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Theorem pell1234qrdich 29199
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 29189 . . 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 766 . . . . . . . . 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 6096 . . . . . . . . . . . . . 14  |-  ( c  =  a  ->  (
c  +  ( ( sqr `  D )  x.  b ) )  =  ( a  +  ( ( sqr `  D
)  x.  b ) ) )
43eqeq2d 2452 . . . . . . . . . . . . 13  |-  ( c  =  a  ->  ( A  =  ( c  +  ( ( sqr `  D )  x.  b
) )  <->  A  =  ( a  +  ( ( sqr `  D
)  x.  b ) ) ) )
5 oveq1 6096 . . . . . . . . . . . . . . 15  |-  ( c  =  a  ->  (
c ^ 2 )  =  ( a ^
2 ) )
65oveq1d 6104 . . . . . . . . . . . . . 14  |-  ( c  =  a  ->  (
( c ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  ( ( a ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) ) )
76eqeq1d 2449 . . . . . . . . . . . . 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 2734 . . . . . . . . . . 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 3071 . . . . . . . . . 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 29187 . . . . . . . . . 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 913 . . . . . . . 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 768 . . . . . . . . . . . 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 9773 . . . . . . . . . . 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 758 . . . . . . . . . . . 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 10678 . . . . . . . . . . . . 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 755 . . . . . . . . . . . . . 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 9602 . . . . . . . . . . . . 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 10649 . . . . . . . . . . . . . . . 16  |-  ( a  e.  ZZ  ->  a  e.  CC )
2524adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  ->  a  e.  CC )
2625ad3antrrr 729 . . . . . . . . . . . . . 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 )
27 eldifi 3476 . . . . . . . . . . . . . . . . . 18  |-  ( D  e.  ( NN  \NN )  ->  D  e.  NN )
2827nncnd 10336 . . . . . . . . . . . . . . . . 17  |-  ( D  e.  ( NN  \NN )  ->  D  e.  CC )
2928ad5antr 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 )
3029sqrcld 12921 . . . . . . . . . . . . . . 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 )
31 zcn 10649 . . . . . . . . . . . . . . . 16  |-  ( b  e.  ZZ  ->  b  e.  CC )
3231ad2antlr 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 )
3330, 32mulcld 9404 . . . . . . . . . . . . . 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 )
3426, 33negdid 9730 . . . . . . . . . . . . 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 ) ) )
35 mulneg2 9780 . . . . . . . . . . . . . . . 16  |-  ( ( ( sqr `  D
)  e.  CC  /\  b  e.  CC )  ->  ( ( sqr `  D
)  x.  -u b
)  =  -u (
( sqr `  D
)  x.  b ) )
3635eqcomd 2446 . . . . . . . . . . . . . . 15  |-  ( ( ( sqr `  D
)  e.  CC  /\  b  e.  CC )  -> 
-u ( ( sqr `  D )  x.  b
)  =  ( ( sqr `  D )  x.  -u b ) )
3730, 32, 36syl2anc 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
) )
3837oveq2d 6105 . . . . . . . . . . . . 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 ) ) )
3923, 34, 383eqtrd 2477 . . . . . . . . . . . 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 ) ) )
40 sqneg 11924 . . . . . . . . . . . . . . 15  |-  ( a  e.  CC  ->  ( -u a ^ 2 )  =  ( a ^
2 ) )
4126, 40syl 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 ) )
42 sqneg 11924 . . . . . . . . . . . . . . . 16  |-  ( b  e.  CC  ->  ( -u b ^ 2 )  =  ( b ^
2 ) )
4332, 42syl 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 ) )
4443oveq2d 6105 . . . . . . . . . . . . . 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 ) ) )
4541, 44oveq12d 6107 . . . . . . . . . . . . 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 ) ) ) )
46 simprr 756 . . . . . . . . . . . . 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 )
4745, 46eqtrd 2473 . . . . . . . . . . . 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 )
48 oveq1 6096 . . . . . . . . . . . . . . 15  |-  ( c  =  -u a  ->  (
c  +  ( ( sqr `  D )  x.  d ) )  =  ( -u a  +  ( ( sqr `  D )  x.  d
) ) )
4948eqeq2d 2452 . . . . . . . . . . . . . 14  |-  ( c  =  -u a  ->  ( -u A  =  ( c  +  ( ( sqr `  D )  x.  d
