Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pellexlem3 Unicode version

Theorem pellexlem3 26422
Description: Lemma for pellex 26426. To each good rational approximation of  ( sqr `  D
), there exists a near-solution. (Contributed by Stefan O'Rear, 14-Sep-2014.)
Assertion
Ref Expression
pellexlem3  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) }  ~<_  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) } )
Distinct variable group:    x, D, y, z

Proof of Theorem pellexlem3
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnex 9899 . . . 4  |-  NN  e.  _V
21, 1xpex 4904 . . 3  |-  ( NN 
X.  NN )  e. 
_V
3 opabssxp 4865 . . 3  |-  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) }  C_  ( NN  X.  NN )
42, 3ssexi 4261 . 2  |-  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) }  e.  _V
5 simprl 732 . . . . . . . 8  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  a  e.  QQ )
6 simprrl 740 . . . . . . . 8  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  0  <  a )
7 qgt0numnn 13030 . . . . . . . 8  |-  ( ( a  e.  QQ  /\  0  <  a )  -> 
(numer `  a )  e.  NN )
85, 6, 7syl2anc 642 . . . . . . 7  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  (numer `  a )  e.  NN )
9 qdencl 13020 . . . . . . . 8  |-  ( a  e.  QQ  ->  (denom `  a )  e.  NN )
105, 9syl 15 . . . . . . 7  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  (denom `  a )  e.  NN )
118, 10jca 518 . . . . . 6  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  (
(numer `  a )  e.  NN  /\  (denom `  a )  e.  NN ) )
12 simpll 730 . . . . . . 7  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  D  e.  NN )
13 simplr 731 . . . . . . 7  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  -.  ( sqr `  D )  e.  QQ )
14 pellexlem1 26420 . . . . . . 7  |-  ( ( ( D  e.  NN  /\  (numer `  a )  e.  NN  /\  (denom `  a )  e.  NN )  /\  -.  ( sqr `  D )  e.  QQ )  ->  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) )  =/=  0 )
1512, 8, 10, 13, 14syl31anc 1186 . . . . . 6  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  (
( (numer `  a
) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) )  =/=  0 )
16 simprrr 741 . . . . . . . 8  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) )
17 qeqnumdivden 13025 . . . . . . . . . . . 12  |-  ( a  e.  QQ  ->  a  =  ( (numer `  a )  /  (denom `  a ) ) )
1817oveq1d 5996 . . . . . . . . . . 11  |-  ( a  e.  QQ  ->  (
a  -  ( sqr `  D ) )  =  ( ( (numer `  a )  /  (denom `  a ) )  -  ( sqr `  D ) ) )
1918fveq2d 5636 . . . . . . . . . 10  |-  ( a  e.  QQ  ->  ( abs `  ( a  -  ( sqr `  D ) ) )  =  ( abs `  ( ( (numer `  a )  /  (denom `  a )
)  -  ( sqr `  D ) ) ) )
2019breq1d 4135 . . . . . . . . 9  |-  ( a  e.  QQ  ->  (
( abs `  (
a  -  ( sqr `  D ) ) )  <  ( (denom `  a ) ^ -u 2
)  <->  ( abs `  (
( (numer `  a
)  /  (denom `  a ) )  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) )
215, 20syl 15 . . . . . . . 8  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  (
( abs `  (
a  -  ( sqr `  D ) ) )  <  ( (denom `  a ) ^ -u 2
)  <->  ( abs `  (
( (numer `  a
)  /  (denom `  a ) )  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) )
2216, 21mpbid 201 . . . . . . 7  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  ( abs `  ( ( (numer `  a )  /  (denom `  a ) )  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) )
23 pellexlem2 26421 . . . . . . 7  |-  ( ( ( D  e.  NN  /\  (numer `  a )  e.  NN  /\  (denom `  a )  e.  NN )  /\  ( abs `  (
( (numer `  a
)  /  (denom `  a ) )  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) )  ->  ( abs `  (
( (numer `  a
) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) )
