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Theorem pellex 27023
Description: Every Pell equation has a nontrivial solution. Theorem 62 in [vandenDries] p. 43. (Contributed by Stefan O'Rear, 19-Oct-2014.)
Assertion
Ref Expression
pellex  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  E. x  e.  NN  E. y  e.  NN  (
( x ^ 2 )  -  ( D  x.  ( y ^
2 ) ) )  =  1 )
Distinct variable group:    x, D, y

Proof of Theorem pellex
Dummy variables  a 
b  c  d  e  f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pellexlem5 27021 . 2  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  E. a  e.  ZZ  ( a  =/=  0  /\  { <. b ,  c
>.  |  ( (
b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^
2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  a ) }  ~~  NN ) )
2 fzfi 11050 . . . . . . . . . 10  |-  ( 0 ... ( ( abs `  a )  -  1 ) )  e.  Fin
3 xpfi 7144 . . . . . . . . . 10  |-  ( ( ( 0 ... (
( abs `  a
)  -  1 ) )  e.  Fin  /\  ( 0 ... (
( abs `  a
)  -  1 ) )  e.  Fin )  ->  ( ( 0 ... ( ( abs `  a
)  -  1 ) )  X.  ( 0 ... ( ( abs `  a )  -  1 ) ) )  e. 
Fin )
42, 2, 3mp2an 653 . . . . . . . . 9  |-  ( ( 0 ... ( ( abs `  a )  -  1 ) )  X.  ( 0 ... ( ( abs `  a
)  -  1 ) ) )  e.  Fin
5 isfinite 7369 . . . . . . . . 9  |-  ( ( ( 0 ... (
( abs `  a
)  -  1 ) )  X.  ( 0 ... ( ( abs `  a )  -  1 ) ) )  e. 
Fin 
<->  ( ( 0 ... ( ( abs `  a
)  -  1 ) )  X.  ( 0 ... ( ( abs `  a )  -  1 ) ) )  ~<  om )
64, 5mpbi 199 . . . . . . . 8  |-  ( ( 0 ... ( ( abs `  a )  -  1 ) )  X.  ( 0 ... ( ( abs `  a
)  -  1 ) ) )  ~<  om
7 nnenom 11058 . . . . . . . . 9  |-  NN  ~~  om
87ensymi 6927 . . . . . . . 8  |-  om  ~~  NN
9 sdomentr 7011 . . . . . . . 8  |-  ( ( ( ( 0 ... ( ( abs `  a
)  -  1 ) )  X.  ( 0 ... ( ( abs `  a )  -  1 ) ) )  ~<  om  /\  om  ~~  NN )  ->  ( ( 0 ... ( ( abs `  a )  -  1 ) )  X.  (
0 ... ( ( abs `  a )  -  1 ) ) )  ~<  NN )
106, 8, 9mp2an 653 . . . . . . 7  |-  ( ( 0 ... ( ( abs `  a )  -  1 ) )  X.  ( 0 ... ( ( abs `  a
)  -  1 ) ) )  ~<  NN
11 ensym 6926 . . . . . . . 8  |-  ( {
<. b ,  c >.  |  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) } 
~~  NN  ->  NN  ~~  {
<. b ,  c >.  |  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) } )
1211ad2antll 709 . . . . . . 7  |-  ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  (
a  =/=  0  /\ 
{ <. b ,  c
>.  |  ( (
b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^
2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  a ) }  ~~  NN ) )  ->  NN  ~~  { <. b ,  c >.  |  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) } )
13 sdomentr 7011 . . . . . . 7  |-  ( ( ( ( 0 ... ( ( abs `  a
)  -  1 ) )  X.  ( 0 ... ( ( abs `  a )  -  1 ) ) )  ~<  NN  /\  NN  ~~  { <. b ,  c >.  |  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) } )  ->  ( (
0 ... ( ( abs `  a )  -  1 ) )  X.  (
0 ... ( ( abs `  a )  -  1 ) ) )  ~<  { <. b ,  c
>.  |  ( (
b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^
2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  a ) } )
1410, 12, 13sylancr 644 . . . . . 6  |-  ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  (
a  =/=  0  /\ 
{ <. b ,  c
>.  |  ( (
b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^
2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  a ) }  ~~  NN ) )  ->  ( (
0 ... ( ( abs `  a )  -  1 ) )  X.  (
0 ... ( ( abs `  a )  -  1 ) ) )  ~<  { <. b ,  c
>.  |  ( (
b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^
2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  a ) } )
15 opabssxp 4778 . . . . . . . . . 10  |-  { <. b ,  c >.  |  ( ( b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^ 2 )  -  ( D  x.  (
c ^ 2 ) ) )  =  a ) }  C_  ( NN  X.  NN )
1615sseli 3189 . . . . . . . . 9  |-  ( d  e.  { <. b ,  c >.  |  ( ( b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^ 2 )  -  ( D  x.  (
c ^ 2 ) ) )  =  a ) }  ->  d  e.  ( NN  X.  NN ) )
17 simprrl 740 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  e.  ( _V  X.  _V )  /\  ( ( 1st `  d )  e.  NN  /\  ( 2nd `  d
)  e.  NN ) ) )  ->  ( 1st `  d )  e.  NN )
1817nnzd 10132 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  e.  ( _V  X.  _V )  /\  ( ( 1st `  d )  e.  NN  /\  ( 2nd `  d
)  e.  NN ) ) )  ->  ( 1st `  d )  e.  ZZ )
19 simpllr 735 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  e.  ( _V  X.  _V )  /\  ( ( 1st `  d )  e.  NN  /\  ( 2nd `  d
)  e.  NN ) ) )  ->  a  e.  ZZ )
20 simplr 731 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  e.  ( _V  X.  _V )  /\  ( ( 1st `  d )  e.  NN  /\  ( 2nd `  d
)  e.  NN ) ) )  ->  a  =/=  0 )
21 nnabscl 11825 . . . . . . . . . . . . . 14  |-  ( ( a  e.  ZZ  /\  a  =/=  0 )  -> 
( abs `  a
)  e.  NN )
2219, 20, 21syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  e.  ( _V  X.  _V )  /\  ( ( 1st `  d )  e.  NN  /\  ( 2nd `  d
)  e.  NN ) ) )  ->  ( abs `  a )  e.  NN )
