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Theorem opnrebl2 30379
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 2454 . . . . 5  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
21rexmet 21462 . . . 4  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( *Met `  RR )
3 eqid 2454 . . . . . 6  |-  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )
41, 3tgioo 21467 . . . . 5  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) )
54mopnss 21115 . . . 4  |-  ( ( ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )  e.  ( *Met `  RR )  /\  A  e.  ( topGen `  ran  (,) )
)  ->  A  C_  RR )
62, 5mpan 668 . . 3  |-  ( A  e.  ( topGen `  ran  (,) )  ->  A  C_  RR )
74mopni3 21163 . . . . . . . 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 432 . . . . . . 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 1309 . . . . . 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 3489 . . . . . . 7  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  x  e.  A )  ->  x  e.  RR )
11 rpre 11227 . . . . . . . . . . . . 13  |-  ( z  e.  RR+  ->  z  e.  RR )
121bl2ioo 21463 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( x ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) z )  =  ( ( x  -  z ) (,) (
x  +  z ) ) )
1311, 12sylan2 472 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  z  e.  RR+ )  -> 
( x ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) z )  =  ( ( x  -  z ) (,) (
x  +  z ) ) )
1413sseq1d 3516 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  z  e.  RR+ )  -> 
( ( x (
ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) z )  C_  A  <->  ( (
x  -  z ) (,) ( x  +  z ) )  C_  A ) )
1514anbi2d 701 . . . . . . . . . 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 2962 . . . . . . . . 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 11227 . . . . . . . . . . 11  |-  ( y  e.  RR+  ->  y  e.  RR )
19 ltle 9662 . . . . . . . . . . 11  |-  ( ( z  e.  RR  /\  y  e.  RR )  ->  ( z  <  y  ->  z  <_  y )
)
2011, 18, 19syl2anr 476 . . . . . . . . . 10  |-  ( ( y  e.  RR+  /\  z  e.  RR+ )  ->  (
z  <  y  ->  z  <_  y ) )
2120anim1d 562 . . . . . . . . 9  |-  ( ( y  e.  RR+  /\  z  e.  RR+ )  ->  (
( z  <  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )  ->  ( z  <_  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )
) )
2221reximdva 2929 . . . . . . . 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 601 . . . 4  |-  ( A  e.  ( topGen `  ran  (,) )  ->  ( (
x  e.  A  /\  y  e.  RR+ )  ->  E. z  e.  RR+  (
z  <_  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )
) )
2726ralrimivv 2874 . . 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 530 . 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 3484 . . . . . 6  |-  ( ( A  C_  RR  /\  x  e.  A )  ->  x  e.  RR )
30 1rp 11225 . . . . . . . 8  |-  1  e.  RR+
31 simpr 459 . . . . . . . . . 10  |-  ( ( z  <_  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )  ->  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )
3231reximi 2922 . . . . . . . . 9  |-  ( E. z  e.  RR+  (
z  <_  y  /\  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )  ->  E. z  e.  RR+  ( ( x  -  z ) (,) (
x  +  z ) )  C_  A )
3332ralimi 2847 . . . . . . . 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 3203 . . . . . . . 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 2962 . . . . . . 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 2862 . . . 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 688 . . 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 21114 . . . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805    C_ wss 3461   class class class wbr 4439    X. cxp 4986   ran crn 4989    |` cres 4990    o. ccom 4992   ` cfv 5570  (class class class)co 6270   RRcr 9480   1c1 9482    + caddc 9484    < clt 9617    <_ cle 9618    - cmin 9796   RR+crp 11221   (,)cioo 11532   abscabs 13149   topGenctg 14927   *Metcxmt 18598   ballcbl 18600   MetOpencmopn 18603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-seq 12090  df-exp 12149  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-topgen 14933  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-top 19566  df-bases 19568  df-topon 19569
This theorem is referenced by: (None)
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