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Theorem pellex 26319
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
Dummy variables  a 
b  c  d  e  f  g are mutually distinct and distinct from all other variables.

Proof of Theorem pellex
StepHypRef Expression
1 pellexlem5 26317 . 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 11028 . . . . . . . . . 10  |-  ( 0 ... ( ( abs `  a )  -  1 ) )  e.  Fin
3 xpfi 7123 . . . . . . . . . 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 655 . . . . . . . . 9  |-  ( ( 0 ... ( ( abs `  a )  -  1 ) )  X.  ( 0 ... ( ( abs `  a
)  -  1 ) ) )  e.  Fin
5 isfinite 7348 . . . . . . . . 9  |-  ( ( ( 0 ... (
( abs `  a
)  -  1 ) )  X.  ( 0 ... ( ( abs `  a )  -  1 ) ) )  e. 
Fin 
<->  ( ( 0 ... ( ( abs `  a
)  -  1 ) )  X.  ( 0 ... ( ( abs `  a )  -  1 ) ) )  ~<  om )
64, 5mpbi 201 . . . . . . . 8  |-  ( ( 0 ... ( ( abs `  a )  -  1 ) )  X.  ( 0 ... ( ( abs `  a
)  -  1 ) ) )  ~<  om
7 nnenom 11036 . . . . . . . . 9  |-  NN  ~~  om
87ensymi 6906 . . . . . . . 8  |-  om  ~~  NN
9 sdomentr 6990 . . . . . . . 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 655 . . . . . . 7  |-  ( ( 0 ... ( ( abs `  a )  -  1 ) )  X.  ( 0 ... ( ( abs `  a
)  -  1 ) ) )  ~<  NN
11 ensym 6905 . . . . . . . 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 711 . . . . . . 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 6990 . . . . . . 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 646 . . . . . 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 4761 . . . . . . . . . 10  |-  { <. b ,  c >.  |  ( ( b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^ 2 )  -  ( D  x.  (
c ^ 2 ) ) )  =  a ) }  C_  ( NN  X.  NN )
1615sseli 3177 . . . . . . . . 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 742 . . . . . . . . . . . . . 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 10111 . . . . . . . . . . . . 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 737 . . . . . . . . . . . . . 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 733 . . . . . . . . . . . . . 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 11803 . . . . . . . . . . . . . 14  |-  ( ( a  e.  ZZ  /\  a  =/=  0 )  -> 
( abs `  a
)  e.  NN )
2219, 20, 21syl2anc 644 . . . . . . . . . . . . 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 10985 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  d
)  e.  ZZ  /\  ( abs `  a )  e.  NN )  -> 
( ( 1st `  d
)  mod  ( abs `  a ) )  e.  ( 0 ... (
( abs `  a
)  -  1 ) ) )
2418, 22, 23syl2anc 644 . . . . . . . . . . . 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 743 . . . . . . . . . . . . . 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 10111 . . . . . . . . . . . . 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 10985 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  d
)  e.  ZZ  /\  ( abs `  a )  e.  NN )  -> 
( ( 2nd `  d
)  mod  ( abs `  a ) )  e.  ( 0 ... (
( abs `  a
)  -  1 ) ) )
2826, 22, 27syl2anc 644 . . . . . . . . . . . 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 520 . . . . . . . . . . 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 425 . . . . . . . . . 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 6113 . . . . . . . . . 10  |-  ( d  e.  ( NN  X.  NN )  <->  ( d  e.  ( _V  X.  _V )  /\  ( ( 1st `  d )  e.  NN  /\  ( 2nd `  d
)  e.  NN ) ) )
32 opelxp 4718 . . . . . . . . . 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 263 . . . . . . . . 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 30 . . . . . . . 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 420 . . . . . . 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 703 . . . . . 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 5485 . . . . . . . 8  |-  ( d  =  e  ->  ( 1st `  d )  =  ( 1st `  e
) )
3837oveq1d 5834 . . . . . . 7  |-  ( d  =  e  ->  (
( 1st `  d
)  mod  ( abs `  a ) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) ) )
39 fveq2 5485 . . . . . . . 8  |-  ( d  =  e  ->  ( 2nd `  d )  =  ( 2nd `  e
) )
4039oveq1d 5834 . . . . . . 7  |-  ( d  =  e  ->  (
( 2nd `  d
)  mod  ( abs `  a ) )  =  ( ( 2nd `  e
)  mod  ( abs `  a ) ) )
4138, 40opeq12d 3805 . . . . . 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 26298 . . . . 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 2344 . . . . . . . . . . . . . 14  |-  ( b  =  f  ->  (