23 zmodfz 11007 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  d
)  e.  ZZ  /\  ( abs `  a )  e.  NN )  -> 
( ( 1st `  d
)  mod  ( abs `  a ) )  e.  ( 0 ... (
( abs `  a
)  -  1 ) ) )
2418, 22, 23syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  e.  ( _V  X.  _V )  /\  ( ( 1st `  d )  e.  NN  /\  ( 2nd `  d
)  e.  NN ) ) )  ->  (
( 1st `  d
)  mod  ( abs `  a ) )  e.  ( 0 ... (
( abs `  a
)  -  1 ) ) )
25 simprrr 741 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  e.  ( _V  X.  _V )  /\  ( ( 1st `  d )  e.  NN  /\  ( 2nd `  d
)  e.  NN ) ) )  ->  ( 2nd `  d )  e.  NN )
2625nnzd 10132 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  e.  ( _V  X.  _V )  /\  ( ( 1st `  d )  e.  NN  /\  ( 2nd `  d
)  e.  NN ) ) )  ->  ( 2nd `  d )  e.  ZZ )
27 zmodfz 11007 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  d
)  e.  ZZ  /\  ( abs `  a )  e.  NN )  -> 
( ( 2nd `  d
)  mod  ( abs `  a ) )  e.  ( 0 ... (
( abs `  a
)  -  1 ) ) )
2826, 22, 27syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  e.  ( _V  X.  _V )  /\  ( ( 1st `  d )  e.  NN  /\  ( 2nd `  d
)  e.  NN ) ) )  ->  (
( 2nd `  d
)  mod  ( abs `  a ) )  e.  ( 0 ... (
( abs `  a
)  -  1 ) ) )
2924, 28jca 518 . . . . . . . . . . 11  |-  ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  e.  ( _V  X.  _V )  /\  ( ( 1st `  d )  e.  NN  /\  ( 2nd `  d
)  e.  NN ) ) )  ->  (
( ( 1st `  d
)  mod  ( abs `  a ) )  e.  ( 0 ... (
( abs `  a
)  -  1 ) )  /\  ( ( 2nd `  d )  mod  ( abs `  a
) )  e.  ( 0 ... ( ( abs `  a )  -  1 ) ) ) )
3029ex 423 . . . . . . . . . 10  |-  ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0 )  ->  (
( d  e.  ( _V  X.  _V )  /\  ( ( 1st `  d
)  e.  NN  /\  ( 2nd `  d )  e.  NN ) )  ->  ( ( ( 1st `  d )  mod  ( abs `  a
) )  e.  ( 0 ... ( ( abs `  a )  -  1 ) )  /\  ( ( 2nd `  d )  mod  ( abs `  a ) )  e.  ( 0 ... ( ( abs `  a
)  -  1 ) ) ) ) )
31 elxp7 6168 . . . . . . . . . 10  |-  ( d  e.  ( NN  X.  NN )  <->  ( d  e.  ( _V  X.  _V )  /\  ( ( 1st `  d )  e.  NN  /\  ( 2nd `  d
)  e.  NN ) ) )
32 opelxp 4735 . . . . . . . . . 10  |-  ( <.
( ( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  e.  ( ( 0 ... ( ( abs `  a
)  -  1 ) )  X.  ( 0 ... ( ( abs `  a )  -  1 ) ) )  <->  ( (
( 1st `  d
)  mod  ( abs `  a ) )  e.  ( 0 ... (
( abs `  a
)  -  1 ) )  /\  ( ( 2nd `  d )  mod  ( abs `  a
) )  e.  ( 0 ... ( ( abs `  a )  -  1 ) ) ) )
3330, 31, 323imtr4g 261 . . . . . . . . 9  |-  ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0 )  ->  (
d  e.  ( NN 
X.  NN )  ->  <. ( ( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  e.  ( ( 0 ... ( ( abs `  a
)  -  1 ) )  X.  ( 0 ... ( ( abs `  a )  -  1 ) ) ) ) )
3416, 33syl5 28 . . . . . . . 8  |-  ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0 )  ->  (
d  e.  { <. b ,  c >.  |  ( ( b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^ 2 )  -  ( D  x.  (
c ^ 2 ) ) )  =  a ) }  ->  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  e.  ( ( 0 ... ( ( abs `  a
)  -  1 ) )  X.  ( 0 ... ( ( abs `  a )  -  1 ) ) ) ) )
3534imp 418 . . . . . . 7  |-  ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  d  e.  {
<. b ,  c >.  |  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) } )  ->  <. ( ( 1st `  d )  mod  ( abs `  a
) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  e.  ( ( 0 ... ( ( abs `  a
)  -  1 ) )  X.  ( 0 ... ( ( abs `  a )  -  1 ) ) ) )
3635adantlrr 701 . . . . . 6  |-  ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  ( a  =/=  0  /\  { <. b ,  c
>.  |  ( (
b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^
2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  a ) }  ~~  NN ) )  /\  d  e. 
{ <. b ,  c
>.  |  ( (
b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^
2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  a ) } )  ->  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  e.  ( ( 0 ... ( ( abs `  a
)  -  1 ) )  X.  ( 0 ... ( ( abs `  a )  -  1 ) ) ) )
37 fveq2 5541 . . . . . . . 8  |-  ( d  =  e  ->  ( 1st `  d )  =  ( 1st `  e
) )
3837oveq1d 5889 . . . . . . 7  |-  ( d  =  e  ->  (
( 1st `  d
)  mod  ( abs `  a ) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) ) )
39 fveq2 5541 . . . . . . . 8  |-  ( d  =  e  ->  ( 2nd `  d )  =  ( 2nd `  e
) )
4039oveq1d 5889 . . . . . . 7  |-  ( d  =  e  ->  (
( 2nd `  d
)  mod  ( abs `  a ) )  =  ( ( 2nd `  e
)  mod  ( abs `  a ) ) )
4138, 40opeq12d 3820 . . . . . 6  |-  ( d  =  e  ->  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. )
4214, 36, 41fphpd 27002 . . . . 5  |-  ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  (
a  =/=  0  /\ 
{ <. b ,  c
>.  |  ( (
b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^
2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  a ) }  ~~  NN ) )  ->  E. d  e.  { <. b ,  c
>.  |  ( (
b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^
2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  a ) } E. e  e. 
