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Theorem pellex 35750
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 12223 . . . . . . . 8  |-  ( 0 ... ( ( abs `  a )  -  1 ) )  e.  Fin
2 xpfi 7860 . . . . . . . 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 686 . . . . . . 7  |-  ( ( 0 ... ( ( abs `  a )  -  1 ) )  X.  ( 0 ... ( ( abs `  a
)  -  1 ) ) )  e.  Fin
4 isfinite 8175 . . . . . . 7  |-  ( ( ( 0 ... (
( abs `  a
)  -  1 ) )  X.  ( 0 ... ( ( abs `  a )  -  1 ) ) )  e. 
Fin 
<->  ( ( 0 ... ( ( abs `  a
)  -  1 ) )  X.  ( 0 ... ( ( abs `  a )  -  1 ) ) )  ~<  om )
53, 4mpbi 213 . . . . . 6  |-  ( ( 0 ... ( ( abs `  a )  -  1 ) )  X.  ( 0 ... ( ( abs `  a
)  -  1 ) ) )  ~<  om
6 nnenom 12231 . . . . . . 7  |-  NN  ~~  om
76ensymi 7637 . . . . . 6  |-  om  ~~  NN
8 sdomentr 7724 . . . . . 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 686 . . . . 5  |-  ( ( 0 ... ( ( abs `  a )  -  1 ) )  X.  ( 0 ... ( ( abs `  a
)  -  1 ) ) )  ~<  NN
10 ensym 7636 . . . . . 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 743 . . . . 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 7724 . . . . 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 676 . . . 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 4914 . . . . . . . 8  |-  { <. b ,  c >.  |  ( ( b  e.  NN  /\  c  e.  NN )  /\  ( ( b ^ 2 )  -  ( D  x.  (
c ^ 2 ) ) )  =  a ) }  C_  ( NN  X.  NN )
1514sseli 3414 . . . . . . 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 782 . . . . . . . . . . . 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 11062 . . . . . . . . . . 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 777 . . . . . . . . . . . 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 770 . . . . . . . . . . . 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 13465 . . . . . . . . . . . 12  |-  ( ( a  e.  ZZ  /\  a  =/=  0 )  -> 
( abs `  a
)  e.  NN )
2118, 19, 20syl2anc 673 . . . . . . . . . . 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 12151 . . . . . . . . . . 11  |-  ( ( ( 1st `  d
)  e.  ZZ  /\  ( abs `  a )  e.  NN )  -> 
( ( 1st `  d
)  mod  ( abs `  a ) )  e.  ( 0 ... (
( abs `  a
)  -  1 ) ) )
2317, 21, 22syl2anc 673 . . . . . . . . . 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 783 . . . . . . . . . . . 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 11062 . . . . . . . . . . 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 12151 . . . . . . . . . . 11  |-  ( ( ( 2nd `  d
)  e.  ZZ  /\  ( abs `  a )  e.  NN )  -> 
( ( 2nd `  d
)  mod  ( abs `  a ) )  e.  ( 0 ... (
( abs `  a
)  -  1 ) ) )
2725, 21, 26syl2anc 673 . . . . . . . . . 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 541 . . . . . . . . 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 441 . . . . . . . 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 6845 . . . . . . . 8  |-  ( d  e.  ( NN  X.  NN )  <->  ( d  e.  ( _V  X.  _V )  /\  ( ( 1st `  d )  e.  NN  /\  ( 2nd `  d
)  e.  NN ) ) )
31 opelxp 4869 . . . . . . . 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 278 . . . . . . 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 32 . . . . . 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 436 . . . . 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 735 . . . 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 5879 . . . . . 6  |-  ( d  =  e  ->  ( 1st `  d )  =  ( 1st `  e
) )
3736oveq1d 6323 . . . . 5  |-  ( d  =  e  ->  (
( 1st `  d
)  mod  ( abs `  a ) )  =  ( ( 1st `  e
)  mod  ( abs `  a ) ) )
38 fveq2 5879 . . . . . 6  |-  ( d  =  e  ->  ( 2nd `  d )  =  ( 2nd `  e
) )
3938oveq1d 6323 . . . . 5  |-  ( d  =  e  ->  (
( 2nd `  d
)  mod  ( abs `  a ) )  =  ( ( 2nd `  e
)  mod  ( abs `  a ) ) )
4037, 39opeq12d 4166 . . . 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 35730 . . 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 2537 . . . . . . . . . . . 12  |-  ( b  =  f  ->  (
