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Theorem irrapxlem6 35124
Description: Lemma for irrapx1 35125. 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 simplr 754 . . . 4  |-  ( ( ( ( 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 )
2 simpr1 1003 . . . . 5  |-  ( ( ( ( A  e.  RR+  /\  B  e.  RR+ )  /\  a  e.  QQ )  /\  ( 0  < 
a  /\  ( abs `  ( a  -  A
) )  <  B  /\  ( abs `  (
a  -  A ) )  <  ( (denom `  a ) ^ -u 2
) ) )  -> 
0  <  a )
3 simpr3 1005 . . . . 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 ) )  <  ( (denom `  a ) ^ -u 2
) )
42, 3jca 530 . . . 4  |-  ( ( ( ( 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
) ) )
5 breq2 4399 . . . . . 6  |-  ( y  =  a  ->  (
0  <  y  <->  0  <  a ) )
6 oveq1 6285 . . . . . . . 8  |-  ( y  =  a  ->  (
y  -  A )  =  ( a  -  A ) )
76fveq2d 5853 . . . . . . 7  |-  ( y  =  a  ->  ( abs `  ( y  -  A ) )  =  ( abs `  (
a  -  A ) ) )
8 fveq2 5849 . . . . . . . 8  |-  ( y  =  a  ->  (denom `  y )  =  (denom `  a ) )
98oveq1d 6293 . . . . . . 7  |-  ( y  =  a  ->  (
(denom `  y ) ^ -u 2 )  =  ( (denom `  a
) ^ -u 2
) )
107, 9breq12d 4408 . . . . . 6  |-  ( y  =  a  ->  (
( abs `  (
y  -  A ) )  <  ( (denom `  y ) ^ -u 2
)  <->  ( abs `  (
a  -  A ) )  <  ( (denom `  a ) ^ -u 2
) ) )
115, 10anbi12d 709 . . . . 5  |-  ( y  =  a  ->  (
( 0  <  y  /\  ( abs `  (
y  -  A ) )  <  ( (denom `  y ) ^ -u 2
) )  <->  ( 0  <  a  /\  ( abs `  ( a  -  A ) )  < 
( (denom `  a
) ^ -u 2
) ) ) )
1211elrab 3207 . . . 4  |-  ( 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
) ) ) )
131, 4, 12sylanbrc 662 . . 3  |-  ( ( ( ( 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
) ) } )
14 simpr2 1004 . . 3  |-  ( ( ( ( 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 )
15 oveq1 6285 . . . . . 6  |-  ( x  =  a  ->  (
x  -  A )  =  ( a  -  A ) )
1615fveq2d 5853 . . . . 5  |-  ( x  =  a  ->  ( abs `  ( x  -  A ) )  =  ( abs `  (
a  -  A ) ) )
1716breq1d 4405 . . . 4  |-  ( x  =  a  ->  (
( abs `  (
x  -  A ) )  <  B  <->  ( abs `  ( a  -  A
) )  <  B
) )
1817rspcev 3160 . . 3  |-  ( ( 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 )
1913, 14, 18syl2anc 659 . 2  |-  ( ( ( ( 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 )
20 irrapxlem5 35123 . 2  |-  ( ( A  e.  RR+  /\  B  e.  RR+ )  ->  E. a  e.  QQ  ( 0  < 
a  /\  ( abs `  ( a  -  A
) )  <  B  /\  ( abs `  (
a  -  A ) )  <  ( (denom `  a ) ^ -u 2
) ) )
2119, 20r19.29a 2949 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    e. wcel 1842   E.wrex 2755   {crab 2758   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   0cc0 9522    < clt 9658    - cmin 9841   -ucneg 9842   2c2 10626   QQcq 11227   RR+crp 11265   ^cexp 12210   abscabs 13216  denomcdenom 14476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-card 8352  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-q 11228  df-rp 11266  df-ico 11588  df-fz 11727  df-fl 11966  df-mod 12035  df-seq 12152  df-exp 12211  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-dvds 14196  df-gcd 14354  df-numer 14477  df-denom 14478
This theorem is referenced by:  irrapx1  35125
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