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Theorem opnrebl2 26315
Description: A set is open in the standard topology of the reals precisely when every point can be enclosed in an arbitrarily small ball. (Contributed by Jeff Hankins, 22-Sep-2013.) (Proof shortened by Mario Carneiro, 30-Jan-2014.)
Assertion
Ref Expression
opnrebl2  |-  ( A  e.  ( topGen `  ran  (,) )  <->  ( A  C_  RR  /\  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  ( z  <_  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )
) )
Distinct variable group:    x, y, z, A

Proof of Theorem opnrebl2
StepHypRef Expression
1 eqid 2435 . . . . 5  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
21rexmet 18814 . . . 4  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( * Met `  RR )
3 eqid 2435 . . . . . 6  |-  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )
41, 3tgioo 18819 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) )
54mopnss 18468 . . . 4  |-  ( ( ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )  e.  ( * Met `  RR )  /\  A  e.  ( topGen `  ran  (,) )
)  ->  A  C_  RR )
62, 5mpan 652 . . 3  |-  ( A  e.  ( topGen `  ran  (,) )  ->  A  C_  RR )
74mopni3 18516 . . . . . . . 8  |-  ( ( ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) )  e.  ( * Met `  RR )  /\  A  e.  ( topGen `  ran  (,) )  /\  x  e.  A
)  /\  y  e.  RR+ )  ->  E. z  e.  RR+  ( z  < 
y  /\  ( x
( ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) z )  C_  A )
)
87ex 424 . . . . . . 7  |-  ( ( ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )  e.  ( * Met `  RR )  /\  A  e.  ( topGen `  ran  (,) )  /\  x  e.  A
)  ->  ( y  e.  RR+  ->  E. z  e.  RR+  ( z  < 
y  /\  ( x
( ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) z )  C_  A )
) )
92, 8mp3an1 1266 . . . . . 6  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  x  e.  A )  ->  (
y  e.  RR+  ->  E. z  e.  RR+  (
z  <  y  /\  ( x ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) z )  C_  A ) ) )
106sselda 3340 . . . . . . 7  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  x  e.  A )  ->  x  e.  RR )
11 rpre 10610 . . . . . . . . . . . . 13  |-  ( z  e.  RR+  ->  z  e.  RR )
121bl2ioo 18815 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( x ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) z )  =  ( ( x  -  z ) (,) (
x  +  z ) ) )
1311, 12sylan2 461 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  z  e.  RR+ )  -> 
( x ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) z )  =  ( ( x  -  z ) (,) (
x  +  z ) ) )
1413sseq1d 3367 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  z  e.  RR+ )  -> 
( ( x (
ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) z )  C_  A  <->  ( (
x  -  z ) (,) ( x  +  z ) )  C_  A ) )
1514anbi2d 685 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  z  e.  RR+ )  -> 
( ( z  < 
y  /\  ( x
( ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) z )  C_  A )  <->  ( z  <  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )
) )
1615rexbidva 2714 . . . . . . . . 9  |-  ( x  e.  RR  ->  ( E. z  e.  RR+  (
z  <  y  /\  ( x ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) z )  C_  A )  <->  E. z  e.  RR+  ( z  < 
y  /\  ( (
x  -  z ) (,) ( x  +  z ) )  C_  A ) ) )
1716biimpd 199 . . . . . . . 8  |-  ( x  e.  RR  ->  ( E. z  e.  RR+  (
z  <  y  /\  ( x ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) z )  C_  A )  ->  E. z  e.  RR+  ( z  < 
y  /\  ( (
x  -  z ) (,) ( x  +  z ) )  C_  A ) ) )
18 rpre 10610 . . . . . . . . . . 11  |-  ( y  e.  RR+  ->  y  e.  RR )
19 ltle 9155 . . . . . . . . . . 11  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( z  <  y  ->  z  <_  y )
)
2011, 18, 19syl2anr 465 . . . . . . . . . 10  |-  ( ( y  e.  RR+  /\  z  e.  RR+ )  ->  (
z  <  y  ->  z  <_  y ) )
2120anim1d 548 . . . . . . . . 9  |-  ( ( y  e.  RR+  /\  z  e.  RR+ )  ->  (
( z  <  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )  ->  ( z  <_  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )
) )
2221reximdva 2810 . . . . . . . 8  |-  ( y  e.  RR+  ->  ( E. z  e.  RR+  (
z  <  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )  ->  E. z  e.  RR+  ( z  <_  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )
) )
2317, 22syl9 68 . . . . . . 7  |-  ( x  e.  RR  ->  (
y  e.  RR+  ->  ( E. z  e.  RR+  ( z  <  y  /\  ( x ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) z )  C_  A )  ->  E. z  e.  RR+  ( z  <_ 
y  /\  ( (
x  -  z ) (,) ( x  +  z ) )  C_  A ) ) ) )
2410, 23syl 16 . . . . . 6  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  x  e.  A )  ->  (
y  e.  RR+  ->  ( E. z  e.  RR+  ( z  <  y  /\  ( x ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) z )  C_  A )  ->  E. z  e.  RR+  ( z  <_ 
y  /\  ( (
x  -  z ) (,) ( x  +  z ) )  C_  A ) ) ) )
