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Theorem pellex 35650
Description: Every Pell equation has a nontrivial solution. Theorem 62 in [vandenDries] p. 43. (Contributed by Stefan O'Rear, 19-Oct-2014.)
Assertion
Ref Expression
pellex  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  E. x  e.  NN  E. y  e.  NN  (
( x ^ 2 )  -  ( D  x.  ( y ^
2 ) ) )  =  1 )
Distinct variable group:    x, D, y

Proof of Theorem pellex
Dummy variables  a 
b  c  d  e  f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fzfi 12192 . . . . . . . 8  |-  ( 0 ... ( ( abs `  a )  -  1 ) )  e.  Fin
2 xpfi 7852 . . . . . . . 8  |-  ( ( ( 0 ... (
( abs `  a
)  -  1 ) )  e.  Fin  /\  ( 0 ... (
( abs `  a
)  -  1 ) )  e.  Fin )  ->  ( ( 0 ... ( ( abs `  a
)  -  1 ) )  X.  ( 0 ... ( ( abs `  a )  -  1 ) ) )  e. 
Fin )
31, 1, 2mp2an 676 . . . . . . 7  |-  ( ( 0 ... ( ( abs `  a )  -  1 ) )  X.  ( 0 ... ( ( abs `  a
)  -  1 ) ) )  e.  Fin
4 isfinite 8167 . . . . . . 7  |-  ( ( ( 0 ... (
( abs `  a
)  -  1 ) )  X.  ( 0 ... ( ( abs `  a )  -  1 ) ) )  e. 
Fin 
<->  ( ( 0 ... ( ( abs `  a
)  -  1 ) )  X.  ( 0 ... ( ( abs `  a )  -  1 ) ) )  ~<  om )
53, 4mpbi 211 . . . . . 6  |-  ( ( 0 ... ( ( abs `  a )  -  1 ) )  X.  ( 0 ... ( ( abs `  a
)  -  1 ) ) )  ~<  om
6 nnenom 12200 . . . . . . 7  |-  NN  ~~  om
76ensymi 7630 . . . . . 6  |-  om  ~~  NN
8 sdomentr 7716 . . . . . 6  |-  ( ( ( ( 0 ... ( ( abs `  a
)  -  1 ) )  X.  ( 0 ... ( ( abs `  a )  -  1 ) ) )  ~<  om  /\  om  ~~  NN )  ->  ( ( 0 ... ( ( abs `  a )  -  1 ) )  X.  (
0 ... ( ( abs `  a )  -  1 ) ) )  ~<  NN )
95, 7, 8mp2an 676 . . . . 5  |-  ( ( 0 ... ( ( abs `  a )  -  1 ) )  X.  ( 0 ... ( ( abs `  a
)  -  1 ) ) )  ~<  NN
10 ensym 7629 . . . . . 6  |-  ( {
<. 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 ) } )
1110ad2antll 733 . . . . 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 ) )  ->  NN  ~~  { <. b ,  c >.  |  ( ( b  e.  NN  /\  c  e.  NN )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a ) } )
12 sdomentr 7716 . . . . 5  |-  ( ( ( ( 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 ) } )
139, 11, 12sylancr 667 . . . 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 ) )  ->  ( (
0 ... ( ( abs `  a )  -  1 ) )  X.  (
0 ... ( ( abs `  a )  -  1 ) ) )  ~<  { <. b ,  c
>.  |  ( (
b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^
2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  a ) } )
14 opabssxp 4928 . . . . . . . 8  |-  { <. b ,  c >.  |  ( ( b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^ 2 )  -  ( D  x.  (
c ^ 2 ) ) )  =  a ) }  C_  ( NN  X.  NN )
1514sseli 3460 . . . . . . 7  |-  ( d  e.  { <. b ,  c >.  |  ( ( b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^ 2 )  -  ( D  x.  (
c ^ 2 ) ) )  =  a ) }  ->  d  e.  ( NN  X.  NN ) )
16 simprrl 772 . . . . . . . . . . . 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 )  e.  NN )
1716nnzd 11047 . . . . . . . . . . 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 )  e.  ZZ )
18 simpllr 767 . . . . . . . . . . . 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 ) ) )  ->  a  e.  ZZ )
19 simplr 760 . . . . . . . . . . . 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 ) ) )  ->  a  =/=  0 )
20 nnabscl 13389 . . . . . . . . . . . 12  |-  ( ( a  e.  ZZ  /\  a  =/=  0 )  -> 
( abs `  a
)  e.  NN )
2118, 19, 20syl2anc 665 . . . . . . . . . . 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 ) ) )  ->  ( abs `  a )  e.  NN )
