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Theorem dya2icobrsiga 26825
Description: Dyadic intervals are Borel sets of  RR. (Contributed by Thierry Arnoux, 22-Sep-2017.) (Revised by Thierry Arnoux, 13-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
dya2icobrsiga  |-  ran  I  C_ 𝔅
Distinct variable group:    x, n
Allowed substitution hints:    I( x, n)    J( x, n)

Proof of Theorem dya2icobrsiga
Dummy variable  d is distinct from all other variables.
StepHypRef Expression
1 dya2ioc.1 . . . 4  |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) ) )
2 ovex 6215 . . . 4  |-  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  +  1 )  / 
( 2 ^ n
) ) )  e. 
_V
31, 2elrnmpt2 6303 . . 3  |-  ( d  e.  ran  I  <->  E. x  e.  ZZ  E. n  e.  ZZ  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )
4 simpr 461 . . . . . 6  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  d  =  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) ) )
5 mnfxr 11195 . . . . . . . . . 10  |- -oo  e.  RR*
65a1i 11 . . . . . . . . 9  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  -> -oo  e.  RR* )
7 simpl 457 . . . . . . . . . . . 12  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  x  e.  ZZ )
87zred 10848 . . . . . . . . . . 11  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  x  e.  RR )
9 2rp 11097 . . . . . . . . . . . . 13  |-  2  e.  RR+
109a1i 11 . . . . . . . . . . . 12  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  2  e.  RR+ )
11 simpr 461 . . . . . . . . . . . 12  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  n  e.  ZZ )
1210, 11rpexpcld 12132 . . . . . . . . . . 11  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( 2 ^ n
)  e.  RR+ )
138, 12rerpdivcld 11155 . . . . . . . . . 10  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( x  /  (
2 ^ n ) )  e.  RR )
1413rexrd 9534 . . . . . . . . 9  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( x  /  (
2 ^ n ) )  e.  RR* )
15 1re 9486 . . . . . . . . . . . . 13  |-  1  e.  RR
1615a1i 11 . . . . . . . . . . . 12  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  1  e.  RR )
178, 16readdcld 9514 . . . . . . . . . . 11  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( x  +  1 )  e.  RR )
1817, 12rerpdivcld 11155 . . . . . . . . . 10  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( x  + 
1 )  /  (
2 ^ n ) )  e.  RR )
1918rexrd 9534 . . . . . . . . 9  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( x  + 
1 )  /  (
2 ^ n ) )  e.  RR* )
20 mnflt 11205 . . . . . . . . . 10  |-  ( ( x  /  ( 2 ^ n ) )  e.  RR  -> -oo  <  ( x  /  ( 2 ^ n ) ) )
2113, 20syl 16 . . . . . . . . 9  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  -> -oo  <  ( x  /  ( 2 ^ n ) ) )
22 difioo 26206 . . . . . . . . 9  |-  ( ( ( -oo  e.  RR*  /\  ( x  /  (
2 ^ n ) )  e.  RR*  /\  (
( x  +  1 )  /  ( 2 ^ n ) )  e.  RR* )  /\ -oo  <  ( x  /  (
2 ^ n ) ) )  ->  (
( -oo (,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) 
\  ( -oo (,) ( x  /  (
2 ^ n ) ) ) )  =  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) ) )
236, 14, 19, 21, 22syl31anc 1222 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( -oo (,) ( ( x  + 
1 )  /  (
2 ^ n ) ) )  \  ( -oo (,) ( x  / 
( 2 ^ n
) ) ) )  =  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  + 
1 )  /  (
2 ^ n ) ) ) )
24 brsigarn 26732 . . . . . . . . . 10  |- 𝔅  e.  (sigAlgebra `  RR )
25 elrnsiga 26703 . . . . . . . . . 10  |-  (𝔅  e.  (sigAlgebra `  RR )  -> 𝔅  e.  U. ran sigAlgebra )
2624, 25ax-mp 5 . . . . . . . . 9  |- 𝔅  e.  U. ran sigAlgebra
27 retop 20456 . . . . . . . . . . 11  |-  ( topGen ` 
