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Theorem irrapxlem6 26582
Description: Lemma for irrapx1 26583. 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 26581 . 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 732 . . . . . 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 963 . . . . . . 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 965 . . . . . . 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 519 . . . . . 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 4158 . . . . . . . 8  |-  ( y  =  a  ->  (
0  <  y  <->  0  <  a ) )
7 oveq1 6028 . . . . . . . . . 10  |-  ( y  =  a  ->  (
y  -  A )  =  ( a  -  A ) )
87fveq2d 5673 . . . . . . . . 9  |-  ( y  =  a  ->  ( abs `  ( y  -  A ) )  =  ( abs `  (
a  -  A ) ) )
9 fveq2 5669 . . . . . . . . . 10  |-  ( y  =  a  ->  (denom `  y )  =  (denom `  a ) )
109oveq1d 6036 . . . . . . . . 9  |-  ( y  =  a  ->  (
(denom `  y ) ^ -u 2 )  =  ( (denom `  a
) ^ -u 2
) )
118, 10breq12d 4167 . . . . . . . 8  |-  ( y  =  a  ->  (
( abs `  (
y  -  A ) )  <  ( (denom `  y ) ^ -u 2
)  <->  ( abs `  (
a  -  A ) )  <  ( (denom `  a ) ^ -u 2
) ) )
126, 11anbi12d 692 . . . . . . 7  |-  ( y  =  a  ->  (
( 0  <  y  /\  ( abs `  (
y  -  A ) )  <  ( (denom `  y ) ^ -u 2
) )  <->  ( 0  <  a  /\  ( abs `  ( a  -  A ) )  < 
( (denom `  a
) ^ -u 2
) ) ) )
1312elrab 3036 . . . . . 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 646 . . . . 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 964 . . . . 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 6028 . . . . . . . 8  |-  ( x  =  a  ->  (
x  -  A )  =  ( a  -  A ) )
1716fveq2d 5673 . . . . . . 7  |-  ( x  =  a  ->  ( abs `  ( x  -  A ) )  =  ( abs `  (
a  -  A ) ) )
1817breq1d 4164 . . . . . 6  |-  ( x  =  a  ->  (
( abs `  (
x  -  A ) )  <  B  <->  ( abs `  ( a  -  A
) )  <  B
) )
1918rspcev 2996 . . . . 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 643 . . . 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 424 . . 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 2774 . 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 15 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 359    /\ w3a 936    e. wcel 1717   E.wrex 2651   {crab 2654   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   0cc0 8924    < clt 9054    - cmin 9224   -ucneg 9225   2c2 9982   QQcq 10507   RR+crp 10545   ^cexp 11310   abscabs 11967  denomcdenom 13054
This theorem is referenced by:  irrapx1  26583
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-sup 7382  df-card 7760  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-q 10508  df-rp 10546  df-ico 10855  df-fz 10977  df-fl 11130  df-mod 11179  df-seq 11252  df-exp 11311  df-hash 11547  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-dvds 12781  df-gcd 12935  df-numer 13055  df-denom 13056
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