22 zmodfz 12125 . . . . . . . . . . 11  |-  ( ( ( 1st `  d
)  e.  ZZ  /\  ( abs `  a )  e.  NN )  -> 
( ( 1st `  d
)  mod  ( abs `  a ) )  e.  ( 0 ... (
( abs `  a
)  -  1 ) ) )
2317, 21, 22syl2anc 665 . . . . . . . . . 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 ) ) )
24 simprrr 773 . . . . . . . . . . . 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 )  e.  NN )
2524nnzd 11047 . . . . . . . . . . 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 ) ) )  ->  ( 2nd `  d )  e.  ZZ )
26 zmodfz 12125 . . . . . . . . . . 11  |-  ( ( ( 2nd `  d
)  e.  ZZ  /\  ( abs `  a )  e.  NN )  -> 
( ( 2nd `  d
)  mod  ( abs `  a ) )  e.  ( 0 ... (
( abs `  a
)  -  1 ) ) )
2725, 21, 26syl2anc 665 . . . . . . . . . 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 ) ) )  ->  (
( 2nd `  d
)  mod  ( abs `  a ) )  e.  ( 0 ... (
( abs `  a
)  -  1 ) ) )
2823, 27jca 534 . . . . . . . . 9  |-  ( ( ( ( ( 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 ) ) ) )
2928ex 435 . . . . . . . 8  |-  ( ( ( ( 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 ) ) ) ) )
30 elxp7 6841 . . . . . . . 8  |-  ( d  e.  ( NN  X.  NN )  <->  ( d  e.  ( _V  X.  _V )  /\  ( ( 1st `  d )  e.  NN  /\  ( 2nd `  d
)  e.  NN ) ) )
31 opelxp 4883 . . . . . . . 8  |-  ( <.
( ( 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 ) ) ) )
3229, 30, 313imtr4g 273 . . . . . . 7  |-  ( ( ( ( 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 ) ) ) ) )
3315, 32syl5 33 . . . . . 6  |-  ( ( ( ( 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 ) ) ) ) )
3433imp 430 . . . . 5  |-  ( ( ( ( ( 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 ) ) ) )
3534adantlrr 725 . . . 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 ) )  /\  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 ) ) ) )
36 fveq2 5882 . . . . . 6  |-  ( d  =  e  ->  ( 1st `  d )  =  ( 1st `  e
) )
3736oveq1d 6321 . . . . 5  |-  ( d  =  e  ->  (
( 1st `  d
)  mod  ( abs `  a ) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) ) )
38 fveq2 5882 . . . . . 6  |-  ( d  =  e  ->  ( 2nd `  d )  =  ( 2nd `  e
) )
3938oveq1d 6321 . . . . 5  |-  ( d  =  e  ->  (
( 2nd `  d
)  mod  ( abs `  a ) )  =  ( ( 2nd `  e
)  mod  ( abs `  a ) ) )
4037, 39opeq12d 4195 . . . 4  |-  ( d  =  e  ->  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. )
4113, 35, 40fphpd 35629 . . 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. 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 ) )
>. ) )
42 eleq1 2495 . . . . . . . . . . . 12  |-  ( b  =  f  ->  (
b  e.  NN  <->  f  e.  NN ) )
43 eleq1 2495 . . . . . . . . . . . 12  |-  ( c  =  g  ->  (
c  e.  NN  <->  g  e.  NN ) )
4442, 43bi2anan9 881 . . . . . . . . . . 11  |-  ( ( b  =  f  /\  c  =  g )  ->  ( ( b  e.  NN  /\  c  e.  NN )  <->  ( f  e.  NN  /\  g  e.  NN ) ) )
45 oveq1 6313 . . . . . . . . . . . . 13  |-  ( b  =  f  ->  (
b ^ 2 )  =  ( f ^
2 ) )
46 oveq1 6313 . . . . . . . . . . . . . 14  |-  ( c  =  g  ->  (
c ^ 2 )  =  ( g ^
2 ) )
4746oveq2d 6322 . . . . . . . . . . . . 13  |-  ( c  =  g  ->  ( D  x.  ( c ^ 2 ) )  =  ( D  x.  ( g ^ 2 ) ) )
4845, 47oveqan12d 6325 . . . . . . . . . . . 12  |-  ( ( b  =  f  /\  c  =  g )  ->  ( ( b ^
2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  ( ( f ^ 2 )  -  ( D  x.  ( g ^ 2 ) ) ) )
4948eqeq1d 2424 . . . . . . . . . . 11  |-  ( ( b  =  f  /\  c  =  g )  ->  ( ( ( b ^ 2 )  -  ( D  x.  (
c ^ 2 ) ) )  =  a  <-> 
( ( f ^
