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Theorem pellex 26252
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
StepHypRef Expression
1 pellexlem5 26250 . 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 10965 . . . . . . . . . 10  |-  ( 0 ... ( ( abs `  a )  -  1 ) )  e.  Fin
3 xpfi 7061 . . . . . . . . . 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 656 . . . . . . . . 9  |-  ( ( 0 ... ( ( abs `  a )  -  1 ) )  X.  ( 0 ... ( ( abs `  a
)  -  1 ) ) )  e.  Fin
5 isfinite 7286 . . . . . . . . 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 10973 . . . . . . . . 9  |-  NN  ~~  om
87ensymi 6844 . . . . . . . 8  |-  om  ~~  NN
9 sdomentr 6928 . . . . . . . 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 656 . . . . . . 7  |-  ( ( 0 ... ( ( abs `  a )  -  1 ) )  X.  ( 0 ... ( ( abs `  a
)  -  1 ) ) )  ~<  NN
11 ensym 6843 . . . . . . . 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 712 . . . . . . 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 6928 . . . . . . 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 647 . . . . . 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 4715 . . . . . . . . . 10  |-  { <. b ,  c >.  |  ( ( b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^ 2 )  -  ( D  x.  (
c ^ 2 ) ) )  =  a ) }  C_  ( NN  X.  NN )
1615sseli 3118 . . . . . . . . 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 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 ) ) )  ->  ( 1st `  d )  e.  NN )
1817nnzd 10048 . . . . . . . . . . . . 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 738 . . . . . . . . . . . . . 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 734 . . . . . . . . . . . . . 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 11739 . . . . . . . . . . . . . 14  |-  ( ( a  e.  ZZ  /\  a  =/=  0 )  -> 
( abs `  a
)  e.  NN )
2219, 20, 21syl2anc 645 . . . . . . . . . . . . 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 10922 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  d
)  e.  ZZ  /\  ( abs `  a )  e.  NN )  -> 
( ( 1st `  d
)  mod  ( abs `  a ) )  e.  ( 0 ... (
( abs `  a
)  -  1 ) ) )
2418, 22, 23syl2anc 645 . . . . . . . . . . . 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 744 . . . . . . . . . . . . . 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 10048 . . . . . . . . . . . . 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 10922 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  d
)  e.  ZZ  /\  ( abs `  a )  e.  NN )  -> 
( ( 2nd `  d
)  mod  ( abs `  a ) )  e.  ( 0 ... (
( abs `  a
)  -  1 ) ) )
2826, 22, 27syl2anc 645 . . . . . . . . . . . 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 6051 . . . . . . . . . 10  |-  ( d  e.  ( NN  X.  NN )  <->  ( d  e.  ( _V  X.  _V )  /\  ( ( 1st `  d )  e.  NN  /\  ( 2nd `  d
)  e.  NN ) ) )
32 opelxp 4672 . . . . . . . . . 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 704 . . . . . 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 5423 . . . . . . . 8  |-  ( d  =  e  ->  ( 1st `  d )  =  ( 1st `  e
) )
3837oveq1d 5772 . . . . . . 7  |-  ( d  =  e  ->  (
( 1st `  d
)  mod  ( abs `  a ) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) ) )
39 fveq2 5423 . . . . . . . 8  |-  ( d  =  e  ->  ( 2nd `  d )  =  ( 2nd `  e
) )
4039oveq1d 5772 . . . . . . 7  |-  ( d  =  e  ->  (
( 2nd `  d
)  mod  ( abs `  a ) )  =  ( ( 2nd `  e
)  mod  ( abs `  a ) ) )
4138, 40opeq12d 3745 . . . . . 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 26231 . . . . 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 2316 . . . . . . . . . . . . . 14  |-  ( b  =  f  ->  (
b  e.  NN  <->  f  e.  NN ) )
44 eleq1 2316 . . . . . . . . . . . . . 14  |-  ( c  =  g  ->  (
c  e.  NN  <->  g  e.  NN ) )
