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Theorem opnrebl2 28659
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 2452 . . . . 5  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
21rexmet 20495 . . . 4  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( *Met `  RR )
3 eqid 2452 . . . . . 6  |-  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )
41, 3tgioo 20500 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) )
54mopnss 20148 . . . 4  |-  ( ( ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )  e.  ( *Met `  RR )  /\  A  e.  ( topGen `  ran  (,) )
)  ->  A  C_  RR )
62, 5mpan 670 . . 3  |-  ( A  e.  ( topGen `  ran  (,) )  ->  A  C_  RR )
74mopni3 20196 . . . . . . . 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 434 . . . . . . 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 1302 . . . . . 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 3459 . . . . . . 7  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  x  e.  A )  ->  x  e.  RR )
11 rpre 11103 . . . . . . . . . . . . 13  |-  ( z  e.  RR+  ->  z  e.  RR )
121bl2ioo 20496 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( x ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) z )  =  ( ( x  -  z ) (,) (
x  +  z ) ) )
1311, 12sylan2 474 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  z  e.  RR+ )  -> 
( x ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) z )  =  ( ( x  -  z ) (,) (
x  +  z ) ) )
1413sseq1d 3486 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  z  e.  RR+ )  -> 
( ( x (
ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) z )  C_  A  <->  ( (
x  -  z ) (,) ( x  +  z ) )  C_  A ) )
1514anbi2d 703 . . . . . . . . . 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 2854 . . . . . . . . 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 207 . . . . . . . 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 11103 . . . . . . . . . . 11  |-  ( y  e.  RR+  ->  y  e.  RR )
19 ltle 9569 . . . . . . . . . . 11  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( z  <  y  ->  z  <_  y )
)
2011, 18, 19syl2anr 478 . . . . . . . . . 10  |-  ( ( y  e.  RR+  /\  z  e.  RR+ )  ->  (
z  <  y  ->  z  <_  y ) )
2120anim1d 564 . . . . . . . . 9  |-  ( ( y  e.  RR+  /\  z  e.  RR+ )  ->  (
( z  <  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )  ->  ( z  <_  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )
) )
2221reximdva 2928 . . . . . . . 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 71 . . . . . . 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 40 . . . . 5  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  x  e.  A )  ->  (
y  e.  RR+  ->  E. z  e.  RR+  (
z  <_  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )
) )
2625expimpd 603 . . . 4  |-  ( A  e.  ( topGen `  ran  (,) )  ->  ( (
x  e.  A  /\  y  e.  RR+ )  ->  E. z  e.  RR+  (
z  <_  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )
) )
2726ralrimivv 2907 . . 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 532 . 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 3454 . . . . . 6  |-  ( ( A  C_  RR  /\  x  e.  A )  ->  x  e.  RR )
30 1rp 11101 . . . . . . . 8  |-  1  e.  RR+
31 simpr 461 . . . . . . . . . 10  |-  ( ( z  <_  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )  ->  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )
3231reximi 2923 . . . . . . . . 9  |-  ( E. z  e.  RR+  (
z  <_  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )  ->  E. z  e.  RR+  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )
3332ralimi 2816 . . . . . . . 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 237 . . . . . . . . 9  |-  ( y  =  1  ->  ( E. z  e.  RR+  (
( x  -  z
) (,) ( x  +  z ) ) 
C_  A  <->  E. z  e.  RR+  ( ( x  -  z ) (,) ( x  +  z ) )  C_  A
) )
3534rspcv 3169 . . . . . . . 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 63 . . . . . . 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 2854 . . . . . . 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 221 . . . . . 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 2829 . . . 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 690 . . 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 20147 . . . 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 5 . . 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 212 . 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 188 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2796   E.wrex 2797    C_ wss 3431   class class class wbr 4395    X. cxp 4941   ran crn 4944    |` cres 4945    o. ccom 4947   ` cfv 5521  (class class class)co 6195   RRcr 9387   1c1 9389    + caddc 9391    < clt 9524    <_ cle 9525    - cmin 9701   RR+crp 11097   (,)cioo 11406   abscabs 12836   topGenctg 14490   *Metcxmt 17921   ballcbl 17923   MetOpencmopn 17926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-er 7206  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-sup 7797  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-n0 10686  df-z 10753  df-uz 10968  df-q 11060  df-rp 11098  df-xneg 11195  df-xadd 11196  df-xmul 11197  df-ioo 11410  df-seq 11919  df-exp 11978  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837  df-abs 12838  df-topgen 14496  df-psmet 17929  df-xmet 17930  df-met 17931  df-bl 17932  df-mopn 17933  df-top 18630  df-bases 18632  df-topon 18633
This theorem is referenced by: (None)
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