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Theorem dya2icoseg2 26708
Description: For any point and any open interval of  RR containing that point, there is a closed-below open-above dyadic rational interval which contains that point and is included in the original interval. (Contributed by Thierry Arnoux, 12-Oct-2017.)
Hypotheses
Ref Expression
sxbrsiga.0  |-  J  =  ( topGen `  ran  (,) )
dya2ioc.1  |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) ) )
Assertion
Ref Expression
dya2icoseg2  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  E. b  e.  ran  I ( X  e.  b  /\  b  C_  E ) )
Distinct variable groups:    x, n    x, I    n, b, x    E, b, x    I, b    X, b, x
Allowed substitution hints:    E( n)    I( n)    J( x, n, b)    X( n)

Proof of Theorem dya2icoseg2
Dummy variable  d is distinct from all other variables.
StepHypRef Expression
1 sxbrsiga.0 . . . . . 6  |-  J  =  ( topGen `  ran  (,) )
2 dya2ioc.1 . . . . . 6  |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) ) )
3 eqid 2443 . . . . . 6  |-  ( |_
`  ( 1  -  ( 2logb d ) ) )  =  ( |_ `  ( 1  -  ( 2logb d ) ) )
41, 2, 3dya2icoseg 26707 . . . . 5  |-  ( ( X  e.  RR  /\  d  e.  RR+ )  ->  E. b  e.  ran  I ( X  e.  b  /\  b  C_  ( ( X  -  d ) (,) ( X  +  d )
) ) )
54ralrimiva 2814 . . . 4  |-  ( X  e.  RR  ->  A. d  e.  RR+  E. b  e. 
ran  I ( X  e.  b  /\  b  C_  ( ( X  -  d ) (,) ( X  +  d )
) ) )
653ad2ant1 1009 . . 3  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  A. d  e.  RR+  E. b  e.  ran  I
( X  e.  b  /\  b  C_  (
( X  -  d
) (,) ( X  +  d ) ) ) )
7 simp3 990 . . . . 5  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  X  e.  E )
8 iooex 11338 . . . . . . . . . 10  |-  (,)  e.  _V
98rnex 6527 . . . . . . . . 9  |-  ran  (,)  e.  _V
10 bastg 18586 . . . . . . . . 9  |-  ( ran 
(,)  e.  _V  ->  ran 
(,)  C_  ( topGen `  ran  (,) ) )
119, 10ax-mp 5 . . . . . . . 8  |-  ran  (,)  C_  ( topGen `  ran  (,) )
12 simp2 989 . . . . . . . 8  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  E  e.  ran  (,) )
1311, 12sseldi 3369 . . . . . . 7  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  E  e.  ( topGen ` 
ran  (,) ) )
1413, 1syl6eleqr 2534 . . . . . 6  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  E  e.  J )
15 eqid 2443 . . . . . . . . 9  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
1615rexmet 20383 . . . . . . . 8  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( *Met `  RR )
17 recms 20899 . . . . . . . . . . 11  |- RRfld  e. CMetSp
18 cmsms 20874 . . . . . . . . . . 11  |-  (RRfld  e. CMetSp  -> RRfld  e. 
MetSp )
19 msxms 20044 . . . . . . . . . . 11  |-  (RRfld  e.  MetSp  -> RRfld 
e.  *MetSp )
2017, 18, 19mp2b 10 . . . . . . . . . 10  |- RRfld  e.  *MetSp
21 retopn 20898 . . . . . . . . . . . 12  |-  ( topGen ` 
ran  (,) )  =  (
TopOpen ` RRfld )
221, 21eqtri 2463 . . . . . . . . . . 11  |-  J  =  ( TopOpen ` RRfld )
23 rebase 18051 . . . . . . . . . . 11  |-  RR  =  ( Base ` RRfld )
24 reds 18061 . . . . . . . . . . . 12  |-  ( abs 
o.  -  )  =  ( dist ` RRfld )
2524reseq1i 5121 . . . . . . . . . . 11  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( dist ` RRfld )  |`  ( RR  X.  RR ) )
2622, 23, 25xmstopn 20041 . . . . . . . . . 10  |-  (RRfld  e.  *MetSp  ->  J  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) ) )
2720, 26ax-mp 5 . . . . . . . . 9  |-  J  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) )
2827elmopn2 20035 . . . . . . . 8  |-  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( *Met `  RR )  ->  ( E  e.  J  <->  ( E  C_  RR  /\  A. x  e.  E  E. d  e.  RR+  ( x ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) d )  C_  E ) ) )
2916, 28ax-mp 5 . . . . . . 7  |-  ( E  e.  J  <->  ( E  C_  RR  /\  A. x  e.  E  E. d  e.  RR+  ( x (
ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) d )  C_  E )
)
3029simprbi 464 . . . . . 6  |-  ( E  e.  J  ->  A. x  e.  E  E. d  e.  RR+  ( x (
ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) d )  C_  E )
3114, 30syl 16 . . . . 5  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  A. x  e.  E  E. d  e.  RR+  (
x ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) d )  C_  E )
32 oveq1 6113 . . . . . . . 8  |-  ( x  =  X  ->  (
x ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) d )  =  ( X ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) d ) )
3332sseq1d 3398 . . . . . . 7  |-  ( x  =  X  ->  (
( x ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) d )  C_  E 
<->  ( X ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) d )  C_  E ) )
3433rexbidv 2751 . . . . . 6  |-  ( x  =  X  ->  ( E. d  e.  RR+  (
x ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) d )  C_  E  <->  E. d  e.  RR+  ( X (
ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) d )  C_  E )
)
3534rspcva 3086 . . . . 5  |-  ( ( X  e.  E  /\  A. x  e.  E  E. d  e.  RR+  ( x ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) d )  C_  E )  ->  E. d  e.  RR+  ( X ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) d )  C_  E )
367, 31, 35syl2anc 661 . . . 4  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  E. d  e.  RR+  ( X ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) d )  C_  E )
37 rpre 11012 . . . . . . 7  |-  ( d  e.  RR+  ->  d  e.  RR )
3815bl2ioo 20384 . . . . . . . 8  |-  ( ( X  e.  RR  /\  d  e.  RR )  ->  ( X ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) d )  =  ( ( X  -  d ) (,) ( X  +  d )
) )
3938sseq1d 3398 . . . . . . 7  |-  ( ( X  e.  RR  /\  d  e.  RR )  ->  ( ( X (
ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) d )  C_  E  <->  ( ( X  -  d ) (,) ( X  +  d ) )  C_  E
) )
4037, 39sylan2 474 . . . . . 6  |-  ( ( X  e.  RR  /\  d  e.  RR+ )  -> 
( ( X (
ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) d )  C_  E  <->  ( ( X  -  d ) (,) ( X  +  d ) )  C_  E
) )
4140rexbidva 2747 . . . . 5  |-  ( X  e.  RR  ->  ( E. d  e.  RR+  ( X ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) d )  C_  E  <->  E. d  e.  RR+  ( ( X  -  d ) (,) ( X  +  d ) )  C_  E
) )
42413ad2ant1 1009 . . . 4  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  ( E. d  e.  RR+  ( X ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) d )  C_  E 
<->  E. d  e.  RR+  ( ( X  -  d ) (,) ( X  +  d )
)  C_  E )
)
4336, 42mpbid 210 . . 3  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  E. d  e.  RR+  ( ( X  -  d ) (,) ( X  +  d )
)  C_  E )
44 r19.29 2872 . . 3  |-  ( ( A. d  e.  RR+  E. b  e.  ran  I
( X  e.  b  /\  b  C_  (
( X  -  d
) (,) ( X  +  d ) ) )  /\  E. d  e.  RR+  ( ( X  -  d ) (,) ( X  +  d ) )  C_  E
)  ->  E. d  e.  RR+  ( E. b  e.  ran  I ( X  e.  b  /\  b  C_  ( ( X  -  d ) (,) ( X  +  d )
) )  /\  (
( X  -  d
) (,) ( X  +  d ) ) 
C_  E ) )
456, 43, 44syl2anc 661 . 2  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  E. d  e.  RR+  ( E. b  e.  ran  I ( X  e.  b  /\  b  C_  ( ( X  -  d ) (,) ( X  +  d )
) )  /\  (
( X  -  d
) (,) ( X  +  d ) ) 
C_  E ) )
46 r19.41v 2888 . . . 4  |-  ( E. b  e.  ran  I
( ( X  e.  b  /\  b  C_  ( ( X  -  d ) (,) ( X  +  d )
) )  /\  (
( X  -  d
) (,) ( X  +  d ) ) 
C_  E )  <->  ( E. b  e.  ran  I ( X  e.  b  /\  b  C_  ( ( X  -  d ) (,) ( X  +  d ) ) )  /\  ( ( X  -  d ) (,) ( X  +  d )
)  C_  E )
)
47 sstr 3379 . . . . . . 7  |-  ( ( b  C_  ( ( X  -  d ) (,) ( X  +  d ) )  /\  (
( X  -  d
) (,) ( X  +  d ) ) 
C_  E )  -> 
b  C_  E )
4847anim2i 569 . . . . . 6  |-  ( ( X  e.  b  /\  ( b  C_  (
( X  -  d
) (,) ( X  +  d ) )  /\  ( ( X  -  d ) (,) ( X  +  d ) )  C_  E
) )  ->  ( X  e.  b  /\  b  C_  E ) )
4948anassrs 648 . . . . 5  |-  ( ( ( X  e.  b  /\  b  C_  (
( X  -  d
) (,) ( X  +  d ) ) )  /\  ( ( X  -  d ) (,) ( X  +  d ) )  C_  E )  ->  ( X  e.  b  /\  b  C_  E ) )
5049reximi 2838 . . . 4  |-  ( E. b  e.  ran  I
( ( X  e.  b  /\  b  C_  ( ( X  -  d ) (,) ( X  +  d )