{ <. b ,  c
>.  |  ( (
b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^
2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  a ) }  ( d  =/=  e  /\  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. ) )
43 eleq1 2356 . . . . . . . . . . . . . 14  |-  ( b  =  f  ->  (
b  e.  NN  <->  f  e.  NN ) )
44 eleq1 2356 . . . . . . . . . . . . . 14  |-  ( c  =  g  ->  (
c  e.  NN  <->  g  e.  NN ) )
4543, 44bi2anan9 843 . . . . . . . . . . . . 13  |-  ( ( b  =  f  /\  c  =  g )  ->  ( ( b  e.  NN  /\  c  e.  NN )  <->  ( f  e.  NN  /\  g  e.  NN ) ) )
46 oveq1 5881 . . . . . . . . . . . . . . 15  |-  ( b  =  f  ->  (
b ^ 2 )  =  ( f ^
2 ) )
47 oveq1 5881 . . . . . . . . . . . . . . . 16  |-  ( c  =  g  ->  (
c ^ 2 )  =  ( g ^
2 ) )
4847oveq2d 5890 . . . . . . . . . . . . . . 15  |-  ( c  =  g  ->  ( D  x.  ( c ^ 2 ) )  =  ( D  x.  ( g ^ 2 ) ) )
4946, 48oveqan12d 5893 . . . . . . . . . . . . . 14  |-  ( ( b  =  f  /\  c  =  g )  ->  ( ( b ^
2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  ( ( f ^ 2 )  -  ( D  x.  ( g ^ 2 ) ) ) )
5049eqeq1d 2304 . . . . . . . . . . . . 13  |-  ( ( b  =  f  /\  c  =  g )  ->  ( ( ( b ^ 2 )  -  ( D  x.  (
c ^ 2 ) ) )  =  a  <-> 
( ( f ^
2 )  -  ( D  x.  ( g ^ 2 ) ) )  =  a ) )
5145, 50anbi12d 691 . . . . . . . . . . . 12  |-  ( ( b  =  f  /\  c  =  g )  ->  ( ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a )  <->  ( (
f  e.  NN  /\  g  e.  NN )  /\  ( ( f ^
2 )  -  ( D  x.  ( g ^ 2 ) ) )  =  a ) ) )
5251cbvopabv 4104 . . . . . . . . . . 11  |-  { <. b ,  c >.  |  ( ( b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^ 2 )  -  ( D  x.  (
c ^ 2 ) ) )  =  a ) }  =  { <. f ,  g >.  |  ( ( f  e.  NN  /\  g  e.  NN )  /\  (
( f ^ 2 )  -  ( D  x.  ( g ^
2 ) ) )  =  a ) }
5352eleq2i 2360 . . . . . . . . . 10  |-  ( e  e.  { <. b ,  c >.  |  ( ( b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^ 2 )  -  ( D  x.  (
c ^ 2 ) ) )  =  a ) }  <->  e  e.  {
<. f ,  g >.  |  ( ( f  e.  NN  /\  g  e.  NN )  /\  (
( f ^ 2 )  -  ( D  x.  ( g ^
2 ) ) )  =  a ) } )
5453biimpi 186 . . . . . . . . 9  |-  ( e  e.  { <. b ,  c >.  |  ( ( b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^ 2 )  -  ( D  x.  (
c ^ 2 ) ) )  =  a ) }  ->  e  e.  { <. f ,  g
>.  |  ( (
f  e.  NN  /\  g  e.  NN )  /\  ( ( f ^
2 )  -  ( D  x.  ( g ^ 2 ) ) )  =  a ) } )
55 elopab 4288 . . . . . . . . . . 11  |-  ( d  e.  { <. b ,  c >.  |  ( ( b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^ 2 )  -  ( D  x.  (
c ^ 2 ) ) )  =  a ) }  <->  E. b E. c ( d  = 
<. b ,  c >.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) ) )
56 elopab 4288 . . . . . . . . . . . . . 14  |-  ( e  e.  { <. f ,  g >.  |  ( ( f  e.  NN  /\  g  e.  NN )  /\  ( ( f ^ 2 )  -  ( D  x.  (
g ^ 2 ) ) )  =  a ) }  <->  E. f E. g ( e  = 
<. f ,  g >.  /\  ( ( f  e.  NN  /\  g  e.  NN )  /\  (
( f ^ 2 )  -  ( D  x.  ( g ^
2 ) ) )  =  a ) ) )
57 simp3ll 1026 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  d  =  <. b ,  c >.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) )  ->  b  e.  NN )
58573expb 1152 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  =  <. b ,  c
>.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) ) )  ->  b  e.  NN )
59583ad2ant1 976 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  =  <. b ,  c
>.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) ) )  /\  ( e  =  <. f ,  g
>.  /\  ( ( f  e.  NN  /\  g  e.  NN )  /\  (
( f ^ 2 )  -  ( D  x.  ( g ^
2 ) ) )  =  a ) )  /\  ( d  =/=  e  /\  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. ) )  ->  b  e.  NN )
60 simp3lr 1027 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  d  =  <. b ,  c >.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) )  ->  c  e.  NN )
61603expb 1152 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  =  <. b ,  c
>.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) ) )  ->  c  e.  NN )
62613ad2ant1 976 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  =  <. b ,  c
>.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) ) )  /\  ( e  =  <. f ,  g
>.  /\  ( ( f  e.  NN  /\  g  e.  NN )  /\  (
( f ^ 2 )  -  ( D  x.  ( g ^
2 ) ) )  =  a ) )  /\  ( d  =/=  e  /\  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. ) )  ->  c  e.  NN )
63 simp1lr 1019 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( e  =  <. f ,  g
>.  /\  ( ( f  e.  NN  /\  g  e.  NN )  /\  (
( f ^ 2 )  -  ( D  x.  ( g ^
2 ) ) )  =  a ) )  /\  ( d  =/=  e  /\  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. ) )  ->  a  e.  ZZ )
64633adant1r 1175 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  =  <. b ,  c
>.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) ) )  /\  ( e  =  <. f ,  g
>.  /\  ( ( f  e.  NN  /\  g  e.  NN )  /\  (
( f ^ 2 )  -  ( D  x.  ( g ^
2 ) ) )  =  a ) )  /\  ( d  =/=  e  /\  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. ) )  ->  a  e.  ZZ )
65 simpl 443 . . . . . . . . . . . . . . . . . . 19  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  D  e.  NN )
6665ad3antrrr 710 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  =  <. b ,  c
>.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) ) )  ->  D  e.  NN )
67663ad2ant1 976 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  =  <. b ,  c
>.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) ) )  /\  ( e  =  <. f ,  g
>.  /\  ( ( f  e.  NN  /\  g  e.  NN )  /\  (
( f ^ 2 )  -  ( D  x.  ( g ^
2 ) ) )  =  a ) )  /\  ( d  =/=  e  /\  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. ) )  ->  D  e.  NN )