ran  (,) )  e.  Top
28 iooretop 20461 . . . . . . . . . . 11  |-  ( -oo (,) ( ( x  + 
1 )  /  (
2 ^ n ) ) )  e.  (
topGen `  ran  (,) )
29 elsigagen 26724 . . . . . . . . . . 11  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( -oo (,) ( ( x  + 
1 )  /  (
2 ^ n ) ) )  e.  (
topGen `  ran  (,) )
)  ->  ( -oo (,) ( ( x  + 
1 )  /  (
2 ^ n ) ) )  e.  (sigaGen `  ( topGen `  ran  (,) )
) )
3027, 28, 29mp2an 672 . . . . . . . . . 10  |-  ( -oo (,) ( ( x  + 
1 )  /  (
2 ^ n ) ) )  e.  (sigaGen `  ( topGen `  ran  (,) )
)
31 df-brsiga 26730 . . . . . . . . . 10  |- 𝔅  =  (sigaGen `  ( topGen `
 ran  (,) )
)
3230, 31eleqtrri 2538 . . . . . . . . 9  |-  ( -oo (,) ( ( x  + 
1 )  /  (
2 ^ n ) ) )  e. 𝔅
33 iooretop 20461 . . . . . . . . . . 11  |-  ( -oo (,) ( x  /  (
2 ^ n ) ) )  e.  (
topGen `  ran  (,) )
34 elsigagen 26724 . . . . . . . . . . 11  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( -oo (,) ( x  /  (
2 ^ n ) ) )  e.  (
topGen `  ran  (,) )
)  ->  ( -oo (,) ( x  /  (
2 ^ n ) ) )  e.  (sigaGen `  ( topGen `  ran  (,) )
) )
3527, 33, 34mp2an 672 . . . . . . . . . 10  |-  ( -oo (,) ( x  /  (
2 ^ n ) ) )  e.  (sigaGen `  ( topGen `  ran  (,) )
)
3635, 31eleqtrri 2538 . . . . . . . . 9  |-  ( -oo (,) ( x  /  (
2 ^ n ) ) )  e. 𝔅
37 difelsiga 26710 . . . . . . . . 9  |-  ( (𝔅  e.  U.
ran sigAlgebra  /\  ( -oo (,) ( ( x  + 
1 )  /  (
2 ^ n ) ) )  e. 𝔅  /\  ( -oo (,) ( x  /  (
2 ^ n ) ) )  e. 𝔅 )  ->  ( ( -oo (,) ( ( x  +  1 )  / 
( 2 ^ n
) ) )  \ 
( -oo (,) ( x  /  ( 2 ^ n ) ) ) )  e. 𝔅 )
3826, 32, 36, 37mp3an 1315 . . . . . . . 8  |-  ( ( -oo (,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) 
\  ( -oo (,) ( x  /  (
2 ^ n ) ) ) )  e. 𝔅
3923, 38syl6eqelr 2548 . . . . . . 7  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) )  e. 𝔅 )
4039adantr 465 . . . . . 6  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( (
x  /  ( 2 ^ n ) ) [,) ( ( x  +  1 )  / 
( 2 ^ n
) ) )  e. 𝔅 )
414, 40eqeltrd 2539 . . . . 5  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  d  e. 𝔅 )
4241ex 434 . . . 4  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) )  ->  d  e. 𝔅 ) )
4342rexlimivv 2942 . . 3  |-  ( E. x  e.  ZZ  E. n  e.  ZZ  d  =  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  + 
1 )  /  (
2 ^ n ) ) )  ->  d  e. 𝔅 )
443, 43sylbi 195 . 2  |-  ( d  e.  ran  I  -> 
d  e. 𝔅 )
4544ssriv 3458 1  |-  ran  I  C_ 𝔅
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2796    \ cdif 3423    C_ wss 3426   U.cuni 4189   class class class wbr 4390   ran crn 4939   ` cfv 5516  (class class class)co 6190    |-> cmpt2 6192   RRcr 9382   1c1 9384    + caddc 9386   -oocmnf 9517   RR*cxr 9518    < clt 9519    / cdiv 10094   2c2 10472   ZZcz 10747   RR+crp 11092   (,)cioo 11401   [,)cico 11403   ^cexp 11966   topGenctg 14478   Topctop 18614  sigAlgebracsiga 26684  sigaGencsigagen 26715  𝔅cbrsiga 26729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-ac2 8733  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-iin 4272  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-2o 7021  df-oadd 7024  df-er 7201  df-map 7316  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-sup 7792  df-oi 7825  df-card 8210  df-acn 8213  df-ac 8387  df-cda 8438  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-n0 10681  df-z 10748  df-uz 10963  df-q 11055  df-rp 11093  df-ioo 11405  df-ico 11407  df-seq 11908  df-exp 11967  df-topgen 14484  df-top 18619  df-bases 18621  df-siga 26685  df-sigagen 26716  df-brsiga 26730
This theorem is referenced by:  sxbrsigalem2  26835  sxbrsigalem5  26837
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