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Theorem irrapxlem6 27015
Description: Lemma for irrapx1 27016. Explicit description of a non-closed set. (Contributed by Stefan O'Rear, 13-Sep-2014.)
Assertion
Ref Expression
irrapxlem6  |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  E. x  e.  { y  e.  QQ  |  ( 0  < 
y  /\  ( abs `  ( y  -  A
) )  <  (
(denom `  y ) ^ -u 2 ) ) }  ( abs `  (
x  -  A ) )  <  B )
Distinct variable groups:    x, A, y    x, B, y

Proof of Theorem irrapxlem6
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 irrapxlem5 27014 . 2  |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  E. a  e.  QQ  ( 0  < 
a  /\  ( abs `  ( a  -  A
) )  <  B  /\  ( abs `  (
a  -  A ) )  <  ( (denom `  a ) ^ -u 2
) ) )
2 simplr 731 . . . . . 6  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+ )  /\  a  e.  QQ )  /\  ( 0  < 
a  /\  ( abs `  ( a  -  A
) )  <  B  /\  ( abs `  (
a  -  A ) )  <  ( (denom `  a ) ^ -u 2
) ) )  -> 
a  e.  QQ )
3 simpr1 961 . . . . . . 7  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+ )  /\  a  e.  QQ )  /\  ( 0  < 
a  /\  ( abs `  ( a  -  A
) )  <  B  /\  ( abs `  (
a  -  A ) )  <  ( (denom `  a ) ^ -u 2
) ) )  -> 
0  <  a )
4 simpr3 963 . . . . . . 7  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+ )  /\  a  e.  QQ )  /\  ( 0  < 
a  /\  ( abs `  ( a  -  A
) )  <  B  /\  ( abs `  (
a  -  A ) )  <  ( (denom `  a ) ^ -u 2
) ) )  -> 
( abs `  (
a  -  A ) )  <  ( (denom `  a ) ^ -u 2
) )
53, 4jca 518 . . . . . 6  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+ )  /\  a  e.  QQ )  /\  ( 0  < 
a  /\  ( abs `  ( a  -  A
) )  <  B  /\  ( abs `  (
a  -  A ) )  <  ( (denom `  a ) ^ -u 2
) ) )  -> 
( 0  <  a  /\  ( abs `  (
a  -  A ) )  <  ( (denom `  a ) ^ -u 2
) ) )
6 breq2 4043 . . . . . . . 8  |-  ( y  =  a  ->  (
0  <  y  <->  0  <  a ) )
7 oveq1 5881 . . . . . . . . . 10  |-  ( y  =  a  ->  (
y  -  A )  =  ( a  -  A ) )
87fveq2d 5545 . . . . . . . . 9  |-  ( y  =  a  ->  ( abs `  ( y  -  A ) )  =  ( abs `  (
a  -  A ) ) )
9 fveq2 5541 . . . . . . . . . 10  |-  ( y  =  a  ->  (denom `  y )  =  (denom `  a ) )
109oveq1d 5889 . . . . . . . . 9  |-  ( y  =  a  ->  (
(denom `  y ) ^ -u 2 )  =  ( (denom `  a
) ^ -u 2
) )
118, 10breq12d 4052 . . . . . . . 8  |-  ( y  =  a  ->  (
( abs `  (
y  -  A ) )  <  ( (denom `  y ) ^ -u 2
)  <->  ( abs `  (
a  -  A ) )  <  ( (denom `  a ) ^ -u 2
) ) )
126, 11anbi12d 691 . . . . . . 7  |-  ( y  =  a  ->  (
( 0  <  y  /\  ( abs `  (
y  -  A ) )  <  ( (denom `  y ) ^ -u 2
) )  <->  ( 0  <  a  /\  ( abs `  ( a  -  A ) )  < 
( (denom `  a
) ^ -u 2
) ) ) )
1312elrab 2936 . . . . . 6  |-  ( a  e.  { y  e.  QQ  |  ( 0  <  y  /\  ( abs `  ( y  -  A ) )  < 
( (denom `  y
) ^ -u 2
) ) }  <->  ( a  e.  QQ  /\  ( 0  <  a  /\  ( abs `  ( a  -  A ) )  < 
( (denom `  a
) ^ -u 2
) ) ) )
142, 5, 13sylanbrc 645 . . . . 5  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+ )  /\  a  e.  QQ )  /\  ( 0  < 
