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Theorem pellexlem4 27020
Description: Lemma for pellex 27023. Invoking irrapx1 27016, we have infinitely many near-solutions. (Contributed by Stefan O'Rear, 14-Sep-2014.)
Assertion
Ref Expression
pellexlem4  |-  ( ( 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
) ) ) ) ) }  ~~  NN )
Distinct variable group:    y, D, z

Proof of Theorem pellexlem4
Dummy variable  b is distinct from all other variables.
StepHypRef Expression
1 nnex 9768 . . . . 5  |-  NN  e.  _V
21, 1xpex 4817 . . . 4  |-  ( NN 
X.  NN )  e. 
_V
3 opabssxp 4778 . . . 4  |-  { <. 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 )
4 ssdomg 6923 . . . 4  |-  ( ( NN  X.  NN )  e.  _V  ->  ( { <. 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 )  ->  { <. 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
) ) ) ) ) }  ~<_  ( NN 
X.  NN ) ) )
52, 3, 4mp2 17 . . 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
) ) ) ) ) }  ~<_  ( NN 
X.  NN )
6 xpnnen 12503 . . 3  |-  ( NN 
X.  NN )  ~~  NN
7 domentr 6936 . . 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
) ) ) ) ) }  ~<_  ( NN 
X.  NN )  /\  ( NN  X.  NN )  ~~  NN )  ->  { <. 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
) ) ) ) ) }  ~<_  NN )
85, 6, 7mp2an 653 . 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
) ) ) ) ) }  ~<_  NN
9 nnrp 10379 . . . . . . 7  |-  ( D  e.  NN  ->  D  e.  RR+ )
109rpsqrcld 11910 . . . . . 6  |-  ( D  e.  NN  ->  ( sqr `  D )  e.  RR+ )
1110anim1i 551 . . . . 5  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  ( ( sqr `  D )  e.  RR+  /\ 
-.  ( sqr `  D
)  e.  QQ ) )
12 eldif 3175 . . . . 5  |-  ( ( sqr `  D )  e.  ( RR+  \  QQ ) 
<->  ( ( sqr `  D
)  e.  RR+  /\  -.  ( sqr `  D )  e.  QQ ) )
1311, 12sylibr 203 . . . 4  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  ( sqr `  D
)  e.  ( RR+  \  QQ ) )
14 irrapx1 27016 . . . 4  |-  ( ( sqr `  D )  e.  ( RR+  \  QQ )  ->  { b  e.  QQ  |  ( 0  <  b  /\  ( abs `  ( b  -  ( sqr `  D ) ) )  <  (
(denom `  b ) ^ -u 2 ) ) }  ~~  NN )
15 ensym 6926 . . . 4  |-  ( { b  e.  QQ  | 
( 0  <  b  /\  ( abs `  (
b  -  ( sqr `  D ) ) )  <  ( (denom `  b ) ^ -u 2
) ) }  ~~  NN  ->  NN  ~~  {
b  e.  QQ  | 
( 0  <  b  /\  ( abs `  (
b  -  ( sqr `  D ) ) )  <  ( (denom `  b ) ^ -u 2
) ) } )
1613, 14, 153syl 18 . . 3  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  NN  ~~  {
b  e.  QQ  | 
( 0  <  b  /\  ( abs `  (
b  -  ( sqr `  D ) ) )  <  ( (denom `  b ) ^ -u 2
) ) } )
17 pellexlem3 27019 . . 3  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  { b  e.  QQ  |  ( 0  <  b  /\  ( abs `  ( b  -  ( sqr `  D ) ) )  <  (
(denom `  b ) ^ -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
) ) ) ) ) } )
18 endomtr 6935 . . 3  |-  ( ( NN  ~~  { b  e.  QQ  |  ( 0  <  b  /\  ( abs `  ( b  -  ( sqr `  D
) ) )  < 
( (denom `  b
) ^ -u 2
) ) }  /\  { b  e.  QQ  | 
( 0  <  b  /\  ( abs `  (
b  -  ( sqr `  D ) ) )  <  ( (denom `  b ) ^ -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
) ) ) ) ) } )  ->  NN 
~<_  { <. 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
) ) ) ) ) } )
1916, 17, 18syl2anc 642 . 2  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  NN  ~<_  { <. 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
) ) ) ) ) } )
20 sbth 6997 . 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
) ) ) ) ) }  ~<_  NN  /\  NN 
~<_  { <. 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
) ) ) ) ) } )  ->  { <. 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
) ) ) ) ) }  ~~  NN )
218, 19, 20sylancr 644 1  |-  ( ( 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
) ) ) ) ) }  ~~  NN )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    e. wcel 1696    =/= wne 2459   {crab 2560   _Vcvv 2801    \ cdif 3162    C_ wss 3165   class class class wbr 4039   {copab 4092    X. cxp 4703   ` cfv 5271  (class class class)co 5874    ~~ cen 6876    ~<_ cdom 6877   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    - cmin 9053   -ucneg 9054   NNcn 9762   2c2 9811   QQcq 10332   RR+crp 10370   ^cexp 11120   sqrcsqr 11734   abscabs 11735  denomcdenom 12821
This theorem is referenced by:  pellexlem5  27021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-acn 7591  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-ico 10678  df-fz 10799  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-numer 12822  df-denom 12823
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