68 simpr 447 . . . . . . . . . . . . . . . . . . 19  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  -.  ( sqr `  D )  e.  QQ )
6968ad3antrrr 710 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  =  <. b ,  c
>.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) ) )  ->  -.  ( sqr `  D )  e.  QQ )
70693ad2ant1 976 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  =  <. b ,  c
>.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) ) )  /\  ( e  =  <. f ,  g
>.  /\  ( ( f  e.  NN  /\  g  e.  NN )  /\  (
( f ^ 2 )  -  ( D  x.  ( g ^
2 ) ) )  =  a ) )  /\  ( d  =/=  e  /\  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. ) )  ->  -.  ( sqr `  D )  e.  QQ )
71 simp2ll 1022 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  =  <. b ,  c
>.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) ) )  /\  ( ( f  e.  NN  /\  g  e.  NN )  /\  ( ( f ^
2 )  -  ( D  x.  ( g ^ 2 ) ) )  =  a )  /\  ( d  =/=  e  /\  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. ) )  ->  f  e.  NN )
72713adant2l 1176 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  =  <. b ,  c
>.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) ) )  /\  ( e  =  <. f ,  g
>.  /\  ( ( f  e.  NN  /\  g  e.  NN )  /\  (
( f ^ 2 )  -  ( D  x.  ( g ^
2 ) ) )  =  a ) )  /\  ( d  =/=  e  /\  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. ) )  ->  f  e.  NN )
73 simp2lr 1023 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  =  <. b ,  c
>.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) ) )  /\  ( ( f  e.  NN  /\  g  e.  NN )  /\  ( ( f ^
2 )  -  ( D  x.  ( g ^ 2 ) ) )  =  a )  /\  ( d  =/=  e  /\  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. ) )  ->  g  e.  NN )
74733adant2l 1176 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  =  <. b ,  c
>.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) ) )  /\  ( e  =  <. f ,  g
>.  /\  ( ( f  e.  NN  /\  g  e.  NN )  /\  (
( f ^ 2 )  -  ( D  x.  ( g ^
2 ) ) )  =  a ) )  /\  ( d  =/=  e  /\  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. ) )  ->  g  e.  NN )
75 simp2l 981 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  =  <. b ,  c
>.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) ) )  /\  ( e  =  <. f ,  g
>.  /\  ( ( f  e.  NN  /\  g  e.  NN )  /\  (
( f ^ 2 )  -  ( D  x.  ( g ^
2 ) ) )  =  a ) )  /\  ( d  =/=  e  /\  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. ) )  ->  e  =  <. f ,  g
>. )
76 simp1rl 1020 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  =  <. b ,  c
>.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) ) )  /\  ( e  =  <. f ,  g
>.  /\  ( ( f  e.  NN  /\  g  e.  NN )  /\  (
( f ^ 2 )  -  ( D  x.  ( g ^
2 ) ) )  =  a ) )  /\  ( d  =/=  e  /\  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. ) )  ->  d  =  <. b ,  c
>. )
77 simp3l 983 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  =  <. b ,  c
>.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) ) )  /\  ( e  =  <. f ,  g
>.  /\  ( ( f  e.  NN  /\  g  e.  NN )  /\  (
( f ^ 2 )  -  ( D  x.  ( g ^
2 ) ) )  =  a ) )  /\  ( d  =/=  e  /\  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. ) )  ->  d  =/=  e )
78 simp3 957 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( e  =  <. f ,  g >.  /\  d  =  <. b ,  c
>.  /\  d  =/=  e
)  ->  d  =/=  e )
79 simp2 956 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( e  =  <. f ,  g >.  /\  d  =  <. b ,  c
>.  /\  d  =/=  e
)  ->  d  =  <. b ,  c >.
)
80 simp1 955 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( e  =  <. f ,  g >.  /\  d  =  <. b ,  c
>.  /\  d  =/=  e
)  ->  e  =  <. f ,  g >.
)
8178, 79, 803netr3d 2485 . . . . . . . . . . . . . . . . . . 19  |-  ( ( e  =  <. f ,  g >.  /\  d  =  <. b ,  c
>.  /\  d  =/=  e
)  ->  <. b ,  c >.  =/=  <. f ,  g >. )
82 vex 2804 . . . . . . . . . . . . . . . . . . . . 21  |-  b  e. 
_V
83 vex 2804 . . . . . . . . . . . . . . . . . . . . 21  |-  c  e. 
_V
8482, 83opth 4261 . . . . . . . . . . . . . . . . . . . 20  |-  ( <.
b ,  c >.  =  <. f ,  g
>. 
<->  ( b  =  f  /\  c  =  g ) )
8584necon3abii 2489 . . . . . . . . . . . . . . . . . . 19  |-  ( <.
b ,  c >.  =/=  <. f ,  g
>. 
<->  -.  ( b  =  f  /\  c  =  g ) )
8681, 85sylib 188 . . . . . . . . . . . . . . . . . 18  |-  ( ( e  =  <. f ,  g >.  /\  d  =  <. b ,  c
>.  /\  d  =/=  e
)  ->  -.  (
b  =  f  /\  c  =  g )
)
8775, 76, 77, 86syl3anc 1182 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  =  <. b ,  c
>.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) ) )  /\  ( e  =  <. f ,  g
>.  /\  ( ( f  e.  NN  /\  g  e.  NN )  /\  (
( f ^ 2 )  -  ( D  x.  ( g ^
2 ) ) )  =  a ) )  /\  ( d  =/=  e  /\  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. ) )  ->  -.  ( b  =  f  /\  c  =  g ) )
88 simp1lr 1019 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  =  <. b ,  c
>.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) ) )  /\  ( e  =  <. f ,  g
>.  /\  ( ( f  e.  NN  /\  g  e.  NN )  /\  (
( f ^ 2 )  -  ( D  x.  ( g ^
2 ) ) )  =  a ) )  /\  ( d  =/=  e  /\  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. ) )  ->  a  =/=  0 )
89 simp1rr 1021 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( d  =  <. b ,  c >.  /\  (
( b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^ 2 )  -  ( D  x.  (
c ^ 2 ) ) )  =  a ) )  /\  (
e  =  <. f ,  g >.  /\  (
( f  e.  NN  /\  g  e.  NN )  /\  ( ( f ^ 2 )  -  ( D  x.  (
g ^ 2 ) ) )  =  a ) )  /\  (
d  =/=  e  /\  <.
( ( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. ) )  ->  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a )
90893adant1l 1174 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  =  <. b ,  c
>.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) ) )  /\  ( e  =  <. f ,  g
>.  /\  ( ( f  e.  NN  /\  g  e.  NN )  /\  (
( f ^ 2 )  -  ( D  x.  ( g ^
2 ) ) )  =  a ) )  /\  ( d  =/=  e  /\  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. ) )  ->  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a )
91 simp2rr 1025 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  =  <. b ,  c
>.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) ) )  /\  ( e  =  <. f ,  g
>.  /\  ( ( f  e.  NN  /\  g  e.  NN )  /\  (
( f ^ 2 )  -  ( D  x.  ( g ^
2 ) ) )  =  a ) )  /\  ( d  =/=  e  /\  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. ) )  ->  (
( f ^ 2 )  -  ( D  x.  ( g ^
2 ) ) )  =  a )
92 simp3r 984 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  =  <. b ,  c
>.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) ) )  /\  ( e  =  <. f ,  g
>.  /\  ( ( f  e.  NN  /\  g  e.  NN )  /\  (
( f ^ 2 )  -  ( D  x.  ( g ^
2 ) ) )  =  a ) )  /\  ( d  =/=  e  /\  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. ) )  ->  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. )
93 simp3 957 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( d  =  <. b ,  c >.  /\  e  =  <. f ,  g
>.  /\  <. ( ( 1st `  d )  mod  ( abs `  a ) ) ,  ( ( 2nd `  d )  mod  ( abs `  a ) )
>.  =  <. ( ( 1st `  e )  mod  ( abs `  a
) ) ,  ( ( 2nd `  e
)  mod  ( abs `  a ) ) >.
)  ->  <. ( ( 1st `  d )  mod  ( abs `  a
) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. )
94 ovex 5899 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( 1st `  d )  mod  ( abs `  a
) )  e.  _V
95 ovex 5899 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( 2nd `  d )  mod  ( abs `  a
) )  e.  _V
9694, 95opth 4261 . . . . . . . . . . . . . . . . . . . . 21  |-  ( <.
( ( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. 
<->  ( ( ( 1st `  d )  mod  ( abs `  a ) )  =  ( ( 1st `  e )  mod  ( abs `  a ) )  /\  ( ( 2nd `  d )  mod  ( abs `  a ) )  =  ( ( 2nd `  e )  mod  ( abs `  a ) ) ) )
9793, 96sylib 188 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( d  =  <. b ,  c >.  /\  e  =  <. f ,  g
>.  /\  <. ( ( 1st `  d )  mod  ( abs `  a ) ) ,  ( ( 2nd `  d )  mod  ( abs `  a ) )
>.  =  <. ( ( 1st `  e )  mod  ( abs `  a
) ) ,  ( ( 2nd `  e
)  mod  ( abs `  a ) ) >.
)  ->  ( (
( 1st `  d
)  mod  ( abs `  a ) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) )  /\  ( ( 2nd `  d
)  mod  ( abs `  a ) )  =  ( ( 2nd `  e
)  mod  ( abs `  a ) ) ) )
98 simprl 732 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( d  =  <. b ,  c >.  /\  e  =  <. f ,  g
>. )  /\  (
( ( 1st `  d
)  mod  ( abs `  a ) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) )  /\  ( ( 2nd `  d
)  mod  ( abs `  a ) )  =  ( ( 2nd `  e
)  mod  ( abs `  a ) ) ) )  ->  ( ( 1st `  d )  mod  ( abs `  a
) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) ) )
99 simpll 730 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( d  =  <. b ,  c >.  /\  e  =  <. f ,  g
>. )  /\  (
( ( 1st `  d
)  mod  ( abs `  a ) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) )  /\  ( ( 2nd `  d
)  mod  ( abs `  a ) )  =  ( ( 2nd `  e
)  mod  ( abs `  a ) ) ) )  ->  d  =  <. b ,  c >.
)
10099fveq2d 5545 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( d  =  <. b ,  c >.  /\  e  =  <. f ,  g
>. )  /\  (
( ( 1st `  d
)  mod  ( abs `  a ) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) )  /\  ( ( 2nd `  d
)  mod  ( abs `  a ) )  =  ( ( 2nd `  e
)  mod  ( abs `  a ) ) ) )  ->  ( 1st `  d )  =  ( 1st `  <. b ,  c >. )
)
10182, 83op1st 6144 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 1st `  <. b ,  c
>. )  =  b
102100, 101syl6eq 2344 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( d  =  <. b ,  c >.  /\  e  =  <. f ,  g
>. )  /\  (
( ( 1st `  d
)  mod  ( abs `  a ) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) )  /\  ( ( 2nd `  d
)  mod  ( abs `  a ) )  =  ( ( 2nd `  e
)  mod  ( abs `  a ) ) ) )  ->  ( 1st `  d )  =  b )
103102oveq1d 5889 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( d  =  <. b ,  c >.  /\  e  =  <. f ,  g
>. )  /\  (
( ( 1st `  d
)  mod  ( abs `  a ) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) )  /\  ( ( 2nd `  d
)  mod  ( abs `  a ) )  =  ( ( 2nd `  e
)  mod  ( abs `  a ) ) ) )  ->  ( ( 1st `  d )  mod  ( abs `  a
) )  =  ( b  mod  ( abs `  a ) ) )
104 simplr 731 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( d  =  <. b ,  c >.  /\  e  =  <. f ,  g
>. )  /\  (
( ( 1st `  d
)  mod  ( abs `  a ) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) )  /\  ( ( 2nd `  d
)  mod  ( abs `  a ) )  =  ( ( 2nd `  e
)  mod  ( abs `  a ) ) ) )  ->  e  =  <. f ,  g >.
)
105104fveq2d 5545 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( d  =  <. b ,  c >.  /\  e  =  <. f ,  g
>. )  /\  (
( ( 1st `  d
)  mod  ( abs `  a ) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) )  /\  ( ( 2nd `  d
)  mod  ( abs `  a ) )  =  ( ( 2nd `  e
)  mod  ( abs `  a ) ) ) )  ->  ( 1st `  e )  =  ( 1st `  <. f ,  g >. )
)
106 vex 2804 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  f  e. 
_V
107 vex 2804 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  g  e. 
