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Theorem irrapxlem6 26912
Description: Lemma for irrapx1 26913. 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 26911 . 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 4027 . . . . . . . 8  |-  ( y  =  a  ->  (
0  <  y  <->  0  <  a ) )
7 oveq1 5865 . . . . . . . . . 10  |-  ( y  =  a  ->  (
y  -  A )  =  ( a  -  A ) )
87fveq2d 5529 . . . . . . . . 9  |-  ( y  =  a  ->  ( abs `  ( y  -  A ) )  =  ( abs `  (
a  -  A ) ) )
9 fveq2 5525 . . . . . . . . . 10  |-  ( y  =  a  ->  (denom `  y )  =  (denom `  a ) )
109oveq1d 5873 . . . . . . . . 9  |-  ( y  =  a  ->  (
(denom `  y ) ^ -u 2 )  =  ( (denom `  a
) ^ -u 2
) )
118, 10breq12d 4036 . . . . . . . 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 2923 . . . . . 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 5865 . . . . . . . 8  |-  ( x  =  a  ->  (
x  -  A )  =  ( a  -  A ) )
1716fveq2d 5529 . . . . . . 7  |-  ( x  =  a  ->  ( abs `  ( x  -  A ) )  =  ( abs `  (
a  -  A ) ) )
1817breq1d 4033 . . . . . 6  |-  ( x  =  a  ->  (
( abs `  (
x  -  A ) )  <  B  <->  ( abs `  ( a  -  A
) )  <  B
) )
1918rspcev 2884 . . . . 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 2667 . 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 1684   E.wrex 2544   {crab 2547   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   0cc0 8737    < clt 8867    - cmin 9037   -ucneg 9038   2c2 9795   QQcq 10316   RR+crp 10354   ^cexp 11104   abscabs 11719  denomcdenom 12805
This theorem is referenced by:  irrapx1  26913
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-ico 10662  df-fz 10783  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-numer 12806  df-denom 12807
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