) )  <->  -u A  =  ( -u a  +  ( ( sqr `  D
)  x.  d ) ) ) )
50 oveq1 6096 . . . . . . . . . . . . . . . 16  |-  ( c  =  -u a  ->  (
c ^ 2 )  =  ( -u a ^ 2 ) )
5150oveq1d 6104 . . . . . . . . . . . . . . 15  |-  ( c  =  -u a  ->  (
( c ^ 2 )  -  ( D  x.  ( d ^
2 ) ) )  =  ( ( -u a ^ 2 )  -  ( D  x.  (
d ^ 2 ) ) ) )
5251eqeq1d 2449 . . . . . . . . . . . . . 14  |-  ( c  =  -u a  ->  (
( ( c ^
2 )  -  ( D  x.  ( d ^ 2 ) ) )  =  1  <->  (
( -u a ^ 2 )  -  ( D  x.  ( d ^
2 ) ) )  =  1 ) )
5349, 52anbi12d 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 ) ) )
54 oveq2 6097 . . . . . . . . . . . . . . . 16  |-  ( d  =  -u b  ->  (
( sqr `  D
)  x.  d )  =  ( ( sqr `  D )  x.  -u b
) )
5554oveq2d 6105 . . . . . . . . . . . . . . 15  |-  ( d  =  -u b  ->  ( -u a  +  ( ( sqr `  D )  x.  d ) )  =  ( -u a  +  ( ( sqr `  D )  x.  -u b
) ) )
5655eqeq2d 2452 . . . . . . . . . . . . . 14  |-  ( d  =  -u b  ->  ( -u A  =  ( -u a  +  ( ( sqr `  D )  x.  d ) )  <->  -u A  =  ( -u a  +  ( ( sqr `  D
)  x.  -u b
) ) ) )
57 oveq1 6096 . . . . . . . . . . . . . . . . 17  |-  ( d  =  -u b  ->  (
d ^ 2 )  =  ( -u b ^ 2 ) )
5857oveq2d 6105 . . . . . . . . . . . . . . . 16  |-  ( d  =  -u b  ->  ( D  x.  ( d ^ 2 ) )  =  ( D  x.  ( -u b ^ 2 ) ) )
5958oveq2d 6105 . . . . . . . . . . . . . . 15  |-  ( d  =  -u b  ->  (
( -u a ^ 2 )  -  ( D  x.  ( d ^
2 ) ) )  =  ( ( -u a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) ) )
6059eqeq1d 2449 . . . . . . . . . . . . . 14  |-  ( d  =  -u b  ->  (
( ( -u a ^ 2 )  -  ( D  x.  (
d ^ 2 ) ) )  =  1  <-> 
( ( -u a ^ 2 )  -  ( D  x.  ( -u b ^ 2 ) ) )  =  1 ) )
6156, 60anbi12d 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 ) ) )
6253, 61rspc2ev 3079 . . . . . . . . . . . 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 ) )
6319, 21, 39, 47, 62syl112anc 1222 . . . . . . . . . . 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 ) )
64 elpell14qr 29187 . . . . . . . . . . . 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 ) ) ) )
6564ad5antr 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 ) ) ) )
6618, 63, 65mpbir2and 913 . . . . . . . . . 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
) )
6766olcd 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 )
) )
6867ex 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 )
) ) )
6968rexlimdva 2839 . . . . . . 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 ) ) ) )
7069ex 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 ) ) ) ) )
71 elznn0 10659 . . . . . . . 8  |-  ( a  e.  ZZ  <->  ( a  e.  RR  /\  ( a  e.  NN0  \/  -u a  e.  NN0 ) ) )
7271simprbi 464 . . . . . . 7  |-  ( a  e.  ZZ  ->  (
a  e.  NN0  \/  -u a  e.  NN0 )
)
7372adantl 466 . . . . . 6  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  a  e.  ZZ )  ->  (
a  e.  NN0  \/  -u a  e.  NN0 )
)
7416, 70, 73mpjaod 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 ) ) ) )
7574rexlimdva 2839 . . . 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 )
) ) )
7675expimpd 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 )
) ) )
771, 76sylbid 215 . 2  |-  ( D  e.  ( NN  \NN )  -> 
( A  e.  (Pell1234QR `  D )  ->  ( A  e.  (Pell14QR `  D
)  \/  -u A  e.  (Pell14QR `  D )
) ) )
7877imp 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 1369    e. wcel 1756   E.wrex 2714    \ cdif 3323   ` cfv 5416  (class class class)co 6089   CCcc 9278   RRcr 9279   1c1 9281    + caddc 9283    x. cmul 9285    - cmin 9593   -ucneg 9594   NNcn 10320   2c2 10369   NN0cn0 10577   ZZcz 10644   ^cexp 11863   sqrcsqr 12720  ◻NNcsquarenn 29174  Pell1234QRcpell1234qr 29176  Pell14QRcpell14qr 29177
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 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-pre-sup 9358
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-2nd 6576  df-recs 6830  df-rdg 6864  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-sup 7689  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-2 10378  df-3 10379  df-n0 10578  df-z 10645  df-uz 10860  df-rp 10990  df-seq 11805  df-exp 11864  df-cj 12586  df-re 12587  df-im 12588  df-sqr 12722  df-abs 12723  df-pell14qr 29181  df-pell1234qr 29182
This theorem is referenced by:  elpell14qr2  29200
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