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Theorem dya2icoseg2 24633
Description: For any point and any opened interval of  RR containing that point, there is a closed below opened 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 2438 . . . . . 6  |-  ( |_
`  ( 1  -  ( 2logb d ) ) )  =  ( |_ `  ( 1  -  ( 2logb d ) ) )
41, 2, 3dya2icoseg 24632 . . . . 5  |-  ( ( X  e.  RR  /\  d  e.  RR+ )  ->  E. b  e.  ran  I ( X  e.  b  /\  b  C_  ( ( X  -  d ) (,) ( X  +  d )
) ) )
54ralrimiva 2791 . . . 4  |-  ( X  e.  RR  ->  A. d  e.  RR+  E. b  e. 
ran  I ( X  e.  b  /\  b  C_  ( ( X  -  d ) (,) ( X  +  d )
) ) )
653ad2ant1 979 . . 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 960 . . . . 5  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  X  e.  E )
8 iooex 10944 . . . . . . . . . 10  |-  (,)  e.  _V
98rnex 5136 . . . . . . . . 9  |-  ran  (,)  e.  _V
10 bastg 17036 . . . . . . . . 9  |-  ( ran 
(,)  e.  _V  ->  ran 
(,)  C_  ( topGen `  ran  (,) ) )
119, 10ax-mp 5 . . . . . . . 8  |-  ran  (,)  C_  ( topGen `  ran  (,) )
12 simp2 959 . . . . . . . 8  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  E  e.  ran  (,) )
1311, 12sseldi 3348 . . . . . . 7  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  E  e.  ( topGen ` 
ran  (,) ) )
1413, 1syl6eleqr 2529 . . . . . 6  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  E  e.  J )
15 eqid 2438 . . . . . . . . 9  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
1615rexmet 18827 . . . . . . . 8  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( * Met `  RR )
17 cnfldxms 18816 . . . . . . . . . . 11  |-fld  e.  * MetSp
18 reex 9086 . . . . . . . . . . 11  |-  RR  e.  _V
19 ressxms 18560 . . . . . . . . . . 11  |-  ( (fld  e. 
* MetSp  /\  RR  e.  _V )  ->  (flds  RR )  e.  * MetSp )
2017, 18, 19mp2an 655 . . . . . . . . . 10  |-  (flds  RR )  e.  * MetSp
21 eqid 2438 . . . . . . . . . . . . 13  |-  ( TopOpen `  (flds  RR ) )  =  (
TopOpen `  (flds  RR ) )
2221tgioo3 18841 . . . . . . . . . . . 12  |-  ( topGen ` 
ran  (,) )  =  (
TopOpen `  (flds  RR ) )
231, 22eqtri 2458 . . . . . . . . . . 11  |-  J  =  ( TopOpen `  (flds  RR ) )
24 eqid 2438 . . . . . . . . . . . 12  |-  (flds  RR )  =  (flds  RR )
2524rebase 24274 . . . . . . . . . . 11  |-  RR  =  ( Base `  (flds  RR ) )
26 cnfldds 16718 . . . . . . . . . . . . . 14  |-  ( abs 
o.  -  )  =  ( dist ` fld )
2724, 26ressds 13646 . . . . . . . . . . . . 13  |-  ( RR  e.  _V  ->  ( abs  o.  -  )  =  ( dist `  (flds  RR )
) )
2818, 27ax-mp 5 . . . . . . . . . . . 12  |-  ( abs 
o.  -  )  =  ( dist `  (flds  RR ) )
2928reseq1i 5145 . . . . . . . . . . 11  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( dist `  (flds  RR )
)  |`  ( RR  X.  RR ) )
3023, 25, 29xmstopn 18486 . . . . . . . . . 10  |-  ( (flds  RR )  e.  * MetSp  ->  J  =  ( MetOpen `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) )
3120, 30ax-mp 5 . . . . . . . . 9  |-  J  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) )
3231elmopn2 18480 . . . . . . . 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 ) ) )
3316, 32ax-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 )
)
3433simprbi 452 . . . . . 6  |-  ( E  e.  J  ->  A. x  e.  E  E. d  e.  RR+  ( x (
ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) d )  C_  E )
3514, 34syl 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 )
36 oveq1 6091 . . . . . . . 8  |-  ( x  =  X  ->  (
x ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) d )  =  ( X ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) d ) )
3736sseq1d 3377 . . . . . . 7  |-  ( x  =  X  ->  (
( x ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) d )  C_  E 
<->  ( X ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) d )  C_  E ) )
3837rexbidv 2728 . . . . . 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 )
)
3938rspcva 3052 . . . . 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 )
407, 35, 39syl2anc 644 . . . 4  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  E. d  e.  RR+  ( X ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) d )  C_  E )
41 rpre 10623 . . . . . . 7  |-  ( d  e.  RR+  ->  d  e.  RR )
