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Theorem pellexlem3 26792
Description: Lemma for pellex 26796. 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 9970 . . . 4  |-  NN  e.  _V
21, 1xpex 4957 . . 3  |-  ( NN 
X.  NN )  e. 
_V
3 opabssxp 4917 . . 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 4316 . 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 733 . . . . . . . 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 741 . . . . . . . 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 13106 . . . . . . . 8  |-  ( ( a  e.  QQ  /\  0  <  a )  -> 
(numer `  a )  e.  NN )
85, 6, 7syl2anc 643 . . . . . . 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 13096 . . . . . . . 8  |-  ( a  e.  QQ  ->  (denom `  a )  e.  NN )
105, 9syl 16 . . . . . . 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 519 . . . . . 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 731 . . . . . . 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 732 . . . . . . 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 26790 . . . . . . 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 1187 . . . . . 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 742 . . . . . . . 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 13101 . . . . . . . . . . . 12  |-  ( a  e.  QQ  ->  a  =  ( (numer `  a )  /  (denom `  a ) ) )
1817oveq1d 6063 . . . . . . . . . . 11  |-  ( a  e.  QQ  ->  (
a  -  ( sqr `  D ) )  =  ( ( (numer `  a )  /  (denom `  a ) )  -  ( sqr `  D ) ) )
1918fveq2d 5699 . . . . . . . . . 10  |-  ( a  e.  QQ  ->  ( abs `  ( a  -  ( sqr `  D ) ) )  =  ( abs `  ( ( (numer `  a )  /  (denom `  a )
)  -  ( sqr `  D ) ) ) )
2019breq1d 4190 . . . . . . . . 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 16 . . . . . . . 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 202 . . . . . . 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 26791 . . . . . . 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 1187 . . . . . 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 522 . . . . 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 424 . . . 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 4184 . . . . . 6  |-  ( x  =  a  ->  (
0  <  x  <->  0  <  a ) )
28 oveq1 6055 . . . . . . . 8  |-  ( x  =  a  ->  (
x  -  ( sqr `  D ) )  =  ( a  -  ( sqr `  D ) ) )
2928fveq2d 5699 . . . . . . 7  |-  ( x  =  a  ->  ( abs `  ( x  -  ( sqr `  D ) ) )  =  ( abs `  ( a  -  ( sqr `  D
) ) ) )
30 fveq2 5695 . . . . . . . 8  |-  ( x  =  a  ->  (denom `  x )  =  (denom `  a ) )
3130oveq1d 6063 . . . . . . 7  |-  ( x  =  a  ->  (
(denom `  x ) ^ -u 2 )  =  ( (denom `  a
) ^ -u 2
) )
3229, 31breq12d 4193 . . . . . 6  |-  ( x  =  a  ->  (
( abs `  (
x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
)  <->  ( abs `  (
a  -  ( sqr `  D ) ) )  <  ( (denom `  a ) ^ -u 2
) ) )
3327, 32anbi12d 692 . . . . 5  |-  ( x  =  a  ->  (
( 0  <  x  /\  ( abs `  (
x  -  ( sqr `  D ) ) )  <  ( (denom `  x ) ^ -u 2
) )  <->  ( 0  <  a  /\  ( abs `  ( a  -  ( sqr `  D ) ) )  <  (
(denom `  a ) ^ -u 2 ) ) ) )
3433elrab 3060 . . . 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 5709 . . . . 5  |-  (numer `  a )  e.  _V
36 fvex 5709 . . . . 5  |-  (denom `  a )  e.  _V
37 eleq1 2472 . . . . . . 7  |-  ( y  =  (numer `  a
)  ->  ( y  e.  NN  <->  (numer `  a )  e.  NN ) )
3837anbi1d 686 . . . . . 6  |-  ( y  =  (numer `  a
)  ->  ( (
y  e.  NN  /\  z  e.  NN )  <->  ( (numer `  a )  e.  NN  /\  z  e.  NN ) ) )
39 oveq1 6055 . . . . . . . . 9  |-  ( y  =  (numer `  a
)  ->  ( y ^ 2 )  =  ( (numer `  a
) ^ 2 ) )
4039oveq1d 6063 . . . . . . . 8  |-  ( y  =  (numer `  a
)  ->  ( (
y ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )
4140neeq1d 2588 . . . . . . 7  |-  ( y  =  (numer `  a
)  ->  ( (
( y ^ 2 )  -  ( D  x.  ( z ^
2 ) ) )  =/=  0  <->  ( (
(numer `  a ) ^ 2 )  -  ( D  x.  (
z ^ 2 ) ) )  =/=  0
) )
4240fveq2d 5699 . . . . . . . 8  |-  ( y  =  (numer `  a
)  ->  ( abs `  ( ( y ^
2 )  -  ( D  x.  ( z ^ 2 ) ) ) )  =  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  (
z ^ 2 ) ) ) ) )
4342breq1d 4190 . . . . . . 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 692 . . . . . 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 692 . . . . 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 2472 . . . . . . 7  |-  ( z  =  (denom `  a
)  ->  ( z  e.  NN  <->  (denom `  a )  e.  NN ) )
4746anbi2d 685 . . . . . 6  |-  ( z  =  (denom `  a
)  ->  ( (
(numer `  a )  e.  NN  /\  z  e.  NN )  <->  ( (numer `  a )  e.  NN  /\  (denom `  a )  e.  NN ) ) )
48 oveq1 6055 . . . . . . . . . 10  |-  ( z  =  (denom `  a
)  ->  ( z ^ 2 )  =  ( (denom `  a
) ^ 2 ) )
