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Theorem opnrebl2 26008
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 2380 . . . . 5  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
21rexmet 18686 . . . 4  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( * Met `  RR )
3 eqid 2380 . . . . . 6  |-  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )
41, 3tgioo 18691 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) )
54mopnss 18359 . . . 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 18407 . . . . . . . 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 3284 . . . . . . 7  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  x  e.  A )  ->  x  e.  RR )
11 rpre 10543 . . . . . . . . . . . . 13  |-  ( z  e.  RR+  ->  z  e.  RR )
121bl2ioo 18687 . . . . . . . . . . . . 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 3311 . . . . . . . . . . 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 2659 . . . . . . . . 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 10543 . . . . . . . . . . 11  |-  ( y  e.  RR+  ->  y  e.  RR )
19 ltle 9089 . . . . . . . . . . 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 2754 . . . . . . . 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 2733 . . 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 3279 . . . . . 6  |-  ( ( A  C_  RR  /\  x  e.  A )  ->  x  e.  RR )
30 1rp 10541 . . . . . . . 8  |-  1  e.  RR+
31 simpr 448 . . . . . . . . . 10  |-  ( ( z  <_  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )  ->  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )
3231reximi 2749 . . . . . . . . 9  |-  ( E. z  e.  RR+  (
z  <_  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )  ->  E. z  e.  RR+  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )
3332ralimi 2717 . . . . . . . 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 2984 . . . . . . . 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 2659 . . . . . . 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 2720 . . . 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 18358 . . . 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 1649    e. wcel 1717   A.wral 2642   E.wrex 2643    C_ wss 3256   class class class wbr 4146    X. cxp 4809   ran crn 4812    |` cres 4813    o. ccom 4815   ` cfv 5387  (class class class)co 6013   RRcr 8915   1c1 8917    + caddc 8919    < clt 9046    <_ cle 9047    - cmin 9216   RR+crp 10537   (,)cioo 10841   abscabs 11959   topGenctg 13585   * Metcxmt 16605   ballcbl 16607   MetOpencmopn 16610
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 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-sup 7374  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-q 10500  df-rp 10538  df-xneg 10635  df-xadd 10636  df-xmul 10637  df-ioo 10845  df-seq 11244  df-exp 11303  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-topgen 13587  df-xmet 16612  df-met 16613  df-bl 16614  df-mopn 16615  df-top 16879  df-bases 16881  df-topon 16882
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