4215bl2ioo 18828 . . . . . . . 8  |-  ( ( X  e.  RR  /\  d  e.  RR )  ->  ( X ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) d )  =  ( ( X  -  d ) (,) ( X  +  d )
) )
4342sseq1d 3377 . . . . . . 7  |-  ( ( X  e.  RR  /\  d  e.  RR )  ->  ( ( X (
ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) d )  C_  E  <->  ( ( X  -  d ) (,) ( X  +  d ) )  C_  E
) )
4441, 43sylan2 462 . . . . . 6  |-  ( ( X  e.  RR  /\  d  e.  RR+ )  -> 
( ( X (
ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) d )  C_  E  <->  ( ( X  -  d ) (,) ( X  +  d ) )  C_  E
) )
4544rexbidva 2724 . . . . 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
) )
46453ad2ant1 979 . . . 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 )
)
4740, 46mpbid 203 . . 3  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  E. d  e.  RR+  ( ( X  -  d ) (,) ( X  +  d )
)  C_  E )
48 r19.29 2848 . . 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 ) )
496, 47, 48syl2anc 644 . 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 ) )
50 r19.41v 2863 . . . 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 )
)
51 sstr 3358 . . . . . . 7  |-  ( ( b  C_  ( ( X  -  d ) (,) ( X  +  d ) )  /\  (
( X  -  d
) (,) ( X  +  d ) ) 
C_  E )  -> 
b  C_  E )
5251anim2i 554 . . . . . 6  |-  ( ( X  e.  b  /\  ( b  C_  (
( X  -  d
) (,) ( X  +  d ) )  /\  ( ( X  -  d ) (,) ( X  +  d ) )  C_  E
) )  ->  ( X  e.  b  /\  b  C_  E ) )
5352anassrs 631 . . . . 5  |-  ( ( ( X  e.  b  /\  b  C_  (
( X  -  d
) (,) ( X  +  d ) ) )  /\  ( ( X  -  d ) (,) ( X  +  d ) )  C_  E )  ->  ( X  e.  b  /\  b  C_  E ) )
5453reximi 2815 . . . 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 ) )
5550, 54sylbir 206 . . 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 ) )
5655rexlimivw 2828 . 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 ) )
5749, 56syl 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 set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   _Vcvv 2958    C_ wss 3322    X. cxp 4879   ran crn 4882    |` cres 4883    o. ccom 4885   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   RRcr 8994   1c1 8996    + caddc 8998    - cmin 9296    / cdiv 9682   2c2 10054   ZZcz 10287   RR+crp 10617   (,)cioo 10921   [,)cico 10923   |_cfl 11206   ^cexp 11387   abscabs 12044   ↾s cress 13475   distcds 13543   TopOpenctopn 13654   topGenctg 13670   * Metcxmt 16691   ballcbl 16693   MetOpencmopn 16696  ℂfldccnfld 16708   *
MetSpcxme 18352  logbclogb 24393
This theorem is referenced by:  dya2iocnrect  24636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074  ax-mulf 9075
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-fi 7419  df-sup 7449  df-oi 7482  df-card 7831  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-q 10580  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-ioo 10925  df-ioc 10926  df-ico 10927  df-icc 10928  df-fz 11049  df-fzo 11141  df-fl 11207  df-mod 11256  df-seq 11329  df-exp 11388  df-fac 11572  df-bc 11599  df-hash 11624  df-shft 11887  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-limsup 12270  df-clim 12287  df-rlim 12288  df-sum 12485  df-ef 12675  df-sin 12677  df-cos 12678  df-pi 12680  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-starv 13549  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-unif 13557  df-hom 13558  df-cco 13559  df-rest 13655  df-topn 13656  df-topgen 13672  df-pt 13673  df-prds 13676  df-xrs 13731  df-0g 13732  df-gsum 13733  df-qtop 13738  df-imas 13739  df-xps 13741  df-mre 13816  df-mrc 13817  df-acs 13819  df-mnd 14695  df-submnd 14744  df-mulg 14820  df-cntz 15121  df-cmn 15419  df-psmet 16699  df-xmet 16700  df-met 16701  df-bl 16702  df-mopn 16703  df-fbas 16704  df-fg 16705  df-cnfld 16709  df-top 16968  df-bases 16970  df-topon 16971  df-topsp 16972  df-cld 17088  df-ntr 17089  df-cls 17090  df-nei 17167  df-lp 17205  df-perf 17206  df-cn 17296  df-cnp 17297  df-haus 17384  df-tx 17599  df-hmeo 17792  df-fil 17883  df-fm 17975  df-flim 17976  df-flf 17977  df-xms 18355  df-ms 18356  df-tms 18357  df-cncf 18913  df-limc 19758  df-dv 19759  df-log 20459  df-cxp 20460  df-logb 24394
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