a  /\  ( abs `  ( a  -  A
) )  <  B  /\  ( abs `  (
a  -  A ) )  <  ( (denom `  a ) ^ -u 2
) ) )  -> 
a  e.  { y  e.  QQ  |  ( 0  <  y  /\  ( abs `  ( y  -  A ) )  <  ( (denom `  y ) ^ -u 2
) ) } )
15 simpr2 962 . . . . 5  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+ )  /\  a  e.  QQ )  /\  ( 0  < 
a  /\  ( abs `  ( a  -  A
) )  <  B  /\  ( abs `  (
a  -  A ) )  <  ( (denom `  a ) ^ -u 2
) ) )  -> 
( abs `  (
a  -  A ) )  <  B )
16 oveq1 5881 . . . . . . . 8  |-  ( x  =  a  ->  (
x  -  A )  =  ( a  -  A ) )
1716fveq2d 5545 . . . . . . 7  |-  ( x  =  a  ->  ( abs `  ( x  -  A ) )  =  ( abs `  (
a  -  A ) ) )
1817breq1d 4049 . . . . . 6  |-  ( x  =  a  ->  (
( abs `  (
x  -  A ) )  <  B  <->  ( abs `  ( a  -  A
) )  <  B
) )
1918rspcev 2897 . . . . 5  |-  ( ( a  e.  { y  e.  QQ  |  ( 0  <  y  /\  ( abs `  ( y  -  A ) )  <  ( (denom `  y ) ^ -u 2
) ) }  /\  ( abs `  ( a  -  A ) )  <  B )  ->  E. x  e.  { y  e.  QQ  |  ( 0  <  y  /\  ( abs `  ( y  -  A ) )  <  ( (denom `  y ) ^ -u 2
) ) }  ( abs `  ( x  -  A ) )  < 
B )
2014, 15, 19syl2anc 642 . . . 4  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+ )  /\  a  e.  QQ )  /\  ( 0  < 
a  /\  ( abs `  ( a  -  A
) )  <  B  /\  ( abs `  (
a  -  A ) )  <  ( (denom `  a ) ^ -u 2
) ) )  ->  E. x  e.  { y  e.  QQ  |  ( 0  <  y  /\  ( abs `  ( y  -  A ) )  <  ( (denom `  y ) ^ -u 2
) ) }  ( abs `  ( x  -  A ) )  < 
B )
2120ex 423 . . 3  |-  ( ( ( A  e.  RR+  /\  B  e.  RR+ )  /\  a  e.  QQ )  ->  ( ( 0  <  a  /\  ( abs `  ( a  -  A ) )  < 
B  /\  ( abs `  ( a  -  A
) )  <  (
(denom `  a ) ^ -u 2 ) )  ->  E. x  e.  {
y  e.  QQ  | 
( 0  <  y  /\  ( abs `  (
y  -  A ) )  <  ( (denom `  y ) ^ -u 2
) ) }  ( abs `  ( x  -  A ) )  < 
B ) )
2221rexlimdva 2680 . 2  |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  ( E. a  e.  QQ  ( 0  <  a  /\  ( abs `  (
a  -  A ) )  <  B  /\  ( abs `  ( a  -  A ) )  <  ( (denom `  a ) ^ -u 2
) )  ->  E. x  e.  { y  e.  QQ  |  ( 0  < 
y  /\  ( abs `  ( y  -  A
) )  <  (
(denom `  y ) ^ -u 2 ) ) }  ( abs `  (
x  -  A ) )  <  B ) )
231, 22mpd 14 1  |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  E. x  e.  { y  e.  QQ  |  ( 0  < 
y  /\  ( abs `  ( y  -  A
) )  <  (
(denom `  y ) ^ -u 2 ) ) }  ( abs `  (
x  -  A ) )  <  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1696   E.wrex 2557   {crab 2560   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   0cc0 8753    < clt 8883    - cmin 9053   -ucneg 9054   2c2 9811   QQcq 10332   RR+crp 10370   ^cexp 11120   abscabs 11735  denomcdenom 12821
This theorem is referenced by:  irrapx1  27016
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-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-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-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-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-card 7588  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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