b  e.  NN  <->  f  e.  NN ) )
44 eleq1 2344 . . . . . . . . . . . . . 14  |-  ( c  =  g  ->  (
c  e.  NN  <->  g  e.  NN ) )
4543, 44bi2anan9 845 . . . . . . . . . . . . 13  |-  ( ( b  =  f  /\  c  =  g )  ->  ( ( b  e.  NN  /\  c  e.  NN )  <->  ( f  e.  NN  /\  g  e.  NN ) ) )
46 oveq1 5826 . . . . . . . . . . . . . . 15  |-  ( b  =  f  ->  (
b ^ 2 )  =  ( f ^
2 ) )
47 oveq1 5826 . . . . . . . . . . . . . . . 16  |-  ( c  =  g  ->  (
c ^ 2 )  =  ( g ^
2 ) )
4847oveq2d 5835 . . . . . . . . . . . . . . 15  |-  ( c  =  g  ->  ( D  x.  ( c ^ 2 ) )  =  ( D  x.  ( g ^ 2 ) ) )
4946, 48oveqan12d 5838 . . . . . . . . . . . . . 14  |-  ( ( b  =  f  /\  c  =  g )  ->  ( ( b ^
2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  ( ( f ^ 2 )  -  ( D  x.  ( g ^ 2 ) ) ) )
5049eqeq1d 2292 . . . . . . . . . . . . 13  |-  ( ( b  =  f  /\  c  =  g )  ->  ( ( ( b ^ 2 )  -  ( D  x.  (
c ^ 2 ) ) )  =  a  <-> 
( ( f ^
2 )  -  ( D  x.  ( g ^ 2 ) ) )  =  a ) )
5145, 50anbi12d 693 . . . . . . . . . . . 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 4089 . . . . . . . . . . 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 2348 . . . . . . . . . 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 188 . . . . . . . . 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 4271 . . . . . . . . . . 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 4271 . . . . . . . . . . . . . 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 1028 . . . . . . . . . . . . . . . . . . 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 1154 . . . . . . . . . . . . . . . . . 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 978 . . . . . . . . . . . . . . . . 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 1029 . . . . . . . . . . . . . . . . . . 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 1154 . . . . . . . . . . . . . . . . . 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 978 . . . . . . . . . . . . . . . . 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 1021 . . . . . . . . . . . . . . . . . 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 1177 . . . . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . . . . . . . 19  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  D  e.  NN )
6665ad3antrrr 712 . . . . . . . . . . . . . . . . . 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 978 . . . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . . . . . . 19  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  -.  ( sqr `  D )  e.  QQ )
6968ad3antrrr 712 . . . . . . . . . . . . . . . . . 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 978 . . . . . . . . . . . . . . . . 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 1024 . . . . . . . . . . . . . . . . . 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 1178 . . . . . . . . . . . . . . . . 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 1025 . . . . . . . . . . . . . . . . . 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 1178 . . . . . . . . . . . . . . . . 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 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 ) )
>. ) )  ->  e  =  <. f ,  g
>. )
76 simp1rl 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 ) ) )  /\  ( 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 985 . . . . . . . . . . . . . . . . . 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 959 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( e  =  <. f ,  g >.  /\  d  =  <. b ,  c
>.  /\  d  =/=  e
)  ->  d  =/=  e )
79 simp2 958 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( e  =  <. f ,  g >.  /\  d  =  <. b ,  c
>.  /\  d  =/=  e
)  ->  d  =  <. b ,  c >.
)
80 simp1 957 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( e  =  <. f ,  g >.  /\  d  =  <. b ,  c
>.  /\  d  =/=  e
)  ->  e  =  <. f ,  g >.
)
8178, 79, 803netr3d 2473 . . . . . . . . . . . . . . . . . . 19  |-  ( ( e  =  <. f ,  g >.  /\  d  =  <. b ,  c
>.  /\  d  =/=  e
)  ->  <. b ,  c >.  =/=  <. f ,  g >. )
82 vex 2792 . . . . . . . . . . . . . . . . . . . . 21  |-  b  e. 
_V
83 vex 2792 . . . . . . . . . . . . . . . . . . . . 21  |-  c  e. 
_V
8482, 83opth 4244 . . . . . . . . . . . . . . . . . . . 20  |-  ( <.
b ,  c >.  =  <. f ,  g
>. 
<->  ( b  =  f  /\  c  =  g ) )
8584necon3abii 2477 . . . . . . . . . . . . . . . . . . 19  |-  ( <.
b ,  c >.  =/=  <. f ,  g
>. 