_V
108106, 107op1st 6144 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 1st `  <. f ,  g
>. )  =  f
109105, 108syl6eq 2344 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( d  =  <. b ,  c >.  /\  e  =  <. f ,  g
>. )  /\  (
( ( 1st `  d
)  mod  ( abs `  a ) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) )  /\  ( ( 2nd `  d
)  mod  ( abs `  a ) )  =  ( ( 2nd `  e
)  mod  ( abs `  a ) ) ) )  ->  ( 1st `  e )  =  f )
110109oveq1d 5889 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( d  =  <. b ,  c >.  /\  e  =  <. f ,  g
>. )  /\  (
( ( 1st `  d
)  mod  ( abs `  a ) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) )  /\  ( ( 2nd `  d
)  mod  ( abs `  a ) )  =  ( ( 2nd `  e
)  mod  ( abs `  a ) ) ) )  ->  ( ( 1st `  e )  mod  ( abs `  a
) )  =  ( f  mod  ( abs `  a ) ) )
11198, 103, 1103eqtr3d 2336 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( d  =  <. b ,  c >.  /\  e  =  <. f ,  g
>. )  /\  (
( ( 1st `  d
)  mod  ( abs `  a ) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) )  /\  ( ( 2nd `  d
)  mod  ( abs `  a ) )  =  ( ( 2nd `  e
)  mod  ( abs `  a ) ) ) )  ->  ( b  mod  ( abs `  a
) )  =  ( f  mod  ( abs `  a ) ) )
112 simprr 733 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( d  =  <. b ,  c >.  /\  e  =  <. f ,  g
>. )  /\  (
( ( 1st `  d
)  mod  ( abs `  a ) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) )  /\  ( ( 2nd `  d
)  mod  ( abs `  a ) )  =  ( ( 2nd `  e
)  mod  ( abs `  a ) ) ) )  ->  ( ( 2nd `  d )  mod  ( abs `  a
) )  =  ( ( 2nd `  e
)  mod  ( abs `  a ) ) )
11399fveq2d 5545 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( d  =  <. b ,  c >.  /\  e  =  <. f ,  g
>. )  /\  (
( ( 1st `  d
)  mod  ( abs `  a ) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) )  /\  ( ( 2nd `  d
)  mod  ( abs `  a ) )  =  ( ( 2nd `  e
)  mod  ( abs `  a ) ) ) )  ->  ( 2nd `  d )  =  ( 2nd `  <. b ,  c >. )
)
11482, 83op2nd 6145 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 2nd `  <. b ,  c
>. )  =  c
115113, 114syl6eq 2344 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( d  =  <. b ,  c >.  /\  e  =  <. f ,  g
>. )  /\  (
( ( 1st `  d
)  mod  ( abs `  a ) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) )  /\  ( ( 2nd `  d
)  mod  ( abs `  a ) )  =  ( ( 2nd `  e
)  mod  ( abs `  a ) ) ) )  ->  ( 2nd `  d )  =  c )
116115oveq1d 5889 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( d  =  <. b ,  c >.  /\  e  =  <. f ,  g
>. )  /\  (
( ( 1st `  d
)  mod  ( abs `  a ) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) )  /\  ( ( 2nd `  d
)  mod  ( abs `  a ) )  =  ( ( 2nd `  e
)  mod  ( abs `  a ) ) ) )  ->  ( ( 2nd `  d )  mod  ( abs `  a
) )  =  ( c  mod  ( abs `  a ) ) )
117104fveq2d 5545 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( d  =  <. b ,  c >.  /\  e  =  <. f ,  g
>. )  /\  (
( ( 1st `  d
)  mod  ( abs `  a ) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) )  /\  ( ( 2nd `  d
)  mod  ( abs `  a ) )  =  ( ( 2nd `  e
)  mod  ( abs `  a ) ) ) )  ->  ( 2nd `  e )  =  ( 2nd `  <. f ,  g >. )
)
118106, 107op2nd 6145 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 2nd `  <. f ,  g
>. )  =  g
119117, 118syl6eq 2344 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( d  =  <. b ,  c >.  /\  e  =  <. f ,  g
>. )  /\  (
( ( 1st `  d
)  mod  ( abs `  a ) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) )  /\  ( ( 2nd `  d
)  mod  ( abs `  a ) )  =  ( ( 2nd `  e
)  mod  ( abs `  a ) ) ) )  ->  ( 2nd `  e )  =  g )
120119oveq1d 5889 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( d  =  <. b ,  c >.  /\  e  =  <. f ,  g
>. )  /\  (
( ( 1st `  d
)  mod  ( abs `  a ) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) )  /\  ( ( 2nd `  d
)  mod  ( abs `  a ) )  =  ( ( 2nd `  e
)  mod  ( abs `  a ) ) ) )  ->  ( ( 2nd `  e )  mod  ( abs `  a
) )  =  ( g  mod  ( abs `  a ) ) )
121112, 116, 1203eqtr3d 2336 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( d  =  <. b ,  c >.  /\  e  =  <. f ,  g
>. )  /\  (
( ( 1st `  d
)  mod  ( abs `  a ) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) )  /\  ( ( 2nd `  d
)  mod  ( abs `  a ) )  =  ( ( 2nd `  e
)  mod  ( abs `  a ) ) ) )  ->  ( c  mod  ( abs `  a
) )  =  ( g  mod  ( abs `  a ) ) )
122111, 121jca 518 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( d  =  <. b ,  c >.  /\  e  =  <. f ,  g
>. )  /\  (
( ( 1st `  d
)  mod  ( abs `  a ) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) )  /\  ( ( 2nd `  d
)  mod  ( abs `  a ) )  =  ( ( 2nd `  e
)  mod  ( abs `  a ) ) ) )  ->  ( (
b  mod  ( abs `  a ) )  =  ( f  mod  ( abs `  a ) )  /\  ( c  mod  ( abs `  a
) )  =  ( g  mod  ( abs `  a ) ) ) )
123122ex 423 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( d  =  <. b ,  c >.  /\  e  =  <. f ,  g
>. )  ->  ( ( ( ( 1st `  d
)  mod  ( abs `  a ) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) )  /\  ( ( 2nd `  d
)  mod  ( abs `  a ) )  =  ( ( 2nd `  e
)  mod  ( abs `  a ) ) )  ->  ( ( b  mod  ( abs `  a
) )  =  ( f  mod  ( abs `  a ) )  /\  ( c  mod  ( abs `  a ) )  =  ( g  mod  ( abs `  a
) ) ) ) )
1241233adant3 975 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( d  =  <. b ,  c >.  /\  e  =  <. f ,  g
>.  /\  <. ( ( 1st `  d )  mod  ( abs `  a ) ) ,  ( ( 2nd `  d )  mod  ( abs `  a ) )
>.  =  <. ( ( 1st `  e )  mod  ( abs `  a
) ) ,  ( ( 2nd `  e
)  mod  ( abs `  a ) ) >.
