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Theorem dya2icoseg2 28122
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 28121 . . . . 5  |-  ( ( X  e.  RR  /\  d  e.  RR+ )  ->  E. b  e.  ran  I ( X  e.  b  /\  b  C_  ( ( X  -  d ) (,) ( X  +  d )
) ) )
54ralrimiva 2857 . . . 4  |-  ( X  e.  RR  ->  A. d  e.  RR+  E. b  e. 
ran  I ( X  e.  b  /\  b  C_  ( ( X  -  d ) (,) ( X  +  d )
) ) )
653ad2ant1 1018 . . 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 999 . . . . 5  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  X  e.  E )
8 iooex 11561 . . . . . . . . . 10  |-  (,)  e.  _V
98rnex 6719 . . . . . . . . 9  |-  ran  (,)  e.  _V
10 bastg 19340 . . . . . . . . 9  |-  ( ran 
(,)  e.  _V  ->  ran 
(,)  C_  ( topGen `  ran  (,) ) )
119, 10ax-mp 5 . . . . . . . 8  |-  ran  (,)  C_  ( topGen `  ran  (,) )
12 simp2 998 . . . . . . . 8  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  E  e.  ran  (,) )
1311, 12sseldi 3487 . . . . . . 7  |-  ( ( X  e.  RR  /\  E  e.  ran  (,)  /\  X  e.  E )  ->  E  e.  ( topGen ` 
ran  (,) ) )
1413, 1syl6eleqr 2542 . . . . . 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 21169 . . . . . . . 8  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( *Met `  RR )
17 recms 21685 . . . . . . . . . . 11  |- RRfld  e. CMetSp
18 cmsms 21660 . . . . . . . . . . 11  |-  (RRfld  e. CMetSp  -> RRfld  e. 
MetSp )
19 msxms 20830 . . . . . . . . . . 11  |-  (RRfld  e.  MetSp  -> RRfld 
e.  *MetSp )
2017, 18, 19mp2b 10 . . . . . . . . . 10  |- RRfld  e.  *MetSp
21 retopn 21684 . . . . . . . . . . . 12  |-  ( topGen ` 
ran  (,) )  =  (
TopOpen ` RRfld )
221, 21eqtri 2472 . . . . . . . . . . 11  |-  J  =  ( TopOpen ` RRfld )
23 rebase 18515 . . . . . . . . . . 11  |-  RR  =  ( Base ` RRfld )
24 reds 18525 . . . . . . . . . . . 12  |-  ( abs 
o.  -  )  =  ( dist ` RRfld )
2524reseq1i 5259 . . . . . . . . . . 11  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( dist ` RRfld )  |`  ( RR  X.  RR ) )
2622, 23, 25xmstopn 20827 . . . . . . . . . 10  |-  (RRfld  e.  *MetSp  ->  J  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) ) )
2720, 26ax-mp 5 . . . . . . . . 9  |-  J  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) )
2827elmopn2 20821 . . . . . . . 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 6288 . . . . . . . 8  |-  ( x  =  X  ->  (
x ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) d )  =  ( X ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) d ) )
3332sseq1d 3516 . . . . . . 7  |-  ( x  =  X  ->  (
( x ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) d )  C_  E 
<->  ( X ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) d )  C_  E ) )
3433rexbidv 2954 . . . . . 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 3194 . . . . 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 11235 . . . . . . 7  |-  ( d  e.  RR+  ->  d  e.  RR )
3815bl2ioo 21170 . . . . . . . 8  |-  ( ( X  e.  RR  /\  d  e.  RR )  ->  ( X ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) d )  =  ( ( X  -  d ) (,) ( X  +  d )
) )
3938sseq1d 3516 . . . . . . 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 2951 . . . . 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 1018 . . . 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 2978 . . 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 2995 . . . 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 3497 . . . . . . 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 2911 . . . 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 2932 . 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 974    = wceq 1383    e. wcel 1804   A.wral 2793   E.wrex 2794   _Vcvv 3095    C_ wss 3461    X. cxp 4987   ran crn 4990    |` cres 4991    o. ccom 4993   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283   RRcr 9494   1c1 9496    + caddc 9498    - cmin 9810    / cdiv 10212   2c2 10591   ZZcz 10870   RR+crp 11229   (,)cioo 11538   [,)cico 11540   |_cfl 11906   ^cexp 12145   abscabs 13046   distcds 14583   TopOpenctopn 14696   topGenctg 14712   *Metcxmt 18277   ballcbl 18279   MetOpencmopn 18282  RRfldcrefld 18513   *MetSpcxme 20693   MetSpcmt 20694  CMetSpccms 21644  logbclogb 27879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-q 11192  df-rp 11230  df-xneg 11327  df-xadd 11328  df-xmul 11329  df-ioo 11542  df-ioc 11543  df-ico 11544  df-icc 11545  df-fz 11682  df-fzo 11804  df-fl 11908  df-mod 11976  df-seq 12087  df-exp 12146  df-fac 12333  df-bc 12360  df-hash 12385  df-shft 12879  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-limsup 13273  df-clim 13290  df-rlim 13291  df-sum 13488  df-ef 13681  df-sin 13683  df-cos 13684  df-pi 13686  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-starv 14589  df-sca 14590  df-vsca 14591  df-ip 14592  df-tset 14593  df-ple 14594  df-ds 14596  df-unif 14597  df-hom 14598  df-cco 14599  df-rest 14697  df-topn 14698  df-0g 14716  df-gsum 14717  df-topgen 14718  df-pt 14719  df-prds 14722  df-xrs 14776  df-qtop 14781  df-imas 14782  df-xps 14784  df-mre 14860  df-mrc 14861  df-acs 14863  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15841  df-mulg 15934  df-cntz 16229  df-cmn 16674  df-psmet 18285  df-xmet 18286  df-met 18287  df-bl 18288  df-mopn 18289  df-fbas 18290  df-fg 18291  df-cnfld 18295  df-refld 18514  df-top 19272  df-bases 19274  df-topon 19275  df-topsp 19276  df-cld 19393  df-ntr 19394  df-cls 19395  df-nei 19472  df-lp 19510  df-perf 19511  df-cn 19601  df-cnp 19602  df-haus 19689  df-cmp 19760  df-tx 19936  df-hmeo 20129  df-fil 20220  df-fm 20312  df-flim 20313  df-flf 20314  df-fcls 20315  df-xms 20696  df-ms 20697  df-tms 20698  df-cncf 21255  df-cfil 21567  df-cmet 21569  df-cms 21647  df-limc 22143  df-dv 22144  df-log 22816  df-cxp 22817  df-logb 27880
This theorem is referenced by:  dya2iocnrect  28125
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