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Theorem pellexlem4 29308
Description: Lemma for pellex 29311. Invoking irrapx1 29304, 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 10426 . . . . 5  |-  NN  e.  _V
21, 1xpex 6605 . . . 4  |-  ( NN 
X.  NN )  e. 
_V
3 opabssxp 5006 . . . 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 7452 . . . 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 9 . . 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 13590 . . 3  |-  ( NN 
X.  NN )  ~~  NN
7 domentr 7465 . . 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 672 . 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 11098 . . . . . . 7  |-  ( D  e.  NN  ->  D  e.  RR+ )
109rpsqrcld 12997 . . . . . 6  |-  ( D  e.  NN  ->  ( sqr `  D )  e.  RR+ )
1110anim1i 568 . . . . 5  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  ( ( sqr `  D )  e.  RR+  /\ 
-.  ( sqr `  D
)  e.  QQ ) )
12 eldif 3433 . . . . 5  |-  ( ( sqr `  D )  e.  ( RR+  \  QQ ) 
<->  ( ( sqr `  D
)  e.  RR+  /\  -.  ( sqr `  D )  e.  QQ ) )
1311, 12sylibr 212 . . . 4  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  ( sqr `  D
)  e.  ( RR+  \  QQ ) )
14 irrapx1 29304 . . . 4  |-  ( ( sqr `  D )  e.  ( RR+  \  QQ )  ->  { b  e.  QQ  |  ( 0  <  b  /\  ( abs `  ( b  -  ( sqr `  D ) ) )  <  (
(denom `  b ) ^ -u 2 ) ) }  ~~  NN )
15 ensym 7455 . . . 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 20 . . 3  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  NN  ~~  {
b  e.  QQ  | 
( 0  <  b  /\  ( abs `  (
b  -  ( sqr `  D ) ) )  <  ( (denom `  b ) ^ -u 2
) ) } )
17 pellexlem3 29307 . . 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 7464 . . 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 661 . 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 7528 . 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 663 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    e. wcel 1758    =/= wne 2642   {crab 2797   _Vcvv 3065    \ cdif 3420    C_ wss 3423   class class class wbr 4387   {copab 4444    X. cxp 4933   ` cfv 5513  (class class class)co 6187    ~~ cen 7404    ~<_ cdom 7405   0cc0 9380   1c1 9381    + caddc 9383    x. cmul 9385    < clt 9516    - cmin 9693   -ucneg 9694   NNcn 10420   2c2 10469   QQcq 11051   RR+crp 11089   ^cexp 11963   sqrcsqr 12821   abscabs 12822  denomcdenom 13911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-inf2 7945  ax-cnex 9436  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457  ax-pre-sup 9458
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-se 4775  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-isom 5522  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-recs 6929  df-rdg 6963  df-1o 7017  df-oadd 7021  df-omul 7022  df-er 7198  df-map 7313  df-en 7408  df-dom 7409  df-sdom 7410  df-fin 7411  df-sup 7789  df-oi 7822  df-card 8207  df-acn 8210  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-div 10092  df-nn 10421  df-2 10478  df-3 10479  df-n0 10678  df-z 10745  df-uz 10960  df-q 11052  df-rp 11090  df-ico 11404  df-fz 11536  df-fl 11740  df-mod 11807  df-seq 11905  df-exp 11964  df-hash 12202  df-cj 12687  df-re 12688  df-im 12689  df-sqr 12823  df-abs 12824  df-dvds 13635  df-gcd 13790  df-numer 13912  df-denom 13913
This theorem is referenced by:  pellexlem5  29309
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