2 )  -  ( D  x.  ( g ^ 2 ) ) )  =  a ) )
5044, 49anbi12d 715 . . . . . . . . . 10  |-  ( ( 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 ) ) )
5150cbvopabv 4493 . . . . . . . . 9  |-  { <. 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 ) }
5251eleq2i 2499 . . . . . . . 8  |-  ( 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 ) } )
5352biimpi 197 . . . . . . 7  |-  ( 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 ) } )
54 elopab 4728 . . . . . . . . 9  |-  ( 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 ) ) )
55 elopab 4728 . . . . . . . . . . . 12  |-  ( 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 ) ) )
56 simp3ll 1076 . . . . . . . . . . . . . . . . 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 ) )  ->  b  e.  NN )
57563expb 1206 . . . . . . . . . . . . . . . 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 ) ) )  ->  b  e.  NN )
58573ad2ant1 1026 . . . . . . . . . . . . . . 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 ) )
>. ) )  ->  b  e.  NN )
59 simp3lr 1077 . . . . . . . . . . . . . . . . 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 ) )  ->  c  e.  NN )
60593expb 1206 . . . . . . . . . . . . . . . 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 ) ) )  ->  c  e.  NN )
61603ad2ant1 1026 . . . . . . . . . . . . . . 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 ) )
>. ) )  ->  c  e.  NN )
62 simp1lr 1069 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( 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 )
63623adant1r 1257 . . . . . . . . . . . . . . 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 ) )
>. ) )  ->  a  e.  ZZ )
64 simp-4l 774 . . . . . . . . . . . . . . . 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 ) ) )  ->  D  e.  NN )
65643ad2ant1 1026 . . . . . . . . . . . . . . 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 ) )
>. ) )  ->  D  e.  NN )
66 simp-4r 775 . . . . . . . . . . . . . . . 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 ) ) )  ->  -.  ( sqr `  D )  e.  QQ )
67663ad2ant1 1026 . . . . . . . . . . . . . . 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 ) )
>. ) )  ->  -.  ( sqr `  D )  e.  QQ )
68 simp2ll 1072 . . . . . . . . . . . . . . . 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 ) ) )  /\  ( ( 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 )
69683adant2l 1258 . . . . . . . . . . . . . . 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 ) )
>. ) )  ->  f  e.  NN )
70 simp2lr 1073 . . . . . . . . . . . . . . . 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 ) ) )  /\  ( ( 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 )
71703adant2l 1258 . . . . . . . . . . . . . . 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 ) )
>. ) )  ->  g  e.  NN )
72 simp2l 1031 . . . . . . . . . . . . . . . 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  =  <. f ,  g
>. )
73 simp1rl 1070 . . . . . . . . . . . . . . . 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 ) )
>. ) )  ->  d  =  <. b ,  c
>. )
74 simp3l 1033 . . . . . . . . . . . . . . . 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 ) )
>. ) )  ->  d  =/=  e )
75 simp3 1007 . . . . . . . . . . . . . . . . . 18  |-  ( ( e  =  <. f ,  g >.  /\  d  =  <. b ,  c
>.  /\  d  =/=  e
)  ->  d  =/=  e )
76 simp2 1006 . . . . . . . . . . . . . . . . . 18  |-  ( ( e  =  <. f ,  g >.  /\  d  =  <. b ,  c
>.  /\  d  =/=  e
)  ->  d  =  <. b ,  c >.
