Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pellexlem4 Structured version   Unicode version

Theorem pellexlem4 35142
Description: Lemma for pellex 35145. Invoking irrapx1 35138, 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 10584 . . . . 5  |-  NN  e.  _V
21, 1xpex 6588 . . . 4  |-  ( NN 
X.  NN )  e. 
_V
3 opabssxp 4900 . . . 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 7601 . . . 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 14155 . . 3  |-  ( NN 
X.  NN )  ~~  NN
7 domentr 7614 . . 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 11276 . . . . . . 7  |-  ( D  e.  NN  ->  D  e.  RR+ )
109rpsqrtcld 13394 . . . . . 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 3426 . . . . 5  |-  ( ( sqr `  D )  e.  ( RR+  \  QQ ) 
<->  ( ( sqr `  D
)  e.  RR+  /\  -.  ( sqr `  D )  e.  QQ ) )
1311, 12sylibr 214 . . . 4  |-  ( ( D  e.  NN  /\  -.  ( sqr `  D
)  e.  QQ )  ->  ( sqr `  D
)  e.  ( RR+  \  QQ ) )
14 irrapx1 35138 . . . 4  |-  ( ( sqr `  D )  e.  ( RR+  \  QQ )  ->  { b  e.  QQ  |  ( 0  <  b  /\  ( abs `  ( b  -  ( sqr `  D ) ) )  <  (
(denom `  b ) ^ -u 2 ) ) }  ~~  NN )
15 ensym 7604 . . . 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 35141 . . 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 7613 . . 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 7677 . 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 1844    =/= wne 2600   {crab 2760   _Vcvv 3061    \ cdif 3413    C_ wss 3416   class class class wbr 4397   {copab 4454    X. cxp 4823   ` cfv 5571  (class class class)co 6280    ~~ cen 7553    ~<_ cdom 7554   0cc0 9524   1c1 9525    + caddc 9527    x. cmul 9529    < clt 9660    - cmin 9843   -ucneg 9844   NNcn 10578   2c2 10628   QQcq 11229   RR+crp 11267   ^cexp 12212   sqrcsqrt 13217   abscabs 13218  denomcdenom 14478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-inf2 8093  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601  ax-pre-sup 9602
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-se 4785  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-isom 5580  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-omul 7174  df-er 7350  df-map 7461  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-sup 7937  df-oi 7971  df-card 8354  df-acn 8357  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-div 10250  df-nn 10579  df-2 10637  df-3 10638  df-n0 10839  df-z 10908  df-uz 11130  df-q 11230  df-rp 11268  df-ico 11590  df-fz 11729  df-fl 11968  df-mod 12037  df-seq 12154  df-exp 12213  df-hash 12455  df-cj 13083  df-re 13084  df-im 13085  df-sqrt 13219  df-abs 13220  df-dvds 14198  df-gcd 14356  df-numer 14479  df-denom 14480
This theorem is referenced by:  pellexlem5  35143
  Copyright terms: Public domain W3C validator