) )  /\  (
( X  -  d
) (,) ( X  +  d ) ) 
C_  E )  ->  E. b  e.  ran  I ( X  e.  b  /\  b  C_  E ) )
5146, 50sylbir 213 . . 3  |-  ( ( E. b  e.  ran  I ( X  e.  b  /\  b  C_  ( ( X  -  d ) (,) ( X  +  d )
) )  /\  (
( X  -  d
) (,) ( X  +  d ) ) 
C_  E )  ->  E. b  e.  ran  I ( X  e.  b  /\  b  C_  E ) )
5251rexlimivw 2852 . 2  |-  ( E. d  e.  RR+  ( E. b  e.  ran  I ( X  e.  b  /\  b  C_  ( ( X  -  d ) (,) ( X  +  d )
) )  /\  (
( X  -  d
) (,) ( X  +  d ) ) 
C_  E )  ->  E. b  e.  ran  I ( X  e.  b  /\  b  C_  E ) )
5345, 52syl 16 1  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  E. b  e.  ran  I ( X  e.  b  /\  b  C_  E ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2730   E.wrex 2731   _Vcvv 2987    C_ wss 3343    X. cxp 4853   ran crn 4856    |` cres 4857    o. ccom 4859   ` cfv 5433  (class class class)co 6106    e. cmpt2 6108   RRcr 9296   1c1 9298    + caddc 9300    - cmin 9610    / cdiv 10008   2c2 10386   ZZcz 10661   RR+crp 11006   (,)cioo 11315   [,)cico 11317   |_cfl 11655   ^cexp 11880   abscabs 12738   distcds 14262   TopOpenctopn 14375   topGenctg 14391   *Metcxmt 17816   ballcbl 17818   MetOpencmopn 17821  RRfldcrefld 18049   *MetSpcxme 19907   MetSpcmt 19908  CMetSpccms 20858  logbclogb 26464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-inf2 7862  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374  ax-pre-sup 9375  ax-addf 9376  ax-mulf 9377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-iin 4189  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-se 4695  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-isom 5442  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-of 6335  df-om 6492  df-1st 6592  df-2nd 6593  df-supp 6706  df-recs 6847  df-rdg 6881  df-1o 6935  df-2o 6936  df-oadd 6939  df-er 7116  df-map 7231  df-pm 7232  df-ixp 7279  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-fsupp 7636  df-fi 7676  df-sup 7706  df-oi 7739  df-card 8124  df-cda 8352  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-div 10009  df-nn 10338  df-2 10395  df-3 10396  df-4 10397  df-5 10398  df-6 10399  df-7 10400  df-8 10401  df-9 10402  df-10 10403  df-n0 10595  df-z 10662  df-dec 10771  df-uz 10877  df-q 10969  df-rp 11007  df-xneg 11104  df-xadd 11105  df-xmul 11106  df-ioo 11319  df-ioc 11320  df-ico 11321  df-icc 11322  df-fz 11453  df-fzo 11564  df-fl 11657  df-mod 11724  df-seq 11822  df-exp 11881  df-fac 12067  df-bc 12094  df-hash 12119  df-shft 12571  df-cj 12603  df-re 12604  df-im 12605  df-sqr 12739  df-abs 12740  df-limsup 12964  df-clim 12981  df-rlim 12982  df-sum 13179  df-ef 13368  df-sin 13370  df-cos 13371  df-pi 13373  df-struct 14191  df-ndx 14192  df-slot 14193  df-base 14194  df-sets 14195  df-ress 14196  df-plusg 14266  df-mulr 14267  df-starv 14268  df-sca 14269  df-vsca 14270  df-ip 14271  df-tset 14272  df-ple 14273  df-ds 14275  df-unif 14276  df-hom 14277  df-cco 14278  df-rest 14376  df-topn 14377  df-0g 14395  df-gsum 14396  df-topgen 14397  df-pt 14398  df-prds 14401  df-xrs 14455  df-qtop 14460  df-imas 14461  df-xps 14463  df-mre 14539  df-mrc 14540  df-acs 14542  df-mnd 15430  df-submnd 15480  df-mulg 15563  df-cntz 15850  df-cmn 16294  df-psmet 17824  df-xmet 17825  df-met 17826  df-bl 17827  df-mopn 17828  df-fbas 17829  df-fg 17830  df-cnfld 17834  df-refld 18050  df-top 18518  df-bases 18520  df-topon 18521  df-topsp 18522  df-cld 18638  df-ntr 18639  df-cls 18640  df-nei 18717  df-lp 18755  df-perf 18756  df-cn 18846  df-cnp 18847  df-haus 18934  df-cmp 19005  df-tx 19150  df-hmeo 19343  df-fil 19434  df-fm 19526  df-flim 19527  df-flf 19528  df-fcls 19529  df-xms 19910  df-ms 19911  df-tms 19912  df-cncf 20469  df-cfil 20781  df-cmet 20783  df-cms 20861  df-limc 21356  df-dv 21357  df-log 22023  df-cxp 22024  df-logb 26465
This theorem is referenced by:  dya2iocnrect  26711
  Copyright terms: Public domain W3C validator