)
77 simp1 1005 . . . . . . . . . . . . . . . . . 18  |-  ( ( e  =  <. f ,  g >.  /\  d  =  <. b ,  c
>.  /\  d  =/=  e
)  ->  e  =  <. f ,  g >.
)
7875, 76, 773netr3d 2723 . . . . . . . . . . . . . . . . 17  |-  ( ( e  =  <. f ,  g >.  /\  d  =  <. b ,  c
>.  /\  d  =/=  e
)  ->  <. b ,  c >.  =/=  <. f ,  g >. )
79 vex 3083 . . . . . . . . . . . . . . . . . . 19  |-  b  e. 
_V
80 vex 3083 . . . . . . . . . . . . . . . . . . 19  |-  c  e. 
_V
8179, 80opth 4695 . . . . . . . . . . . . . . . . . 18  |-  ( <.
b ,  c >.  =  <. f ,  g
>. 
<->  ( b  =  f  /\  c  =  g ) )
8281necon3abii 2680 . . . . . . . . . . . . . . . . 17  |-  ( <.
b ,  c >.  =/=  <. f ,  g
>. 
<->  -.  ( b  =  f  /\  c  =  g ) )
8378, 82sylib 199 . . . . . . . . . . . . . . . 16  |-  ( ( e  =  <. f ,  g >.  /\  d  =  <. b ,  c
>.  /\  d  =/=  e
)  ->  -.  (
b  =  f  /\  c  =  g )
)
8472, 73, 74, 83syl3anc 1264 . . . . . . . . . . . . . . 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 ) )
>. ) )  ->  -.  ( b  =  f  /\  c  =  g ) )
85 simp1lr 1069 . . . . . . . . . . . . . . 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 ) )
>. ) )  ->  a  =/=  0 )
86 simp1rr 1071 . . . . . . . . . . . . . . . 16  |-  ( ( ( 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 )
87863adant1l 1256 . . . . . . . . . . . . . . 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 ) )
>. ) )  ->  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  a )
88 simp2rr 1075 . . . . . . . . . . . . . . 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 ) )
>. ) )  ->  (
( f ^ 2 )  -  ( D  x.  ( g ^
2 ) ) )  =  a )
89 simp3r 1034 . . . . . . . . . . . . . . . . 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 ) )
>. ) )  ->  <. (
( 1st `  d
)  mod  ( abs `  a ) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. )
90 simp3 1007 . . . . . . . . . . . . . . . . . . 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 ) ) >.
)  ->  <. ( ( 1st `  d )  mod  ( abs `  a
) ) ,  ( ( 2nd `  d
)  mod  ( abs `  a ) ) >.  =  <. ( ( 1st `  e )  mod  ( abs `  a ) ) ,  ( ( 2nd `  e )  mod  ( abs `  a ) )
>. )
91 ovex 6334 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 1st `  d )  mod  ( abs `  a
) )  e.  _V
92 ovex 6334 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 2nd `  d )  mod  ( abs `  a
) )  e.  _V
9391, 92opth 4695 . . . . . . . . . . . . . . . . . . 19  |-  ( <.
