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Theorem dya2icoseg2 24581
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 2404 . . . . . 6  |-  ( |_
`  ( 1  -  ( 2logb d ) ) )  =  ( |_ `  ( 1  -  ( 2logb d ) ) )
41, 2, 3dya2icoseg 24580 . . . . 5  |-  ( ( X  e.  RR  /\  d  e.  RR+ )  ->  E. b  e.  ran  I ( X  e.  b  /\  b  C_  ( ( X  -  d ) (,) ( X  +  d )
) ) )
54ralrimiva 2749 . . . 4  |-  ( X  e.  RR  ->  A. d  e.  RR+  E. b  e. 
ran  I ( X  e.  b  /\  b  C_  ( ( X  -  d ) (,) ( X  +  d )
) ) )
653ad2ant1 978 . . 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 959 . . . . 5  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  X  e.  E )
8 iooex 10895 . . . . . . . . . 10  |-  (,)  e.  _V
98rnex 5092 . . . . . . . . 9  |-  ran  (,)  e.  _V
10 bastg 16986 . . . . . . . . 9  |-  ( ran 
(,)  e.  _V  ->  ran 
(,)  C_  ( topGen `  ran  (,) ) )
119, 10ax-mp 8 . . . . . . . 8  |-  ran  (,)  C_  ( topGen `  ran  (,) )
12 simp2 958 . . . . . . . 8  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  E  e.  ran  (,) )
1311, 12sseldi 3306 . . . . . . 7  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  E  e.  ( topGen ` 
ran  (,) ) )
1413, 1syl6eleqr 2495 . . . . . 6  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  E  e.  J )
15 eqid 2404 . . . . . . . . 9  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
1615rexmet 18775 . . . . . . . 8  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( * Met `  RR )
17 cnfldxms 18764 . . . . . . . . . . 11  |-fld  e.  * MetSp
18 reex 9037 . . . . . . . . . . 11  |-  RR  e.  _V
19 ressxms 18508 . . . . . . . . . . 11  |-  ( (fld  e. 
* MetSp  /\  RR  e.  _V )  ->  (flds  RR )  e.  * MetSp )
2017, 18, 19mp2an 654 . . . . . . . . . 10  |-  (flds  RR )  e.  * MetSp
21 eqid 2404 . . . . . . . . . . . . 13  |-  ( TopOpen `  (flds  RR ) )  =  (
TopOpen `  (flds  RR ) )
2221tgioo3 18789 . . . . . . . . . . . 12  |-  ( topGen ` 
ran  (,) )  =  (
TopOpen `  (flds  RR ) )
231, 22eqtri 2424 . . . . . . . . . . 11  |-  J  =  ( TopOpen `  (flds  RR ) )
24 eqid 2404 . . . . . . . . . . . 12  |-  (flds  RR )  =  (flds  RR )
2524rebase 24222 . . . . . . . . . . 11  |-  RR  =  ( Base `  (flds  RR ) )
26 cnfldds 16668 . . . . . . . . . . . . . 14  |-  ( abs 
o.  -  )  =  ( dist ` fld )
2724, 26ressds 13596 . . . . . . . . . . . . 13  |-  ( RR  e.  _V  ->  ( abs  o.  -  )  =  ( dist `  (flds  RR )
) )
2818, 27ax-mp 8 . . . . . . . . . . . 12  |-  ( abs 
o.  -  )  =  ( dist `  (flds  RR ) )
2928reseq1i 5101 . . . . . . . . . . 11  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( dist `  (flds  RR )
)  |`  ( RR  X.  RR ) )
3023, 25, 29xmstopn 18434 . . . . . . . . . 10  |-  ( (flds  RR )  e.  * MetSp  ->  J  =  ( MetOpen `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) )
3120, 30ax-mp 8 . . . . . . . . 9  |-  J  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) )
3231elmopn2 18428 . . . . . . . 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 8 . . . . . . 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 451 . . . . . 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 6047 . . . . . . . 8  |-  ( x  =  X  ->  (
x ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) d )  =  ( X ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) d ) )
3736sseq1d 3335 . . . . . . 7  |-  ( x  =  X  ->  (
( x ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) d )  C_  E 
<->  ( X ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) d )  C_  E ) )
3837rexbidv 2687 . . . . . 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 3010 . . . . 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 643 . . . 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 10574 . . . . . . 7  |-  ( d  e.  RR+  ->  d  e.  RR )
4215bl2ioo 18776 . . . . . . . 8  |-  ( ( X  e.  RR  /\  d  e.  RR )  ->  ( X ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) d )  =  ( ( X  -  d ) (,) ( X  +  d )
) )
4342sseq1d 3335 . . . . . . 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 461 . . . . . 6  |-  ( ( X  e.  RR  /\  d  e.  RR+ )  -> 
( ( X (
ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) d )  C_  E  <->  ( ( X  -  d ) (,) ( X  +  d ) )  C_  E
) )
4544rexbidva 2683 . . . . 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 978 . . . 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 202 . . 3  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  E. d  e.  RR+  ( ( X  -  d ) (,) ( X  +  d )
)  C_  E )
48 r19.29 2806 . . 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 643 . 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 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_  ( ( X  -  d ) (,) ( X  +  d ) ) )  /\  ( ( X  -  d ) (,) ( X  +  d )
)  C_  E )
)
51 sstr 3316 . . . . . . 7  |-  ( ( b  C_  ( ( X  -  d ) (,) ( X  +  d ) )  /\  (
( X  -  d
) (,) ( X  +  d ) ) 
C_  E )  -> 
b  C_  E )
5251anim2i 553 . . . . . 6  |-  ( ( X  e.  b  /\  ( b  C_  (
( X  -  d
) (,) ( X  +  d ) )  /\  ( ( X  -  d ) (,) ( X  +  d ) )  C_  E
) )  ->  ( X  e.  b  /\  b  C_  E ) )
5352anassrs 630 . . . . 5  |-  ( ( ( X  e.  b  /\  b  C_  (
( X  -  d
) (,) ( X  +  d ) ) )  /\  ( ( X  -  d ) (,) ( X  +  d ) )  C_  E )  ->  ( X  e.  b  /\  b  C_  E ) )
5453reximi 2773 . . . 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 205 . . 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 2786 . 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   _Vcvv 2916    C_ wss 3280    X. cxp 4835   ran crn 4838    |` cres 4839    o. ccom 4841   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   RRcr 8945   1c1 8947    + caddc 8949    - cmin 9247    / cdiv 9633   2c2 10005   ZZcz 10238   RR+crp 10568   (,)cioo 10872   [,)cico 10874   |_cfl 11156   ^cexp 11337   abscabs 11994   ↾s cress 13425   distcds 13493   TopOpenctopn 13604   topGenctg 13620   * Metcxmt 16641   ballcbl 16643   MetOpencmopn 16646  ℂfldccnfld 16658   *
MetSpcxme 18300  logbclogb 24341
This theorem is referenced by:  dya2iocnrect  24584
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-cxp 20408  df-logb 24342
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