2412, 8, 10, 22, 23syl31anc 1186 . . . . . 6  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) )
2511, 15, 24jca32 521 . . . . 5  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )  ->  (
( (numer `  a
)  e.  NN  /\  (denom `  a )  e.  NN )  /\  (
( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) ) ) )
2625ex 423 . . . 4  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  ( ( a  e.  QQ  /\  (
0  <  a  /\  ( abs `  ( a  -  ( sqr `  D
) ) )  < 
( (denom `  a
) ^ -u 2
) ) )  -> 
( ( (numer `  a )  e.  NN  /\  (denom `  a )  e.  NN )  /\  (
( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) ) ) ) )
27 breq2 4129 . . . . . 6  |-  ( x  =  a  ->  (
0  <  x  <->  0  <  a ) )
28 oveq1 5988 . . . . . . . 8  |-  ( x  =  a  ->  (
x  -  ( sqr `  D ) )  =  ( a  -  ( sqr `  D ) ) )
2928fveq2d 5636 . . . . . . 7  |-  ( x  =  a  ->  ( abs `  ( x  -  ( sqr `  D ) ) )  =  ( abs `  ( a  -  ( sqr `  D
) ) ) )
30 fveq2 5632 . . . . . . . 8  |-  ( x  =  a  ->  (denom `  x )  =  (denom `  a ) )
3130oveq1d 5996 . . . . . . 7  |-  ( x  =  a  ->  (
(denom `  x ) ^ -u 2 )  =  ( (denom `  a
) ^ -u 2
) )
3229, 31breq12d 4138 . . . . . 6  |-  ( x  =  a  ->  (
( abs `  (
x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
)  <->  ( abs `  (
a  -  ( sqr `  D ) ) )  <  ( (denom `  a ) ^ -u 2
) ) )
3327, 32anbi12d 691 . . . . 5  |-  ( x  =  a  ->  (
( 0  <  x  /\  ( abs `  (
x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) )  <->  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )
3433elrab 3009 . . . 4  |-  ( a  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) }  <->  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )
35 fvex 5646 . . . . 5  |-  (numer `  a )  e.  _V
36 fvex 5646 . . . . 5  |-  (denom `  a )  e.  _V
37 eleq1 2426 . . . . . . 7  |-  ( y  =  (numer `  a
)  ->  ( y  e.  NN  <->  (numer `  a )  e.  NN ) )
3837anbi1d 685 . . . . . 6  |-  ( y  =  (numer `  a
)  ->  ( (
y  e.  NN  /\  z  e.  NN )  <->  ( (numer `  a )  e.  NN  /\  z  e.  NN ) ) )
39 oveq1 5988 . . . . . . . . 9  |-  ( y  =  (numer `  a
)  ->  ( y ^ 2 )  =  ( (numer `  a
) ^ 2 ) )
4039oveq1d 5996 . . . . . . . 8  |-  ( y  =  (numer `  a
)  ->  ( (
y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )
4140neeq1d 2542 . . . . . . 7  |-  ( y  =  (numer `  a
)  ->  ( (
( y ^ 2 )  -  ( D  x.  ( z ^
2 ) ) )  =/=  0  <->  ( (
(numer `  a ) ^ 2 )  -  ( D  x.  (
z ^ 2 ) ) )  =/=  0
) )
4240fveq2d 5636 . . . . . . . 8  |-  ( y  =  (numer `  a
)  ->  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  =  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  (
z ^ 2 ) ) ) ) )
4342breq1d 4135 . . . . . . 7  |-  ( y  =  (numer `  a
)  ->  ( ( abs `  ( ( y ^ 2 )  -  ( D  x.  (
z ^ 2 ) ) ) )  < 
( 1  +  ( 2  x.  ( sqr `  D ) ) )  <-> 
( abs `  (
( (numer `  a
) ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) ) )
4441, 43anbi12d 691 . . . . . 6  |-  ( y  =  (numer `  a
)  ->  ( (
( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  (
( y ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D
) ) ) )  <-> 
( ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) )  =/=  0  /\  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) )
4538, 44anbi12d 691 . . . . 5  |-  ( y  =  (numer `  a
)  ->  ( (
( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) )  <->  ( ( (numer `  a )  e.  NN  /\  z  e.  NN )  /\  ( ( ( (numer `  a ) ^ 2 )  -  ( D  x.  (
z ^ 2 ) ) )  =/=  0  /\  ( abs `  (
( (numer `  a
) ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) ) ) ) )
46 eleq1 2426 . . . . . . 7  |-  ( z  =  (denom `  a
)  ->  ( z  e.  NN  <->  (denom `  a )  e.  NN ) )