259, 24mpdd 38 . . . . 5  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  x  e.  A )  ->  (
y  e.  RR+  ->  E. z  e.  RR+  (
z  <_  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )
) )
2625expimpd 587 . . . 4  |-  ( A  e.  ( topGen `  ran  (,) )  ->  ( (
x  e.  A  /\  y  e.  RR+ )  ->  E. z  e.  RR+  (
z  <_  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )
) )
2726ralrimivv 2789 . . 3  |-  ( A  e.  ( topGen `  ran  (,) )  ->  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  ( z  <_  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )
)
286, 27jca 519 . 2  |-  ( A  e.  ( topGen `  ran  (,) )  ->  ( A  C_  RR  /\  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  ( z  <_  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )
) )
29 ssel2 3335 . . . . . 6  |-  ( ( A  C_  RR  /\  x  e.  A )  ->  x  e.  RR )
30 1rp 10608 . . . . . . . 8  |-  1  e.  RR+
31 simpr 448 . . . . . . . . . 10  |-  ( ( z  <_  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )  ->  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )
3231reximi 2805 . . . . . . . . 9  |-  ( E. z  e.  RR+  (
z  <_  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )  ->  E. z  e.  RR+  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )
3332ralimi 2773 . . . . . . . 8  |-  ( A. y  e.  RR+  E. z  e.  RR+  ( z  <_ 
y  /\  ( (
x  -  z ) (,) ( x  +  z ) )  C_  A )  ->  A. y  e.  RR+  E. z  e.  RR+  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )
34 biidd 229 . . . . . . . . 9  |-  ( y  =  1  ->  ( E. z  e.  RR+  (
( x  -  z
) (,) ( x  +  z ) ) 
C_  A  <->  E. z  e.  RR+  ( ( x  -  z ) (,) ( x  +  z ) )  C_  A
) )
3534rspcv 3040 . . . . . . . 8  |-  ( 1  e.  RR+  ->  ( A. y  e.  RR+  E. z  e.  RR+  ( ( x  -  z ) (,) ( x  +  z ) )  C_  A  ->  E. z  e.  RR+  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )
)
3630, 33, 35mpsyl 61 . . . . . . 7  |-  ( A. y  e.  RR+  E. z  e.  RR+  ( z  <_ 
y  /\  ( (
x  -  z ) (,) ( x  +  z ) )  C_  A )  ->  E. z  e.  RR+  ( ( x  -  z ) (,) ( x  +  z ) )  C_  A
)
3714rexbidva 2714 . . . . . . 7  |-  ( x  e.  RR  ->  ( E. z  e.  RR+  (
x ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) z )  C_  A  <->  E. z  e.  RR+  ( ( x  -  z ) (,) ( x  +  z ) )  C_  A
) )
3836, 37syl5ibr 213 . . . . . 6  |-  ( x  e.  RR  ->  ( A. y  e.  RR+  E. z  e.  RR+  ( z  <_ 
y  /\  ( (
x  -  z ) (,) ( x  +  z ) )  C_  A )  ->  E. z  e.  RR+  ( x (
ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) z )  C_  A )
)
3929, 38syl 16 . . . . 5  |-  ( ( A  C_  RR  /\  x  e.  A )  ->  ( A. y  e.  RR+  E. z  e.  RR+  ( z  <_ 
y  /\  ( (
x  -  z ) (,) ( x  +  z ) )  C_  A )  ->  E. z  e.  RR+  ( x (
ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) z )  C_  A )
)
4039ralimdva 2776 . . . 4  |-  ( A 
C_  RR  ->  ( A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  ( z  <_ 
y  /\  ( (
x  -  z ) (,) ( x  +  z ) )  C_  A )  ->  A. x  e.  A  E. z  e.  RR+  ( x (
ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) z )  C_  A )
)
4140imdistani 672 . . 3  |-  ( ( A  C_  RR  /\  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  ( z  <_ 
y  /\  ( (
x  -  z ) (,) ( x  +  z ) )  C_  A ) )  -> 
( A  C_  RR  /\ 
A. x  e.  A  E. z  e.  RR+  (
x ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) z )  C_  A )
)
424elmopn2 18467 . . . 4  |-  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( * Met `  RR )  ->  ( A  e.  ( topGen `  ran  (,) )  <->  ( A  C_  RR  /\  A. x  e.  A  E. z  e.  RR+  ( x ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) z )  C_  A )
) )
432, 42ax-mp 8 . . 3  |-  ( A  e.  ( topGen `  ran  (,) )  <->  ( A  C_  RR  /\  A. x  e.  A  E. z  e.  RR+  ( x ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) z )  C_  A ) )
4441, 43sylibr 204 . 2  |-  ( ( A  C_  RR  /\  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  ( z  <_ 
y  /\  ( (
x  -  z ) (,) ( x  +  z ) )  C_  A ) )  ->  A  e.  ( topGen ` 
ran  (,) ) )
4528, 44impbii 181 1  |-  ( A  e.  ( topGen `  ran  (,) )  <->  ( A  C_  RR  /\  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  ( z  <_  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698    C_ wss 3312   class class class wbr 4204    X. cxp 4868   ran crn 4871    |` cres 4872    o. ccom 4874   ` cfv 5446  (class class class)co 6073   RRcr 8981   1c1 8983    + caddc 8985    < clt 9112    <_ cle 9113    - cmin 9283   RR+crp 10604   (,)cioo 10908   abscabs 12031   topGenctg 13657   * Metcxmt 16678   ballcbl 16680   MetOpencmopn 16683
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ioo 10912  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-topgen 13659  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-top 16955  df-bases 16957  df-topon 16958
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