( ( 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 ) ) ) )
9490, 93sylib 199 . . . . . . . . . . . . . . . . . 18  |-  ( ( 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 ) ) ) )
95 simprl 762 . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )  ->  ( ( 1st `  d )  mod  ( abs `  a
) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) ) )
96 simpll 758 . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )  ->  d  =  <. b ,  c >.
)
9796fveq2d 5886 . . . . . . . . . . . . . . . . . . . . . . . 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 )  =  ( 1st `  <. b ,  c >. )
)
9879, 80op1st 6816 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 1st `  <. b ,  c
>. )  =  b
9997, 98syl6eq 2479 . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )  ->  ( 1st `  d )  =  b )
10099oveq1d 6321 . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )  ->  ( ( 1st `  d )  mod  ( abs `  a
) )  =  ( b  mod  ( abs `  a ) ) )
101 simplr 760 . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )  ->  e  =  <. f ,  g >.
)
102101fveq2d 5886 . . . . . . . . . . . . . . . . . . . . . . . 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 )  =  ( 1st `  <. f ,  g >. )
)
103 vex 3083 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  f  e. 
_V
104 vex 3083 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  g  e. 
_V
105103, 104op1st 6816 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 1st `  <. f ,  g
>. )  =  f
106102, 105syl6eq 2479 . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )  ->  ( 1st `  e )  =  f )
107106oveq1d 6321 . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )  ->  ( ( 1st `  e )  mod  ( abs `  a
) )  =  ( f  mod  ( abs `  a ) ) )
10895, 100, 1073eqtr3d 2471 . . . . . . . . . . . . . . . . . . . . 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 ) ) )
109 simprr 764 . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )  ->  ( ( 2nd `  d )  mod  ( abs `  a
) )  =  ( ( 2nd `  e
)  mod  ( abs `  a ) ) )
11096fveq2d 5886 . . . . . . . . . . . . . . . . . . . . . . . 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 )  =  ( 2nd `  <. b ,  c >. )
)
11179, 80op2nd 6817 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 2nd `  <. b ,  c
>. )  =  c
112110, 111syl6eq 2479 . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )  ->  ( 2nd `  d )  =  c )
113112oveq1d 6321 . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )  ->  ( ( 2nd `  d )  mod  ( abs `  a
) )  =  ( c  mod  ( abs `  a ) ) )
114101fveq2d 5886 . . . . . . . . . . . . . . . . . . . . . . . 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 )  =  ( 2nd `  <. f ,  g >. )
)
115103, 104op2nd 6817 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 2nd `  <. f ,  g
>. )  =  g
116114, 115syl6eq 2479 . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )  ->  ( 2nd `  e )  =  g )
117116oveq1d 6321 . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )  ->  ( ( 2nd `  e )  mod  ( abs `  a
) )  =  ( g  mod  ( abs `  a ) ) )
118109, 113, 1173eqtr3d 2471 . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )  ->  ( c  mod  ( abs `  a
) )  =  ( g  mod  ( abs `  a ) ) )
119108, 118jca 534 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( 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 ) ) ) )
120119ex 435 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 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
) ) ) ) )
1211203adant3 1025 . . . . . . . . . . . . . . . . . 18  |-  ( ( 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
) ) ) ) )
12294, 121mpd 15 . . . . . . . . . . . . . . . . 17  |-  ( ( 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 ) ) ) )
12373, 72, 89, 122syl3anc 1264 . . . . . . . . . . . . . . . 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 ) )
>. ) )  ->  (
( b  mod  ( abs `  a ) )  =  ( f  mod  ( abs `  a
) )  /\  (
c  mod  ( abs `  a ) )  =  ( g  mod  ( abs `  a ) ) ) )