4948oveq2d 6064 . . . . . . . . 9  |-  ( z  =  (denom `  a
)  ->  ( D  x.  ( z ^ 2 ) )  =  ( D  x.  ( (denom `  a ) ^ 2 ) ) )
5049oveq2d 6064 . . . . . . . 8  |-  ( z  =  (denom `  a
)  ->  ( (
(numer `  a ) ^ 2 )  -  ( D  x.  (
z ^ 2 ) ) )  =  ( ( (numer `  a
) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) )
5150neeq1d 2588 . . . . . . 7  |-  ( z  =  (denom `  a
)  ->  ( (
( (numer `  a
) ^ 2 )  -  ( D  x.  ( z ^ 2 ) ) )  =/=  0  <->  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) )  =/=  0 ) )
5250fveq2d 5699 . . . . . . . 8  |-  ( z  =  (denom `  a
)  ->  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( z ^
2 ) ) ) )  =  ( abs `  ( ( (numer `  a ) ^ 2 )  -  ( D  x.  ( (denom `  a ) ^ 2 ) ) ) ) )
5352breq1d 4190 . . . . . . 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 692 . . . . . 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 692 . . . . 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 4444 . . . 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 262 . . 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 3396 . . . . . 6  |-  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) }  C_  QQ
59 simprl 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 ) ) } ) )  -> 
a  e.  { x  e.  QQ  |  ( 0  <  x  /\  ( abs `  ( x  -  ( sqr `  D ) ) )  <  (
(denom `  x ) ^ -u 2 ) ) } )
6058, 59sseldi 3314 . . . . 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 734 . . . . . 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 3314 . . . . 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 4403 . . . . . . 7  |-  ( <.
(numer `  a ) ,  (denom `  a ) >.  =  <. (numer `  b
) ,  (denom `  b ) >.  <->  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )
64 simprl 733 . . . . . . . . . 10  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  (numer `  a
)  =  (numer `  b ) )
65 simprr 734 . . . . . . . . . 10  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  (denom `  a
)  =  (denom `  b ) )
6664, 65oveq12d 6066 . . . . . . . . 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 731 . . . . . . . . . 10  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  a  e.  QQ )
6867, 17syl 16 . . . . . . . . 9  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  a  =  ( (numer `  a )  /  (denom `  a )
) )
69 simplr 732 . . . . . . . . . 10  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  b  e.  QQ )
70 qeqnumdivden 13101 . . . . . . . . . 10  |-  ( b  e.  QQ  ->  b  =  ( (numer `  b )  /  (denom `  b ) ) )
7169, 70syl 16 . . . . . . . . 9  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  b  =  ( (numer `  b )  /  (denom `  b )
) )
7266, 68, 713eqtr4d 2454 . . . . . . . 8  |-  ( ( ( a  e.  QQ  /\  b  e.  QQ )  /\  ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) ) )  ->  a  =  b )
7372ex 424 . . . . . . 7  |-  ( ( a  e.  QQ  /\  b  e.  QQ )  ->  ( ( (numer `  a )  =  (numer `  b )  /\  (denom `  a )  =  (denom `  b ) )  -> 
a  =  b ) )
7463, 73syl5bi 209 . . . . . 6  |-  ( ( a  e.  QQ  /\  b  e.  QQ )  ->  ( <. (numer `  a
) ,  (denom `  a ) >.  =  <. (numer `  b ) ,  (denom `  b ) >.  ->  a  =  b ) )
75 fveq2 5695 . . . . . . 7  |-  ( a  =  b  ->  (numer `  a )  =  (numer `  b ) )
76 fveq2 5695 . . . . . . 7  |-  ( a  =  b  ->  (denom `  a )  =  (denom `  b ) )
7775, 76opeq12d 3960 . . . . . 6  |-  ( a  =  b  ->  <. (numer `  a ) ,  (denom `  a ) >.  =  <. (numer `  b ) ,  (denom `  b ) >. )
7874, 77impbid1 195 . . . . 5  |-  ( ( a  e.  QQ  /\  b  e.  QQ )  ->  ( <. (numer `  a
) ,  (denom `  a ) >.  =  <. (numer `  b ) ,  (denom `  b ) >.  <->  a  =  b ) )
7960, 62, 78syl2anc 643 . . . 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 424 . . 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 7115 . 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 17 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 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2575   {crab 2678   _Vcvv 2924   <.cop 3785   class class class wbr 4180   {copab 4233    X. cxp 4843   ` cfv 5421  (class class class)co 6048    ~<_ cdom 7074   0cc0 8954   1c1 8955    + caddc 8957    x. cmul 8959    < clt 9084    - cmin 9255   -ucneg 9256    / cdiv 9641   NNcn 9964   2c2 10013   QQcq 10538   ^cexp 11345   sqrcsqr 12001   abscabs 12002  numercnumer 13088  denomcdenom 13089
This theorem is referenced by:  pellexlem4  26793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-n0 10186  df-z 10247  df-uz 10453  df-q 10539  df-rp 10577  df-fl 11165  df-mod 11214  df-seq 11287  df-exp 11346  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-dvds 12816  df-gcd 12970  df-numer 13090  df-denom 13091
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