4543, 44bi2anan9 848 . . . . . . . . . . . . 13  |-  ( ( b  =  f  /\  c  =  g )  ->  ( ( b  e.  NN  /\  c  e.  NN )  <->  ( f  e.  NN  /\  g  e.  NN ) ) )
46 oveq1 5764 . . . . . . . . . . . . . . 15  |-  ( b  =  f  ->  (
b ^ 2 )  =  ( f ^
2 ) )
47 oveq1 5764 . . . . . . . . . . . . . . . 16  |-  ( c  =  g  ->  (
c ^ 2 )  =  ( g ^
2 ) )
4847oveq2d 5773 . . . . . . . . . . . . . . 15  |-  ( c  =  g  ->  ( D  x.  ( c ^ 2 ) )  =  ( D  x.  ( g ^ 2 ) ) )
4946, 48oveqan12d 5776 . . . . . . . . . . . . . 14  |-  ( ( b  =  f  /\  c  =  g )  ->  ( ( b ^
2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  ( ( f ^ 2 )  -  ( D  x.  ( g ^ 2 ) ) ) )
5049eqeq1d 2264 . . . . . . . . . . . . 13  |-  ( ( b  =  f  /\  c  =  g )  ->  ( ( ( b ^ 2 )  -  ( D  x.  (
c ^ 2 ) ) )  =  a  <-> 
( ( f ^
2 )  -  ( D  x.  ( g ^ 2 ) ) )  =  a ) )
5145, 50anbi12d 694 . . . . . . . . . . . 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 4028 . . . . . . . . . . 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 2320 . . . . . . . . . 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 4209 . . . . . . . . . . 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 4209 . . . . . . . . . . . . . 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 1031 . . . . . . . . . . . . . . . . . . 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 1157 . . . . . . . . . . . . . . . . . 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 981 . . . . . . . . . . . . . . . . 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 1032 . . . . . . . . . . . . . . . . . . 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 1157 . . . . . . . . . . . . . . . . . 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 981 . . . . . . . . . . . . . . . . 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 1024 . . . . . . . . . . . . . . . . . 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 1180 . . . . . . . . . . . . . . . . 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 713 . . . . . . . . . . . . . . . . . 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 981 . . . . . . . . . . . . . . . . 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 713 . . . . . . . . . . . . . . . . . 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 981 . . . . . . . . . . . . . . . . 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 1027 . . . . . . . . . . . . . . . . . 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 1181 . . . . . . . . . . . . . . . . 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 1028 . . . . . . . . . . . . . . . . . 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 1181 . . . . . . . . . . . . . . . . 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 986 . . . . . . . . . . . . . . . . . 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 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 ) ) )  /\  ( 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 988 . . . . . . . . . . . . . . . . . 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 962 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( e  =  <. f ,  g >.  /\  d  =  <. b ,  c
>.  /\  d  =/=  e
)  ->  d  =/=  e )
79 simp2 961 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( e  =  <. f ,  g >.  /\  d  =  <. b ,  c
>.  /\  d  =/=  e
)  ->  d  =  <. b ,  c >.
)
80 simp1 960 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( e  =  <. f ,  g >.  /\  d  =  <. b ,  c
>.  /\  d  =/=  e
)  ->  e  =  <. f ,  g >.
)
8178, 79, 803netr3d 2445 . . . . . . . . . . . . . . . . . . 19  |-  ( ( e  =  <. f ,  g >.  /\  d  =  <. b ,  c
>.  /\  d  =/=  e
)  ->  <. b ,  c >.  =/=  <. f ,  g >. )
82 vex 2743 . . . . . . . . . . . . . . . . . . . . 21  |-  b  e. 
_V
83 vex 2743 . . . . . . . . . . . . . . . . . . . . 21  |-  c  e. 
_V
8482, 83opth 4182 . . . . . . . . . . . . . . . . . . . 20  |-  ( <.
b ,  c >.  =  <. f ,  g
>. 
