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Theorem pell1234qrdich 35752
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 35742 . . 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 782 . . . . . . . . 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 6322 . . . . . . . . . . . . . 14  |-  ( c  =  a  ->  (
c  +  ( ( sqr `  D )  x.  b ) )  =  ( a  +  ( ( sqr `  D
)  x.  b ) ) )
43eqeq2d 2472 . . . . . . . . . . . . 13  |-  ( c  =  a  ->  ( A  =  ( c  +  ( ( sqr `  D )  x.  b
) )  <->  A  =  ( a  +  ( ( sqr `  D
)  x.  b ) ) ) )
5 oveq1 6322 . . . . . . . . . . . . . . 15  |-  ( c  =  a  ->  (
c ^ 2 )  =  ( a ^
2 ) )
65oveq1d 6330 . . . . . . . . . . . . . 14  |-  ( c  =  a  ->  (
( c ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) ) )
76eqeq1d 2464 . . . . . . . . . . . . 13  |-  ( c  =  a  ->  (
( ( c ^
2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1  <->  (
( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 ) )
84, 7anbi12d 722 . . . . . . . . . . . 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 2913 . . . . . . . . . . 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 3162 . . . . . . . . . 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 725 . . . . . . . . 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 35740 . . . . . . . . . 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 743 . . . . . . . . 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 938 . . . . . . . 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 398 . . . . . . 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 613 . . . . . 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 784 . . . . . . . . . . . 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 10074 . . . . . . . . . . 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 774 . . . . . . . . . . . 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 11001 . . . . . . . . . . . . 13  |-  ( b  e.  ZZ  ->  -u b  e.  ZZ )
2120ad2antlr 738 . . . . . . . . . . . 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 769 . . . . . . . . . . . . . 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 9895 . . . . . . . . . . . . 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 10971 . . . . . . . . . . . . . . 15  |-  ( a  e.  ZZ  ->  a  e.  CC )
2524ad4antlr 744 . . . . . . . . . . . . . 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 3567 . . . . . . . . . . . . . . . . . 18  |-  ( D  e.  ( NN  \NN )  ->  D  e.  NN )
2726nncnd 10653 . . . . . . . . . . . . . . . . 17  |-  ( D  e.  ( NN  \NN )  ->  D  e.  CC )
2827ad5antr 745 . . . . . . . . . . . . . . . 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 13548 . . . . . . . . . . . . . . 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 10971 . . . . . . . . . . . . . . . 16  |-  ( b  e.  ZZ  ->  b  e.  CC )
3130ad2antlr 738 . . . . . . . . . . . . . . 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 9689 . . . . . . . . . . . . . 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 10025 . . . . . . . . . . . . 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 10084 . . . . . . . . . . . . . . . 16  |-  ( ( ( sqr `  D
)  e.  CC  /\  b  e.  CC )  ->  ( ( sqr `  D
)  x.  -u b
)  =  -u (
( sqr `  D
)  x.  b ) )
3534eqcomd 2468 . . . . . . . . . . . . . . 15  |-  ( ( ( sqr `  D
)  e.  CC  /\  b  e.  CC )  -> 
-u ( ( sqr `  D )  x.  b
)  =  ( ( sqr `  D )  x.  -u b ) )
3629, 31, 35syl2anc 671 . . . . . . . . . . . . . 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 6331 . . . . . . . . . . . . 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 2500 . . . . . . . . . . . 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 12367 . . . . . . . . . . . . . . 15  |-  ( a  e.  CC  ->  ( -u a ^ 2 )  =  ( a ^
2 ) )
4025, 39syl 17 . . . . . . . . . . . . . 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 12367 . . . . . . . . . . . . . . . 16  |-  ( b  e.  CC  ->  ( -u b ^ 2 )  =  ( b ^
2 ) )
4231, 41syl 17 . . . . . . . . . . . . . . 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 6331 . . . . . . . . . . . . . 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 6333 . . . . . . . . . . . . 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 771 . . . . . . . . . . . . 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 2496 . . . . . . . . . . . 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 6322 . . . . . . . . . . . . . . 15  |-  ( c  =  -u a  ->  (