<->  -.  ( b  =  f  /\  c  =  g ) )
8681, 85sylib 190 . . . . . . . . . . . . . . . . . 18  |-  ( ( e  =  <. f ,  g >.  /\  d  =  <. b ,  c
>.  /\  d  =/=  e
)  ->  -.  (
b  =  f  /\  c  =  g )
)
8775, 76, 77, 86syl3anc 1184 . . . . . . . . . . . . . . . . 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 1021 . . . . . . . . . . . . . . . . 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 1023 . . . . . . . . . . . . . . . . . 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 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 ) )
>. ) )  ->  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a )
91 simp2rr 1027 . . . . . . . . . . . . . . . . 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 986 . . . . . . . . . . . . . . . . . . 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 959 . . . . . . . . . . . . . . . . . . . . 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 5844 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( 1st `  d )  mod  ( abs `  a
) )  e.  _V
95 ovex 5844 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( 2nd `  d )  mod  ( abs `  a
) )  e.  _V
9694, 95opth 4244 . . . . . . . . . . . . . . . . . . . . 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 190 . . . . . . . . . . . . . . . . . . . 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 734 . . . . . . . . . . . . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5489 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6089 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 1st `  <. b ,  c
>. )  =  b
102100, 101syl6eq 2332 . . . . . . . . . . . . . . . . . . . . . . . . 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 5834 . . . . . . . . . . . . . . . . . . . . . . . 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 733 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5489 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2792 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  f  e. 
_V
107 vex 2792 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  g  e. 
_V
108106, 107op1st 6089 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 1st `  <. f ,  g
>. )  =  f
109105, 108syl6eq 2332 . . . . . . . . . . . . . . . . . . . . . . . . 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 5834 . . . . . . . . . . . . . . . . . . . . . . . 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 2324 . . . . . . . . . . . . . . . . . . . . . . 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 735 . . . . . . . . . . . . . . . . . . . . . . . 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 5489 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6090 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 2nd `  <. b ,  c
>. )  =  c
115113, 114syl6eq 2332 . . . . . . . . . . . . . . . . . . . . . . . . 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 5834 . . . . . . . . . . . . . . . . . . . . . . . 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 5489 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6090 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 2nd `  <. f ,  g
>. )  =  g
119117, 118syl6eq 2332 . . . . . . . . . . . . . . . . . . . . . . . . 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 5834 . . . . . . . . . . . . . . . . . . . . . . . 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 2324 . . . . . . . . . . . . . . . . . . . . . . 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 520 . . . . . . . . . . . . . . . . . . . . . 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 425 . . . . . . . . . . . . . . . . . . . . 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 977 . . . . . . . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . . . . . . 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 447 . . . . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . . . . . 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 26318 . . . . . . . . . . . . . . . 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 1152 . . . . . . . . . . . . . . 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 1669 . . . . . . . . . . . . . 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 210 . . . . . . . . . . . . 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 425 . . . . . . . . . . . 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 1669 . . . . . . . . . . 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 210 . . . . . . . . . 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 422 . . . . . . . . 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 638 . . . . . . . 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 2674 . . . . . . 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 420 . . . . . 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 703 . . . . 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 651 . . . 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 425 . . 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 2668 . 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 16 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 5    -> wi 6    /\ wa 360    /\ w3a 936   E.wex 1529    = wceq 1624    e. wcel 1685    =/= wne 2447   E.wrex 2545   _Vcvv 2789   <.cop 3644   class class class wbr 4024   {copab 4077   omcom 4655    X. cxp 4686   ` cfv 5221  (class class class)co 5819   1stc1st 6081   2ndc2nd 6082    ~~ cen 6855    ~< csdm 6857   Fincfn 6858   0cc0 8732   1c1 8733    x. cmul 8737    - cmin 9032   NNcn 9741   2c2 9790   ZZcz 10019   QQcq 10311   ...cfz 10776    mod cmo 10967   ^cexp 11098   sqrcsqr 11712   abscabs 11713
This theorem is referenced by:  pellqrex  26363
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-oadd 6478  df-omul 6479  df-er 6655  df-map 6769  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-sup 7189  df-oi 7220  df-card 7567  df-acn 7570  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-n0 9961  df-z 10020  df-uz 10226  df-q 10312  df-rp 10350  df-ico 10656  df-fz 10777  df-fl 10919  df-mod 10968  df-seq 11041  df-exp 11099  df-hash 11332  df-cj 11578  df-re 11579  df-im 11580  df-sqr 11714  df-abs 11715  df-dvds 12526  df-gcd 12680  df-numer 12800  df-denom 12801
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