN0 )  /\  b  e.  ZZ )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( -u A  e.  (Pell14QR `  D )  <->  ( -u A  e.  RR  /\  E. c  e.  NN0  E. d  e.  ZZ  ( -u A  =  ( c  +  ( ( sqr `  D
)  x.  d ) )  /\  ( ( c ^ 2 )  -  ( D  x.  ( d ^ 2 ) ) )  =  1 ) ) ) )
6518, 62, 64mpbir2and 922 . . . . . . . . . 10  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  -u a  e.  NN0 )  /\  b  e.  ZZ )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  ->  -u A  e.  (Pell14QR `  D
) )
6665olcd 393 . . . . . . . . 9  |-  ( ( ( ( ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  -u a  e.  NN0 )  /\  b  e.  ZZ )  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( A  e.  (Pell14QR `  D )  \/  -u A  e.  (Pell14QR `  D )
) )
6766ex 434 . . . . . . . 8  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  -u a  e.  NN0 )  /\  b  e.  ZZ )  ->  ( ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  ->  ( A  e.  (Pell14QR `  D
)  \/  -u A  e.  (Pell14QR `  D )
) ) )
6867rexlimdva 2949 . . . . . . 7  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  /\  -u a  e.  NN0 )  ->  ( E. b  e.  ZZ  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 )  ->  ( A  e.  (Pell14QR `  D )  \/  -u A  e.  (Pell14QR `  D ) ) ) )
6968ex 434 . . . . . 6  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  ->  ( -u a  e.  NN0  ->  ( E. b  e.  ZZ  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 )  ->  ( A  e.  (Pell14QR `  D )  \/  -u A  e.  (Pell14QR `  D ) ) ) ) )
70 elznn0 10900 . . . . . . . 8  |-  ( a  e.  ZZ  <->  ( a  e.  RR  /\  ( a  e.  NN0  \/  -u a  e.  NN0 ) ) )
7170simprbi 464 . . . . . . 7  |-  ( a  e.  ZZ  ->  (
a  e.  NN0  \/  -u a  e.  NN0 )
)
7271adantl 466 . . . . . 6  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  ->  (
a  e.  NN0  \/  -u a  e.  NN0 )
)
7316, 69, 72mpjaod 381 . . . . 5  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  ->  ( E. b  e.  ZZ  ( A  =  (
a  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 )  ->  ( A  e.  (Pell14QR `  D )  \/  -u A  e.  (Pell14QR `  D ) ) ) )
7473rexlimdva 2949 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  ->  ( E. a  e.  ZZ  E. b  e.  ZZ  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  ->  ( A  e.  (Pell14QR `  D
)  \/  -u A  e.  (Pell14QR `  D )
) ) )
7574expimpd 603 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
( ( A  e.  RR  /\  E. a  e.  ZZ  E. b  e.  ZZ  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( A  e.  (Pell14QR `  D )  \/  -u A  e.  (Pell14QR `  D )
) ) )
761, 75sylbid 215 . 2  |-  ( D  e.  ( NN  \NN )  -> 
( A  e.  (Pell1234QR `  D )  ->  ( A  e.  (Pell14QR `  D
)  \/  -u A  e.  (Pell14QR `  D )
) ) )
7776imp 429 1  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  -> 
( A  e.  (Pell14QR `  D )  \/  -u A  e.  (Pell14QR `  D )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819   E.wrex 2808    \ cdif 3468   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   1c1 9510    + caddc 9512    x. cmul 9514    - cmin 9824   -ucneg 9825   NNcn 10556   2c2 10606   NN0cn0 10816   ZZcz 10885   ^cexp 12168   sqrcsqrt 13077  ◻NNcsquarenn 30934  Pell1234QRcpell1234qr 30936  Pell14QRcpell14qr 30937
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-pell14qr 30941  df-pell1234qr 30942
This theorem is referenced by:  elpell14qr2  30960
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