<->  ( b  =  f  /\  c  =  g ) )
8584necon3abii 2449 . . . . . . . . . . . . . . . . . . 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 1187 . . . . . . . . . . . . . . . . 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 1024 . . . . . . . . . . . . . . . . 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 1026 . . . . . . . . . . . . . . . . . 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 1179 . . . . . . . . . . . . . . . . 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 1030 . . . . . . . . . . . . . . . . 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 989 . . . . . . . . . . . . . . . . . . 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 962 . . . . . . . . . . . . . . . . . . . . 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 5782 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( 1st `  d )  mod  ( abs `  a
) )  e.  _V
95 ovex 5782 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( 2nd `  d )  mod  ( abs `  a
) )  e.  _V
9694, 95opth 4182 . . . . . . . . . . . . . . . . . . . . 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 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 ) ) ) )  ->  ( ( 1st `  d )  mod  ( abs `  a
) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) ) )
99 simpll 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 ) ) ) )  ->  d  =  <. b ,  c >.
)
10099fveq2d 5427 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6027 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 1st `  <. b ,  c
>. )  =  b
102100, 101syl6eq 2304 . . . . . . . . . . . . . . . . . . . . . . . . 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 5772 . . . . . . . . . . . . . . . . . . . . . . . 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 734 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5427 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2743 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  f  e. 
_V
107 vex 2743 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  g  e. 
_V
108106, 107op1st 6027 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 1st `  <. f ,  g
>. )  =  f
109105, 108syl6eq 2304 . . . . . . . . . . . . . . . . . . . . . . . . 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 5772 . . . . . . . . . . . . . . . . . . . . . . . 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 2296 . . . . . . . . . . . . . . . . . . . . . . 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 736 . . . . . . . . . . . . . . . . . . . . . . . 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 5427 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6028 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 2nd `  <. b ,  c
>. )  =  c
115113, 114syl6eq 2304 . . . . . . . . . . . . . . . . . . . . . . . . 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 5772 . . . . . . . . . . . . . . . . . . . . . . . 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 5427 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6028 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 2nd `  <. f ,  g
>. )  =  g
119117, 118syl6eq 2304 . . . . . . . . . . . . . . . . . . . . . . . . 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 5772 . . . . . . . . . . . . . . . . . . . . . . . 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 2296 . . . . . . . . . . . . . . . . . . . . . . 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 980 . . . . . . . . . . . . . . . . . . . 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 1187 . . . . . . . . . . . . . . . . . 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 26251 . . . . . . . . . . . . . . . 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 1155 . . . . . . . . . . . . . . 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 2027 . . . . . . . . . . . . . 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 2027 . . . . . . . . . . 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 639 . . . . . . . 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 2644 . . . . . . 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 704 . . . . 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 652 . . . 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 2638 . 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 939   E.wex 1537    = wceq 1619    e. wcel 1621    =/= wne 2419   E.wrex 2517   _Vcvv 2740   <.cop 3584   class class class wbr 3963   {copab 4016   omcom 4593    X. cxp 4624   ` cfv 4638  (class class class)co 5757   1stc1st 6019   2ndc2nd 6020    ~~ cen 6793    ~< csdm 6795   Fincfn 6796   0cc0 8670   1c1 8671    x. cmul 8675    - cmin 8970   NNcn 9679   2c2 9728   ZZcz 9956   QQcq 10248   ...cfz 10713    mod cmo 10904   ^cexp 11035   sqrcsqr 11648   abscabs 11649
This theorem is referenced by:  pellqrex  26296
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-oadd 6416  df-omul 6417  df-er 6593  df-map 6707  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-sup 7127  df-oi 7158  df-card 7505  df-acn 7508  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-n0 9898  df-z 9957  df-uz 10163  df-q 10249  df-rp 10287  df-ico 10593  df-fz 10714  df-fl 10856  df-mod 10905  df-seq 10978  df-exp 11036  df-hash 11269  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651  df-divides 12459  df-gcd 12613  df-numer 12733  df-denom 12734
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