b  e.  NN  <->  f  e.  NN ) )
43 eleq1 2537 . . . . . . . . . . . 12  |-  ( c  =  g  ->  (
c  e.  NN  <->  g  e.  NN ) )
4442, 43bi2anan9 890 . . . . . . . . . . 11  |-  ( ( b  =  f  /\  c  =  g )  ->  ( ( b  e.  NN  /\  c  e.  NN )  <->  ( f  e.  NN  /\  g  e.  NN ) ) )
45 oveq1 6315 . . . . . . . . . . . . 13  |-  ( b  =  f  ->  (
b ^ 2 )  =  ( f ^
2 ) )
46 oveq1 6315 . . . . . . . . . . . . . 14  |-  ( c  =  g  ->  (
c ^ 2 )  =  ( g ^
2 ) )
4746oveq2d 6324 . . . . . . . . . . . . 13  |-  ( c  =  g  ->  ( D  x.  ( c ^ 2 ) )  =  ( D  x.  ( g ^ 2 ) ) )
4845, 47oveqan12d 6327 . . . . . . . . . . . 12  |-  ( ( b  =  f  /\  c  =  g )  ->  ( ( b ^
2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  ( ( f ^ 2 )  -  ( D  x.  ( g ^ 2 ) ) ) )
4948eqeq1d 2473 . . . . . . . . . . 11  |-  ( ( b  =  f  /\  c  =  g )  ->  ( ( ( b ^ 2 )  -  ( D  x.  (
c ^ 2 ) ) )  =  a  <-> 
( ( f ^
2 )  -  ( D  x.  ( g ^ 2 ) ) )  =  a ) )
5044, 49anbi12d 725 . . . . . . . . . 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 4465 . . . . . . . . 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 2541 . . . . . . . 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 199 . . . . . . 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 4709 . . . . . . . . 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 4709 . . . . . . . . . . . 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 1101 . . . . . . . . . . . . . . . . 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 1232 . . . . . . . . . . . . . . . 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 1051 . . . . . . . . . . . . . . 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 1102 . . . . . . . . . . . . . . . . 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 1232 . . . . . . . . . . . . . . . 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 1051 . . . . . . . . . . . . . . 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 1094 . . . . . . . . . . . . . . . 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 1285 . . . . . . . . . . . . . . 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 784 . . . . . . . . . . . . . . . 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 1051 . . . . . . . . . . . . . . 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 785 . . . . . . . . . . . . . . . 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 1051 . . . . . . . . . . . . . . 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 1097 . . . . . . . . . . . . . . . 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 1286 . . . . . . . . . . . . . . 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 1098 . . . . . . . . . . . . . . . 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 1286 . . . . . . . . . . . . . . 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 1056 . . . . . . . . . . . . . . . 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 1095 . . . . . . . . . . . . . . . 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 1058 . . . . . . . . . . . . . . . 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 1032 . . . . . . . . . . . . . . . . . 18  |-  ( ( e  =  <. f ,  g >.  /\  d  =  <. b ,  c
>.  /\  d  =/=  e
)  ->  d  =/=  e )
76 simp2 1031 . . . . . . . . . . . . . . . . . 18  |-  ( ( e  =  <. f ,  g >.  /\  d  =  <. b ,  c
>.  /\  d  =/=  e
)  ->  d  =  <. b ,  c >.
)
77 simp1 1030 . . . . . . . . . . . . . . . . . 18  |-  ( ( e  =  <. f ,  g >.  /\  d  =  <. b ,  c
>.  /\  d  =/=  e
)  ->  e  =  <. f ,  g >.
)
7875, 76, 773netr3d 2719 . . . . . . . . . . . . . . . . 17  |-  ( ( e  =  <. f ,  g >.  /\  d  =  <. b ,  c
>.  /\  d  =/=  e
)  ->  <. b ,  c >.  =/=  <. f ,  g >. )
79 vex 3034 . . . . . . . . . . . . . . . . . . 19  |-  b  e. 
_V
80 vex 3034 . . . . . . . . . . . . . . . . . . 19  |-  c  e. 
_V
8179, 80opth 4676 . . . . . . . . . . . . . . . . . 18  |-  ( <.
b ,  c >.  =  <. f ,  g
>. 