)  ->  ( (
( ( 1st `  d
)  mod  ( abs `  a ) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) )  /\  ( ( 2nd `  d
)  mod  ( abs `  a ) )  =  ( ( 2nd `  e
)  mod  ( abs `  a ) ) )  ->  ( ( b  mod  ( abs `  a
) )  =  ( f  mod  ( abs `  a ) )  /\  ( c  mod  ( abs `  a ) )  =  ( g  mod  ( abs `  a
) ) ) ) )
12597, 124mpd 14 . . . . . . . . . . . . . . . . . . 19  |-  ( ( d  =  <. b ,  c >.  /\  e  =  <. f ,  g
>.  /\  <. ( ( 1st `  d )  mod  ( abs `  a ) ) ,  ( ( 2nd `  d )  mod  ( abs `  a ) )
>.  =  <. ( ( 1st `  e )  mod  ( abs `  a
) ) ,  ( ( 2nd `  e
)  mod  ( abs `  a ) ) >.
)  ->  ( (
b  mod  ( abs `  a ) )  =  ( f  mod  ( abs `  a ) )  /\  ( c  mod  ( abs `  a
) )  =  ( g  mod  ( abs `  a ) ) ) )
12676, 75, 92, 125syl3anc 1182 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  =  <. b ,  c
>.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) ) )  /\  ( e  =  <. f ,  g
>.  /\  ( ( f  e.  NN  /\  g  e.  NN )  /\  (
( f ^ 2 )  -  ( D  x.  ( g ^
2 ) ) )  =  a ) )  /\  ( d  =/=  e  /\  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. ) )  ->  (
( b  mod  ( abs `  a ) )  =  ( f  mod  ( abs `  a
) )  /\  (
c  mod  ( abs `  a ) )  =  ( g  mod  ( abs `  a ) ) ) )
127126simpld 445 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  =  <. b ,  c
>.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) ) )  /\  ( e  =  <. f ,  g
>.  /\  ( ( f  e.  NN  /\  g  e.  NN )  /\  (
( f ^ 2 )  -  ( D  x.  ( g ^
2 ) ) )  =  a ) )  /\  ( d  =/=  e  /\  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. ) )  ->  (
b  mod  ( abs `  a ) )  =  ( f  mod  ( abs `  a ) ) )
128126simprd 449 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  =  <. b ,  c
>.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) ) )  /\  ( e  =  <. f ,  g
>.  /\  ( ( f  e.  NN  /\  g  e.  NN )  /\  (
( f ^ 2 )  -  ( D  x.  ( g ^
2 ) ) )  =  a ) )  /\  ( d  =/=  e  /\  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. ) )  ->  (
c  mod  ( abs `  a ) )  =  ( g  mod  ( abs `  a ) ) )
12959, 62, 64, 67, 70, 72, 74, 87, 88, 90, 91, 127, 128pellexlem6 27022 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  =  <. b ,  c
>.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) ) )  /\  ( e  =  <. f ,  g
>.  /\  ( ( f  e.  NN  /\  g  e.  NN )  /\  (
( f ^ 2 )  -  ( D  x.  ( g ^
2 ) ) )  =  a ) )  /\  ( d  =/=  e  /\  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. ) )  ->  E. x  e.  NN  E. y  e.  NN  ( ( x ^ 2 )  -  ( D  x.  (
y ^ 2 ) ) )  =  1 )
1301293exp 1150 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  =  <. b ,  c
>.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) ) )  ->  ( (
e  =  <. f ,  g >.  /\  (
( f  e.  NN  /\  g  e.  NN )  /\  ( ( f ^ 2 )  -  ( D  x.  (
g ^ 2 ) ) )  =  a ) )  ->  (
( d  =/=  e  /\  <. ( ( 1st `  d )  mod  ( abs `  a ) ) ,  ( ( 2nd `  d )  mod  ( abs `  a ) )
>.  =  <. ( ( 1st `  e )  mod  ( abs `  a
) ) ,  ( ( 2nd `  e
)  mod  ( abs `  a ) ) >.
)  ->  E. x  e.  NN  E. y  e.  NN  ( ( x ^ 2 )  -  ( D  x.  (
y ^ 2 ) ) )  =  1 ) ) )
131130exlimdvv 1627 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  =  <. b ,  c
>.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) ) )  ->  ( E. f E. g ( e  =  <. f ,  g
>.  /\  ( ( f  e.  NN  /\  g  e.  NN )  /\  (
( f ^ 2 )  -  ( D  x.  ( g ^
2 ) ) )  =  a ) )  ->  ( ( d  =/=  e  /\  <. ( ( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. )  ->  E. x  e.  NN  E. y  e.  NN  ( ( x ^ 2 )  -  ( D  x.  (
y ^ 2 ) ) )  =  1 ) ) )
13256, 131syl5bi 208 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  ( d  =  <. b ,  c
>.  /\  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) ) )  ->  ( e  e.  { <. f ,  g
>.  |  ( (
f  e.  NN  /\  g  e.  NN )  /\  ( ( f ^
2 )  -  ( D  x.  ( g ^ 2 ) ) )  =  a ) }  ->  ( (
d  =/=  e  /\  <.
( ( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. )  ->  E. x  e.  NN  E. y  e.  NN  ( ( x ^ 2 )  -  ( D  x.  (
y ^ 2 ) ) )  =  1 ) ) )
133132ex 423 . . . . . . . . . . . 12  |-  ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0 )  ->  (
( d  =  <. b ,  c >.  /\  (
( b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^ 2 )  -  ( D  x.  (
c ^ 2 ) ) )  =  a ) )  ->  (
e  e.  { <. f ,  g >.  |  ( ( f  e.  NN  /\  g  e.  NN )  /\  ( ( f ^ 2 )  -  ( D  x.  (
g ^ 2 ) ) )  =  a ) }  ->  (
( d  =/=  e  /\  <. ( ( 1st `  d )  mod  ( abs `  a ) ) ,  ( ( 2nd `  d )  mod  ( abs `  a ) )
>.  =  <. ( ( 1st `  e )  mod  ( abs `  a
) ) ,  ( ( 2nd `  e
)  mod  ( abs `  a ) ) >.
)  ->  E. x  e.  NN  E. y  e.  NN  ( ( x ^ 2 )  -  ( D  x.  (
y ^ 2 ) ) )  =  1 ) ) ) )
134133exlimdvv 1627 . . . . . . . . . . 11  |-  ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0 )  ->  ( E. b E. c ( d  =  <. b ,  c >.  /\  (
( b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^ 2 )  -  ( D  x.  (
c ^ 2 ) ) )  =  a ) )  ->  (
e  e.  { <. f ,  g >.  |  ( ( f  e.  NN  /\  g  e.  NN )  /\  ( ( f ^ 2 )  -  ( D  x.  (
g ^ 2 ) ) )  =  a ) }  ->  (
( d  =/=  e  /\  <. ( ( 1st `  d )  mod  ( abs `  a ) ) ,  ( ( 2nd `  d )  mod  ( abs `  a ) )
>.  =  <. ( ( 1st `  e )  mod  ( abs `  a
) ) ,  ( ( 2nd `  e
)  mod  ( abs `  a ) ) >.
