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Theorem dya2icoseg2 26629
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 2441 . . . . . 6  |-  ( |_
`  ( 1  -  ( 2logb d ) ) )  =  ( |_ `  ( 1  -  ( 2logb d ) ) )
41, 2, 3dya2icoseg 26628 . . . . 5  |-  ( ( X  e.  RR  /\  d  e.  RR+ )  ->  E. b  e.  ran  I ( X  e.  b  /\  b  C_  ( ( X  -  d ) (,) ( X  +  d )
) ) )
54ralrimiva 2797 . . . 4  |-  ( X  e.  RR  ->  A. d  e.  RR+  E. b  e. 
ran  I ( X  e.  b  /\  b  C_  ( ( X  -  d ) (,) ( X  +  d )
) ) )
653ad2ant1 1004 . . 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 985 . . . . 5  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  X  e.  E )
8 iooex 11319 . . . . . . . . . 10  |-  (,)  e.  _V
98rnex 6511 . . . . . . . . 9  |-  ran  (,)  e.  _V
10 bastg 18530 . . . . . . . . 9  |-  ( ran 
(,)  e.  _V  ->  ran 
(,)  C_  ( topGen `  ran  (,) ) )
119, 10ax-mp 5 . . . . . . . 8  |-  ran  (,)  C_  ( topGen `  ran  (,) )
12 simp2 984 . . . . . . . 8  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  E  e.  ran  (,) )
1311, 12sseldi 3351 . . . . . . 7  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  E  e.  ( topGen ` 
ran  (,) ) )
1413, 1syl6eleqr 2532 . . . . . 6  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  E  e.  J )
15 eqid 2441 . . . . . . . . 9  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
1615rexmet 20327 . . . . . . . 8  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( *Met `  RR )
17 recms 20843 . . . . . . . . . . 11  |- RRfld  e. CMetSp
18 cmsms 20818 . . . . . . . . . . 11  |-  (RRfld  e. CMetSp  -> RRfld  e. 
MetSp )
19 msxms 19988 . . . . . . . . . . 11  |-  (RRfld  e.  MetSp  -> RRfld 
e.  *MetSp )
2017, 18, 19mp2b 10 . . . . . . . . . 10  |- RRfld  e.  *MetSp
21 retopn 20842 . . . . . . . . . . . 12  |-  ( topGen ` 
ran  (,) )  =  (
TopOpen ` RRfld )
221, 21eqtri 2461 . . . . . . . . . . 11  |-  J  =  ( TopOpen ` RRfld )
23 rebase 17995 . . . . . . . . . . 11  |-  RR  =  ( Base ` RRfld )
24 reds 18005 . . . . . . . . . . . 12  |-  ( abs 
o.  -  )  =  ( dist ` RRfld )
2524reseq1i 5102 . . . . . . . . . . 11  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( dist ` RRfld )  |`  ( RR  X.  RR ) )
2622, 23, 25xmstopn 19985 . . . . . . . . . 10  |-  (RRfld  e.  *MetSp  ->  J  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) ) )
2720, 26ax-mp 5 . . . . . . . . 9  |-  J  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) )
2827elmopn2 19979 . . . . . . . 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 461 . . . . . 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 6097 . . . . . . . 8  |-  ( x  =  X  ->  (
x ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) d )  =  ( X ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) d ) )
3332sseq1d 3380 . . . . . . 7  |-  ( x  =  X  ->  (
( x ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) d )  C_  E 
<->  ( X ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) d )  C_  E ) )
3433rexbidv 2734 . . . . . 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 3068 . . . . 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 656 . . . 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 10993 . . . . . . 7  |-  ( d  e.  RR+  ->  d  e.  RR )
3815bl2ioo 20328 . . . . . . . 8  |-  ( ( X  e.  RR  /\  d  e.  RR )  ->  ( X ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) d )  =  ( ( X  -  d ) (,) ( X  +  d )
) )
3938sseq1d 3380 . . . . . . 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 471 . . . . . 6  |-  ( ( X  e.  RR  /\  d  e.  RR+ )  -> 
( ( X (
ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) d )  C_  E  <->  ( ( X  -  d ) (,) ( X  +  d ) )  C_  E
) )
4140rexbidva 2730 . . . . 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 1004 . . . 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 2855 . . 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 656 . 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 2871 . . . 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 3361 . . . . . . 7  |-  ( ( b  C_  ( ( X  -  d ) (,) ( X  +  d ) )  /\  (
( X  -  d
) (,) ( X  +  d ) ) 
C_  E )  -> 
b  C_  E )
4847anim2i 566 . . . . . 6  |-  ( ( X  e.  b  /\  ( b  C_  (
( X  -  d
) (,) ( X  +  d ) )  /\  ( ( X  -  d ) (,) ( X  +  d ) )  C_  E
) )  ->  ( X  e.  b  /\  b  C_  E ) )
4948anassrs 643 . . . . 5  |-  ( ( ( X  e.  b  /\  b  C_  (
( X  -  d
) (,) ( X  +  d ) ) )  /\  ( ( X  -  d ) (,) ( X  +  d ) )  C_  E )  ->  ( X  e.  b  /\  b  C_  E ) )
5049reximi 2821 . . . 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 2835 . 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 960    = wceq 1364    e. wcel 1761   A.wral 2713   E.wrex 2714   _Vcvv 2970    C_ wss 3325    X. cxp 4834   ran crn 4837    |` cres 4838    o. ccom 4840   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092   RRcr 9277   1c1 9279    + caddc 9281    - cmin 9591    / cdiv 9989   2c2 10367   ZZcz 10642   RR+crp 10987   (,)cioo 11296   [,)cico 11298   |_cfl 11636   ^cexp 11861   abscabs 12719   distcds 14243   TopOpenctopn 14356   topGenctg 14372   *Metcxmt 17760   ballcbl 17762   MetOpencmopn 17765  RRfldcrefld 17993   *MetSpcxme 19851   MetSpcmt 19852  CMetSpccms 20802  logbclogb 26385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ioc 11301  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-shft 12552  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-sum 13160  df-ef 13349  df-sin 13351  df-cos 13352  df-pi 13354  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-fbas 17773  df-fg 17774  df-cnfld 17778  df-refld 17994  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-lp 18699  df-perf 18700  df-cn 18790  df-cnp 18791  df-haus 18878  df-cmp 18949  df-tx 19094  df-hmeo 19287  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-fcls 19473  df-xms 19854  df-ms 19855  df-tms 19856  df-cncf 20413  df-cfil 20725  df-cmet 20727  df-cms 20805  df-limc 21300  df-dv 21301  df-log 21967  df-cxp 21968  df-logb 26386
This theorem is referenced by:  dya2iocnrect  26632
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