c  +  ( ( sqr `  D )  x.  d ) )  =  ( -u a  +  ( ( sqr `  D )  x.  d
) ) )
4847eqeq2d 2472 . . . . . . . . . . . . . 14  |-  ( c  =  -u a  ->  ( -u A  =  ( c  +  ( ( sqr `  D )  x.  d
) )  <->  -u A  =  ( -u a  +  ( ( sqr `  D
)  x.  d ) ) ) )
49 oveq1 6322 . . . . . . . . . . . . . . . 16  |-  ( c  =  -u a  ->  (
c ^ 2 )  =  ( -u a ^ 2 ) )
5049oveq1d 6330 . . . . . . . . . . . . . . 15  |-  ( c  =  -u a  ->  (
( c ^ 2 )  -  ( D  x.  ( d ^
2 ) ) )  =  ( ( -u a ^ 2 )  -  ( D  x.  (
d ^ 2 ) ) ) )
5150eqeq1d 2464 . . . . . . . . . . . . . 14  |-  ( c  =  -u a  ->  (
( ( c ^
2 )  -  ( D  x.  ( d ^ 2 ) ) )  =  1  <->  (
( -u a ^ 2 )  -  ( D  x.  ( d ^
2 ) ) )  =  1 ) )
5248, 51anbi12d 722 . . . . . . . . . . . . 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 6323 . . . . . . . . . . . . . . . 16  |-  ( d  =  -u b  ->  (
( sqr `  D
)  x.  d )  =  ( ( sqr `  D )  x.  -u b
) )
5453oveq2d 6331 . . . . . . . . . . . . . . 15  |-  ( d  =  -u b  ->  ( -u a  +  ( ( sqr `  D )  x.  d ) )  =  ( -u a  +  ( ( sqr `  D )  x.  -u b
) ) )
5554eqeq2d 2472 . . . . . . . . . . . . . 14  |-  ( d  =  -u b  ->  ( -u A  =  ( -u a  +  ( ( sqr `  D )  x.  d ) )  <->  -u A  =  ( -u a  +  ( ( sqr `  D
)  x.  -u b
) ) ) )
56 oveq1 6322 . . . . . . . . . . . . . . . . 17  |-  ( d  =  -u b  ->  (
d ^ 2 )  =  ( -u b ^ 2 ) )
5756oveq2d 6331 . . . . . . . . . . . . . . . 16  |-  ( d  =  -u b  ->  ( D  x.  ( d ^ 2 ) )  =  ( D  x.  ( -u b ^ 2 ) ) )
5857oveq2d 6331 . . . . . . . . . . . . . . 15  |-  ( d  =  -u b  ->  (
( -u a ^ 2 )  -  ( D  x.  ( d ^
2 ) ) )  =  ( ( -u a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) ) )
5958eqeq1d 2464 . . . . . . . . . . . . . 14  |-  ( d  =  -u b  ->  (
( ( -u a ^ 2 )  -  ( D  x.  (
d ^ 2 ) ) )  =  1  <-> 
( ( -u a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) )  =  1 ) )
6055, 59anbi12d 722 . . . . . . . . . . . . 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 3173 . . . . . . . . . . . 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 1280 . . . . . . . . . . 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 35740 . . . . . . . . . . . 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 745 . . . . . . . . . . 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 938 . . . . . . . . . 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 399 . . . . . . . . 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 440 . . . . . . . 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 2891 . . . . . . 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 440 . . . . . 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 10981 . . . . . . . 8  |-  ( a  e.  ZZ  <->  ( a  e.  RR  /\  ( a  e.  NN0  \/  -u a  e.  NN0 ) ) )
7170simprbi 470 . . . . . . 7  |-  ( a  e.  ZZ  ->  (
a  e.  NN0  \/  -u a  e.  NN0 )
)
7271adantl 472 . . . . . 6  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  ->  (
a  e.  NN0  \/  -u a  e.  NN0 )
)
7316, 69, 72mpjaod 387 . . . . 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 2891 . . . 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 612 . . 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 223 . 2  |-  ( D  e.  ( NN  \NN )  -> 
( A  e.  (Pell1234QR `  D )  ->  ( A  e.  (Pell14QR `  D
)  \/  -u A  e.  (Pell14QR `  D )
) ) )
7776imp 435 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 189    \/ wo 374    /\ wa 375    = wceq 1455    e. wcel 1898   E.wrex 2750    \ cdif 3413   ` cfv 5601  (class class class)co 6315   CCcc 9563   RRcr 9564   1c1 9566    + caddc 9568    x. cmul 9570    - cmin 9886   -ucneg 9887   NNcn 10637   2c2 10687   NN0cn0 10898   ZZcz 10966   ^cexp 12304   sqrcsqrt 13345  ◻NNcsquarenn 35725  Pell1234QRcpell1234qr 35727  Pell14QRcpell14qr 35728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642  ax-pre-sup 9643
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-2nd 6821  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-er 7389  df-en 7596  df-dom 7597  df-sdom 7598  df-sup 7982  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-div 10298  df-nn 10638  df-2 10696  df-3 10697  df-n0 10899  df-z 10967  df-uz 11189  df-rp 11332  df-seq 12246  df-exp 12305  df-cj 13211  df-re 13212  df-im 13213  df-sqrt 13347  df-abs 13348  df-pell14qr 35733  df-pell1234qr 35734
This theorem is referenced by:  elpell14qr2  35753
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