124123simpld 460 . . . . . . . . . . . . . . 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 ) )
>. ) )  ->  (
b  mod  ( abs `  a ) )  =  ( f  mod  ( abs `  a ) ) )
125123simprd 464 . . . . . . . . . . . . . . 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 ) )
>. ) )  ->  (
c  mod  ( abs `  a ) )  =  ( g  mod  ( abs `  a ) ) )
12658, 61, 63, 65, 67, 69, 71, 84, 85, 87, 88, 124, 125pellexlem6 35649 . . . . . . . . . . . . . 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 ,  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 )
1271263exp 1204 . . . . . . . . . . . . 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  =  <. 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 ) ) )
128127exlimdvv 1773 . . . . . . . . . . . 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. 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 ) ) )
12955, 128syl5bi 220 . . . . . . . . . . 11  |-  ( ( ( ( ( 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 ) ) )
130129ex 435 . . . . . . . . . 10  |-  ( ( ( ( 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 ) ) ) )
131130exlimdvv 1773 . . . . . . . . 9  |-  ( ( ( ( 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 ) ) ) )
13254, 131syl5bi 220 . . . . . . . 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.  { <. 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 ) ) ) )
133132impd 432 . . . . . . 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 ) }  /\  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 ) ) )
13453, 133sylan2i 659 . . . . . 6  |-  ( ( ( ( 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 ) ) )
135134rexlimdvv 2920 . . . . 5  |-  ( ( ( ( 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 ) )
136135imp 430 . . . 4  |-  ( ( ( ( ( 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 )
137136adantlrr 725 . . 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. 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 )
13841, 137mpdan 672 . 2  |-  ( ( ( ( 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 )
139 pellexlem5 35648 . 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 ) )
140138, 139r19.29a 2967 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1657    e. wcel 1872    =/= wne 2614   E.wrex 2772   _Vcvv 3080   <.cop 4004   class class class wbr 4423   {copab 4481    X. cxp 4851   ` cfv 5601  (class class class)co 6306   omcom 6707   1stc1st 6806   2ndc2nd 6807    ~~ cen 7578    ~< csdm 7580   Fincfn 7581   0cc0 9547   1c1 9548    x. cmul 9552    - cmin 9868   NNcn 10617   2c2 10667   ZZcz 10945   QQcq 11272   ...cfz 11792    mod cmo 12103   ^cexp 12279   sqrcsqrt 13297   abscabs 13298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-inf2 8156  ax-cnex 9603  ax-resscn 9604  ax-1cn 9605  ax-icn 9606  ax-addcl 9607  ax-addrcl 9608  ax-mulcl 9609  ax-mulrcl 9610  ax-mulcom 9611  ax-addass 9612  ax-mulass 9613  ax-distr 9614  ax-i2m1 9615  ax-1ne0 9616  ax-1rid 9617  ax-rnegex 9618  ax-rrecex 9619  ax-cnre 9620  ax-pre-lttri 9621  ax-pre-lttrn 9622  ax-pre-ltadd 9623  ax-pre-mulgt0 9624  ax-pre-sup 9625
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-om 6708  df-1st 6808  df-2nd 6809  df-wrecs 7040  df-recs 7102  df-rdg 7140  df-1o 7194  df-oadd 7198  df-omul 7199  df-er 7375  df-map 7486  df-en 7582  df-dom 7583  df-sdom 7584  df-fin 7585  df-sup 7966  df-inf 7967  df-oi 8035  df-card 8382  df-acn 8385  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-le 9689  df-sub 9870  df-neg 9871  df-div 10278  df-nn 10618  df-2 10676  df-3 10677  df-n0 10878  df-z 10946  df-uz 11168  df-q 11273  df-rp 11311  df-ico 11649  df-fz 11793  df-fl 12035  df-mod 12104  df-seq 12221  df-exp 12280  df-hash 12523  df-cj 13163  df-re 13164  df-im 13165  df-sqrt 13299  df-abs 13300  df-dvds 14306  df-gcd 14469  df-numer 14684  df-denom 14685
This theorem is referenced by:  pellqrex  35697
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