)  ->  E. x  e.  NN  E. y  e.  NN  ( ( x ^ 2 )  -  ( D  x.  (
y ^ 2 ) ) )  =  1 ) ) ) )
13555, 134syl5bi 208 . . . . . . . . . 10  |-  ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0 )  ->  (
d  e.  { <. b ,  c >.  |  ( ( b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^ 2 )  -  ( D  x.  (
c ^ 2 ) ) )  =  a ) }  ->  (
e  e.  { <. f ,  g >.  |  ( ( f  e.  NN  /\  g  e.  NN )  /\  ( ( f ^ 2 )  -  ( D  x.  (
g ^ 2 ) ) )  =  a ) }  ->  (
( d  =/=  e  /\  <. ( ( 1st `  d )  mod  ( abs `  a ) ) ,  ( ( 2nd `  d )  mod  ( abs `  a ) )
>.  =  <. ( ( 1st `  e )  mod  ( abs `  a
) ) ,  ( ( 2nd `  e
)  mod  ( abs `  a ) ) >.
)  ->  E. x  e.  NN  E. y  e.  NN  ( ( x ^ 2 )  -  ( D  x.  (
y ^ 2 ) ) )  =  1 ) ) ) )
136135imp3a 420 . . . . . . . . 9  |-  ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0 )  ->  (
( d  e.  { <. b ,  c >.  |  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) }  /\  e  e.  { <. f ,  g >.  |  ( ( f  e.  NN  /\  g  e.  NN )  /\  (
( f ^ 2 )  -  ( D  x.  ( g ^
2 ) ) )  =  a ) } )  ->  ( (
d  =/=  e  /\  <.
( ( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. )  ->  E. x  e.  NN  E. y  e.  NN  ( ( x ^ 2 )  -  ( D  x.  (
y ^ 2 ) ) )  =  1 ) ) )
13754, 136sylan2i 636 . . . . . . . 8  |-  ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0 )  ->  (
( d  e.  { <. b ,  c >.  |  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) }  /\  e  e.  { <. b ,  c >.  |  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) } )  ->  ( (
d  =/=  e  /\  <.
( ( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. )  ->  E. x  e.  NN  E. y  e.  NN  ( ( x ^ 2 )  -  ( D  x.  (
y ^ 2 ) ) )  =  1 ) ) )
138137rexlimdvv 2686 . . . . . . 7  |-  ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0 )  ->  ( E. d  e.  { <. b ,  c >.  |  ( ( b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^ 2 )  -  ( D  x.  (
c ^ 2 ) ) )  =  a ) } E. e  e.  { <. b ,  c
>.  |  ( (
b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^
2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  a ) }  ( d  =/=  e  /\  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. )  ->  E. x  e.  NN  E. y  e.  NN  ( ( x ^ 2 )  -  ( D  x.  (
y ^ 2 ) ) )  =  1 ) )
139138imp 418 . . . . . 6  |-  ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  a  =/=  0
)  /\  E. d  e.  { <. b ,  c
>.  |  ( (
b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^
2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  a ) } E. e  e. 
{ <. b ,  c
>.  |  ( (
b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^
2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  a ) }  ( d  =/=  e  /\  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. ) )  ->  E. x  e.  NN  E. y  e.  NN  ( ( x ^ 2 )  -  ( D  x.  (
y ^ 2 ) ) )  =  1 )
140139adantlrr 701 . . . . 5  |-  ( ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  ( a  =/=  0  /\  { <. b ,  c
>.  |  ( (
b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^
2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  a ) }  ~~  NN ) )  /\  E. d  e.  { <. b ,  c
>.  |  ( (
b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^
2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  a ) } E. e  e. 
{ <. b ,  c
>.  |  ( (
b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^
2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  a ) }  ( d  =/=  e  /\  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. ) )  ->  E. x  e.  NN  E. y  e.  NN  ( ( x ^ 2 )  -  ( D  x.  (
y ^ 2 ) ) )  =  1 )
14142, 140mpdan 649 . . . 4  |-  ( ( ( ( D  e.  NN  /\  -.  ( sqr `  D )  e.  QQ )  /\  a  e.  ZZ )  /\  (
a  =/=  0  /\ 
{ <. b ,  c
>.  |  ( (
b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^
2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  a ) }  ~~  NN ) )  ->  E. x  e.  NN  E. y  e.  NN  ( ( x ^ 2 )  -  ( D  x.  (
y ^ 2 ) ) )  =  1 )
142141ex 423 . . 3  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  a  e.  ZZ )  ->  ( ( a  =/=  0  /\  { <. b ,  c >.  |  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) } 
~~  NN )  ->  E. x  e.  NN  E. y  e.  NN  (
( x ^ 2 )  -  ( D  x.  ( y ^
2 ) ) )  =  1 ) )
143142rexlimdva 2680 . 2  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  ( E. a  e.  ZZ  ( a  =/=  0  /\  { <. b ,  c >.  |  ( ( b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^ 2 )  -  ( D  x.  (
c ^ 2 ) ) )  =  a ) }  ~~  NN )  ->  E. x  e.  NN  E. y  e.  NN  (
( x ^ 2 )  -  ( D  x.  ( y ^
2 ) ) )  =  1 ) )
1441, 143mpd 14 1  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  E. x  e.  NN  E. y  e.  NN  (
( x ^ 2 )  -  ( D  x.  ( y ^
2 ) ) )  =  1 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   _Vcvv 2801   <.cop 3656   class class class wbr 4039   {copab 4092   omcom 4672    X. cxp 4703   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137    ~~ cen 6876    ~< csdm 6878   Fincfn 6879   0cc0 8753   1c1 8754    x. cmul 8758    - cmin 9053   NNcn 9762   2c2 9811   ZZcz 10040   QQcq 10332   ...cfz 10798    mod cmo 10989   ^cexp 11120   sqrcsqr 11734   abscabs 11735
This theorem is referenced by:  pellqrex  27067
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  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-int 3879  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-se 4369  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-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-acn 7591  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-q 10333  df-rp 10371  df-ico 10678  df-fz 10799  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-numer 12822  df-denom 12823
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