4746anbi2d 684 . . . . . 6  |-  ( z  =  (denom `  a
)  ->  ( (
(numer `  a )  e.  NN  /\  z  e.  NN )  <->  ( (numer `  a )  e.  NN  /\  (denom `  a )  e.  NN ) ) )
48 oveq1 5988 . . . . . . . . . 10  |-  ( z  =  (denom `  a
)  ->  ( z ^ 2 )  =  ( (denom `  a
) ^ 2 ) )
4948oveq2d 5997 . . . . . . . . 9  |-  ( z  =  (denom `  a
)  ->  ( D  x.  ( z ^ 2 ) )  =  ( D  x.  ( (denom `  a ) ^ 2 ) ) )
5049oveq2d 5997 . . . . . . . 8  |-  ( z  =  (denom `  a
)  ->  ( (
(numer `  a ) ^ 2 )  -  ( D  x.  (
z ^ 2 ) ) )  =  ( ( (numer `  a
) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )
5150neeq1d 2542 . . . . . . 7  |-  ( z  =  (denom `  a
)  ->  ( (
( (numer `  a
) ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  <->  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) )  =/=  0 ) )
5250fveq2d 5636 . . . . . . . 8  |-  ( z  =  (denom `  a
)  ->  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )  =  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) ) )
5352breq1d 4135 . . . . . . 7  |-  ( z  =  (denom `  a
)  ->  ( ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D
) ) )  <->  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) ) )
5451, 53anbi12d 691 . . . . . 6  |-  ( z  =  (denom `  a
)  ->  ( (
( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) )  =/=  0  /\  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D
) ) ) )  <-> 
( ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) ) ) )
5547, 54anbi12d 691 . . . . 5  |-  ( z  =  (denom `  a
)  ->  ( (
( (numer `  a
)  e.  NN  /\  z  e.  NN )  /\  ( ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) )  =/=  0  /\  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D
) ) ) ) )  <->  ( ( (numer `  a )  e.  NN  /\  (denom `  a )  e.  NN )  /\  (
( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  ( 1  +  ( 2  x.  ( sqr `  D ) ) ) ) ) ) )
5635, 36, 45, 55opelopab 4389 . . . 4  |-  ( <.
(numer `  a ) ,  (denom `  a ) >.  e.  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) }  <->  ( (
(numer `  a )  e.  NN  /\  (denom `  a )  e.  NN )  /\  ( ( ( (numer `  a ) ^ 2 )  -  ( D  x.  (
(denom `  a ) ^ 2 ) ) )  =/=  0  /\  ( abs `  (
( (numer `  a
) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) )
5726, 34, 563imtr4g 261 . . 3  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  ( a  e. 
{ x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) }  ->  <.
(numer `  a ) ,  (denom `  a ) >.  e.  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) } ) )
58 ssrab2 3344 . . . . . 6  |-  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) }  C_  QQ
59 simprl 732 . . . . . 6  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e. 
{ x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) }  /\  b  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) } ) )  -> 
a  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) } )
6058, 59sseldi 3264 . . . . 5  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e. 
{ x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) }  /\  b  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) } ) )  -> 
a  e.  QQ )
61 simprr 733 . . . . . 6  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e. 
{ x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) }  /\  b  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) } ) )  -> 
b  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) } )
6258, 61sseldi 3264 . . . . 5  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e. 
{ x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) }  /\  b  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) } ) )  -> 
b  e.  QQ )
6335, 36opth 4348 . . . . . . 7  |-  ( <.
