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Theorem irrapxlem6 30356
Description: Lemma for irrapx1 30357. 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 30355 . 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 754 . . . . . 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 997 . . . . . . 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 999 . . . . . . 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 532 . . . . . 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 4446 . . . . . . . 8  |-  ( y  =  a  ->  (
0  <  y  <->  0  <  a ) )
7 oveq1 6284 . . . . . . . . . 10  |-  ( y  =  a  ->  (
y  -  A )  =  ( a  -  A ) )
87fveq2d 5863 . . . . . . . . 9  |-  ( y  =  a  ->  ( abs `  ( y  -  A ) )  =  ( abs `  (
a  -  A ) ) )
9 fveq2 5859 . . . . . . . . . 10  |-  ( y  =  a  ->  (denom `  y )  =  (denom `  a ) )
109oveq1d 6292 . . . . . . . . 9  |-  ( y  =  a  ->  (
(denom `  y ) ^ -u 2 )  =  ( (denom `  a
) ^ -u 2
) )
118, 10breq12d 4455 . . . . . . . 8  |-  ( y  =  a  ->  (
( abs `  (
y  -  A ) )  <  ( (denom `  y ) ^ -u 2
)  <->  ( abs `  (
a  -  A ) )  <  ( (denom `  a ) ^ -u 2
) ) )
126, 11anbi12d 710 . . . . . . 7  |-  ( y  =  a  ->  (
( 0  <  y  /\  ( abs `  (
y  -  A ) )  <  ( (denom `  y ) ^ -u 2
) )  <->  ( 0  <  a  /\  ( abs `  ( a  -  A ) )  < 
( (denom `  a
) ^ -u 2
) ) ) )
1312elrab 3256 . . . . . 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 664 . . . . 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 998 . . . . 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 6284 . . . . . . . 8  |-  ( x  =  a  ->  (
x  -  A )  =  ( a  -  A ) )
1716fveq2d 5863 . . . . . . 7  |-  ( x  =  a  ->  ( abs `  ( x  -  A ) )  =  ( abs `  (
a  -  A ) ) )
1817breq1d 4452 . . . . . 6  |-  ( x  =  a  ->  (
( abs `  (
x  -  A ) )  <  B  <->  ( abs `  ( a  -  A
) )  <  B
) )
1918rspcev 3209 . . . . 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 661 . . . 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 434 . . 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 2950 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    e. wcel 1762   E.wrex 2810   {crab 2813   class class class wbr 4442   ` cfv 5581  (class class class)co 6277   0cc0 9483    < clt 9619    - cmin 9796   -ucneg 9797   2c2 10576   QQcq 11173   RR+crp 11211   ^cexp 12124   abscabs 13019  denomcdenom 14117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-sup 7892  df-card 8311  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-n0 10787  df-z 10856  df-uz 11074  df-q 11174  df-rp 11212  df-ico 11526  df-fz 11664  df-fl 11888  df-mod 11955  df-seq 12066  df-exp 12125  df-hash 12363  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-dvds 13839  df-gcd 13995  df-numer 14118  df-denom 14119
This theorem is referenced by:  irrapx1  30357
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