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Theorem opnrebl2 28357
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 2433 . . . . 5  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
21rexmet 20209 . . . 4  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( *Met `  RR )
3 eqid 2433 . . . . . 6  |-  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )
41, 3tgioo 20214 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) )
54mopnss 19862 . . . 4  |-  ( ( ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )  e.  ( *Met `  RR )  /\  A  e.  ( topGen `  ran  (,) )
)  ->  A  C_  RR )
62, 5mpan 663 . . 3  |-  ( A  e.  ( topGen `  ran  (,) )  ->  A  C_  RR )
74mopni3 19910 . . . . . . . 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 1294 . . . . . 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 3344 . . . . . . 7  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  x  e.  A )  ->  x  e.  RR )
11 rpre 10984 . . . . . . . . . . . . 13  |-  ( z  e.  RR+  ->  z  e.  RR )
121bl2ioo 20210 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( x ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) z )  =  ( ( x  -  z ) (,) (
x  +  z ) ) )
1311, 12sylan2 471 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  z  e.  RR+ )  -> 
( x ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) z )  =  ( ( x  -  z ) (,) (
x  +  z ) ) )
1413sseq1d 3371 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  z  e.  RR+ )  -> 
( ( x (
ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) z )  C_  A  <->  ( (
x  -  z ) (,) ( x  +  z ) )  C_  A ) )
1514anbi2d 696 . . . . . . . . . 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 2722 . . . . . . . . 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 10984 . . . . . . . . . . 11  |-  ( y  e.  RR+  ->  y  e.  RR )
19 ltle 9450 . . . . . . . . . . 11  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( z  <  y  ->  z  <_  y )
)
2011, 18, 19syl2anr 475 . . . . . . . . . 10  |-  ( ( y  e.  RR+  /\  z  e.  RR+ )  ->  (
z  <  y  ->  z  <_  y ) )
2120anim1d 559 . . . . . . . . 9  |-  ( ( y  e.  RR+  /\  z  e.  RR+ )  ->  (
( z  <  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )  ->  ( z  <_  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )
) )
2221reximdva 2818 . . . . . . . 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 598 . . . 4  |-  ( A  e.  ( topGen `  ran  (,) )  ->  ( (
x  e.  A  /\  y  e.  RR+ )  ->  E. z  e.  RR+  (
z  <_  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )
) )
2726ralrimivv 2797 . . 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 529 . 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 3339 . . . . . 6  |-  ( ( A  C_  RR  /\  x  e.  A )  ->  x  e.  RR )
30 1rp 10982 . . . . . . . 8  |-  1  e.  RR+
31 simpr 458 . . . . . . . . . 10  |-  ( ( z  <_  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )  ->  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )
3231reximi 2813 . . . . . . . . 9  |-  ( E. z  e.  RR+  (
z  <_  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )  ->  E. z  e.  RR+  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )
3332ralimi 2781 . . . . . . . 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 3058 . . . . . . . 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 2722 . . . . . . 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 2784 . . . 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 683 . . 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 19861 . . . 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 958    = wceq 1362    e. wcel 1755   A.wral 2705   E.wrex 2706    C_ wss 3316   class class class wbr 4280    X. cxp 4825   ran crn 4828    |` cres 4829    o. ccom 4831   ` cfv 5406  (class class class)co 6080   RRcr 9268   1c1 9270    + caddc 9272    < clt 9405    <_ cle 9406    - cmin 9582   RR+crp 10978   (,)cioo 11287   abscabs 12706   topGenctg 14358   *Metcxmt 17644   ballcbl 17646   MetOpencmopn 17649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346  ax-pre-sup 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-sup 7679  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-div 9981  df-nn 10310  df-2 10367  df-3 10368  df-n0 10567  df-z 10634  df-uz 10849  df-q 10941  df-rp 10979  df-xneg 11076  df-xadd 11077  df-xmul 11078  df-ioo 11291  df-seq 11790  df-exp 11849  df-cj 12571  df-re 12572  df-im 12573  df-sqr 12707  df-abs 12708  df-topgen 14364  df-psmet 17652  df-xmet 17653  df-met 17654  df-bl 17655  df-mopn 17656  df-top 18344  df-bases 18346  df-topon 18347
This theorem is referenced by: (None)
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