<->  ( b  =  f  /\  c  =  g ) )
8281necon3abii 2689 . . . . . . . . . . . . . . . . 17  |-  ( <.
b ,  c >.  =/=  <. f ,  g
>. 
<->  -.  ( b  =  f  /\  c  =  g ) )
8378, 82sylib 201 . . . . . . . . . . . . . . . 16  |-  ( ( e  =  <. f ,  g >.  /\  d  =  <. b ,  c
>.  /\  d  =/=  e
)  ->  -.  (
b  =  f  /\  c  =  g )
)
8472, 73, 74, 83syl3anc 1292 . . . . . . . . . . . . . . 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 1094 . . . . . . . . . . . . . . 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 1096 . . . . . . . . . . . . . . . 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 1284 . . . . . . . . . . . . . . 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 1100 . . . . . . . . . . . . . . 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 1059 . . . . . . . . . . . . . . . . 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 1032 . . . . . . . . . . . . . . . . . . 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 6336 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 1st `  d )  mod  ( abs `  a
) )  e.  _V
92 ovex 6336 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 2nd `  d )  mod  ( abs `  a
) )  e.  _V
9391, 92opth 4676 . . . . . . . . . . . . . . . . . . 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 201 . . . . . . . . . . . . . . . . . 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 772 . . . . . . . . . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . . . . . . . . . . . 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 5883 . . . . . . . . . . . . . . . . . . . . . . . 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 6820 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 1st `  <. b ,  c
>. )  =  b
9997, 98syl6eq 2521 . . . . . . . . . . . . . . . . . . . . . . 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 6323 . . . . . . . . . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . . . . . . . . . . . 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 5883 . . . . . . . . . . . . . . . . . . . . . . . 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 3034 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  f  e. 
_V
104 vex 3034 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  g  e. 
_V
105103, 104op1st 6820 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 1st `  <. f ,  g
>. )  =  f
106102, 105syl6eq 2521 . . . . . . . . . . . . . . . . . . . . . . 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 6323 . . . . . . . . . . . . . . . . . . . . . 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 2513 . . . . . . . . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . . . . . . . . . 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 5883 . . . . . . . . . . . . . . . . . . . . . . . 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 6821 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 2nd `  <. b ,  c
>. )  =  c
112110, 111syl6eq 2521 . . . . . . . . . . . . . . . . . . . . . . 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 6323 . . . . . . . . . . . . . . . . . . . . . 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 5883 . . . . . . . . . . . . . . . . . . . . . . . 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 6821 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 2nd `  <. f ,  g
>. )  =  g
116114, 115syl6eq 2521 . . . . . . . . . . . . . . . . . . . . . . 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 6323 . . . . . . . . . . . . . . . . . . . . . 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 2513 . . . . . . . . . . . . . . . . . . . . 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 541 . . . . . . . . . . . . . . . . . . . 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 441 . . . . . . . . . . . . . . . . . . 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 1050 . . . . . . . . . . . . . . . . . 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 1292 . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . 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 470 . . . . . . . . . . . . . . 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 35749 . . . . . . . . . . . . . 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 1230 . . . . . . . . . . . . 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 1788 . . . . . . . . . . . 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 225 . . . . . . . . . . 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 441 . . . . . . . . . 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 1788 . . . . . . . . 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 225 . . . . . . . 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 438 . . . . . . 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 667 . . . . . 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 2877 . . . . 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 436 . . . 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 735 . . 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 681 . 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 35748 . 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 2918 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 376    /\ w3a 1007    = wceq 1452   E.wex 1671    e. wcel 1904    =/= wne 2641   E.wrex 2757   _Vcvv 3031   <.cop 3965   class class class wbr 4395   {copab 4453    X. cxp 4837   ` cfv 5589  (class class class)co 6308   omcom 6711   1stc1st 6810   2ndc2nd 6811    ~~ cen 7584    ~< csdm 7586   Fincfn 7587   0cc0 9557   1c1 9558    x. cmul 9562    - cmin 9880   NNcn 10631   2c2 10681   ZZcz 10961   QQcq 11287   ...cfz 11810    mod cmo 12129   ^cexp 12310   sqrcsqrt 13373   abscabs 13374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-omul 7205  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-acn 8394  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-ico 11666  df-fz 11811  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-dvds 14383  df-gcd 14548  df-numer 14763  df-denom 14764
This theorem is referenced by:  pellqrex  35797
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