(numer `  a ) ,  (denom `  a ) >.  =  <. (numer `  b
) ,  (denom `  b ) >.  <->  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )
64 simprl 732 . . . . . . . . . 10  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  (numer `  a
)  =  (numer `  b ) )
65 simprr 733 . . . . . . . . . 10  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  (denom `  a
)  =  (denom `  b ) )
6664, 65oveq12d 5999 . . . . . . . . 9  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  ( (numer `  a )  /  (denom `  a ) )  =  ( (numer `  b
)  /  (denom `  b ) ) )
67 simpll 730 . . . . . . . . . 10  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  a  e.  QQ )
6867, 17syl 15 . . . . . . . . 9  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  a  =  ( (numer `  a )  /  (denom `  a )
) )
69 simplr 731 . . . . . . . . . 10  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  b  e.  QQ )
70 qeqnumdivden 13025 . . . . . . . . . 10  |-  ( b  e.  QQ  ->  b  =  ( (numer `  b )  /  (denom `  b ) ) )
7169, 70syl 15 . . . . . . . . 9  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  b  =  ( (numer `  b )  /  (denom `  b )
) )
7266, 68, 713eqtr4d 2408 . . . . . . . 8  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  a  =  b )
7372ex 423 . . . . . . 7  |-  ( ( a  e.  QQ  /\  b  e.  QQ )  ->  ( ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) )  -> 
a  =  b ) )
7463, 73syl5bi 208 . . . . . 6  |-  ( ( a  e.  QQ  /\  b  e.  QQ )  ->  ( <. (numer `  a
) ,  (denom `  a ) >.  =  <. (numer `  b ) ,  (denom `  b ) >.  ->  a  =  b ) )
75 fveq2 5632 . . . . . . 7  |-  ( a  =  b  ->  (numer `  a )  =  (numer `  b ) )
76 fveq2 5632 . . . . . . 7  |-  ( a  =  b  ->  (denom `  a )  =  (denom `  b ) )
7775, 76opeq12d 3906 . . . . . 6  |-  ( a  =  b  ->  <. (numer `  a ) ,  (denom `  a ) >.  =  <. (numer `  b ) ,  (denom `  b ) >. )
7874, 77impbid1 194 . . . . 5  |-  ( ( a  e.  QQ  /\  b  e.  QQ )  ->  ( <. (numer `  a
) ,  (denom `  a ) >.  =  <. (numer `  b ) ,  (denom `  b ) >.  <->  a  =  b ) )
7960, 62, 78syl2anc 642 . . . 4  |-  ( ( ( D  e.  NN  /\ 
-.  ( sqr `  D
)  e.  QQ )  /\  ( a  e. 
{ x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) }  /\  b  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) } ) )  -> 
( <. (numer `  a
) ,  (denom `  a ) >.  =  <. (numer `  b ) ,  (denom `  b ) >.  <->  a  =  b ) )
8079ex 423 . . 3  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  ( ( a  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) }  /\  b  e. 
{ x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) } )  ->  ( <. (numer `  a ) ,  (denom `  a ) >.  =  <. (numer `  b ) ,  (denom `  b ) >.  <->  a  =  b ) ) )
8157, 80dom2d 7045 . 2  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  ( { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) }  e.  _V  ->  { x  e.  QQ  |  ( 0  < 
x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) ) }  ~<_  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) } ) )
824, 81mpi 16 1  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) }  ~<_  { <. y ,  z >.  |  ( ( y  e.  NN  /\  z  e.  NN )  /\  ( ( ( y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  /\  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  <  (
1  +  ( 2  x.  ( sqr `  D
) ) ) ) ) } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715    =/= wne 2529   {crab 2632   _Vcvv 2873   <.cop 3732   class class class wbr 4125   {copab 4178    X. cxp 4790   ` cfv 5358  (class class class)co 5981    ~<_ cdom 7004   0cc0 8884   1c1 8885    + caddc 8887    x. cmul 8889    < clt 9014    - cmin 9184   -ucneg 9185    / cdiv 9570   NNcn 9893   2c2 9942   QQcq 10467   ^cexp 11269   sqrcsqr 11925   abscabs 11926  numercnumer 13012  denomcdenom 13013
This theorem is referenced by:  pellexlem4  26423
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-sup 7341  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-2 9951  df-3 9952  df-n0 10115  df-z 10176  df-uz 10382  df-q 10468  df-rp 10506  df-fl 11089  df-mod 11138  df-seq 11211  df-exp 11270  df-cj 11791  df-re 11792  df-im 11793  df-sqr 11927  df-abs 11928  df-dvds 12740  df-gcd 12894  df-numer 13014  df-denom 13015
  Copyright terms: Public domain W3C validator