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Theorem dya2icobrsiga 28408
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 6324 . . . 4  |-  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  +  1 )  / 
( 2 ^ n
) ) )  e. 
_V
31, 2elrnmpt2 6414 . . 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 11348 . . . . . . . . . 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 10990 . . . . . . . . . . 11  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  x  e.  RR )
9 2rp 11250 . . . . . . . . . . . . 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 12335 . . . . . . . . . . 11  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( 2 ^ n
)  e.  RR+ )
138, 12rerpdivcld 11308 . . . . . . . . . 10  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( x  /  (
2 ^ n ) )  e.  RR )
1413rexrd 9660 . . . . . . . . 9  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( x  /  (
2 ^ n ) )  e.  RR* )
15 1red 9628 . . . . . . . . . . . 12  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  1  e.  RR )
168, 15readdcld 9640 . . . . . . . . . . 11  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( x  +  1 )  e.  RR )
1716, 12rerpdivcld 11308 . . . . . . . . . 10  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( x  + 
1 )  /  (
2 ^ n ) )  e.  RR )
1817rexrd 9660 . . . . . . . . 9  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( x  + 
1 )  /  (
2 ^ n ) )  e.  RR* )
19 mnflt 11358 . . . . . . . . . 10  |-  ( ( x  /  ( 2 ^ n ) )  e.  RR  -> -oo  <  ( x  /  ( 2 ^ n ) ) )
2013, 19syl 16 . . . . . . . . 9  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  -> -oo  <  ( x  /  ( 2 ^ n ) ) )
21 difioo 27745 . . . . . . . . 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 ) ) ) )
226, 14, 18, 20, 21syl31anc 1231 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( -oo (,) ( ( x  + 
1 )  /  (
2 ^ n ) ) )  \  ( -oo (,) ( x  / 
( 2 ^ n
) ) ) )  =  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  + 
1 )  /  (
2 ^ n ) ) ) )
23 brsigarn 28316 . . . . . . . . . 10  |- 𝔅  e.  (sigAlgebra `  RR )
24 elrnsiga 28287 . . . . . . . . . 10  |-  (𝔅  e.  (sigAlgebra `  RR )  -> 𝔅  e.  U. ran sigAlgebra )
2523, 24ax-mp 5 . . . . . . . . 9  |- 𝔅  e.  U. ran sigAlgebra
26 retop 21393 . . . . . . . . . . 11  |-  ( topGen ` 
ran  (,) )  e.  Top
27 iooretop 21398 . . . . . . . . . . 11  |-  ( -oo (,) ( ( x  + 
1 )  /  (
2 ^ n ) ) )  e.  (
topGen `  ran  (,) )
28 elsigagen 28308 . . . . . . . . . . 11  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( -oo (,) ( ( x  + 
1 )  /  (
2 ^ n ) ) )  e.  (
topGen `  ran  (,) )
)  ->  ( -oo (,) ( ( x  + 
1 )  /  (
2 ^ n ) ) )  e.  (sigaGen `  ( topGen `  ran  (,) )
) )
2926, 27, 28mp2an 672 . . . . . . . . . 10  |-  ( -oo (,) ( ( x  + 
1 )  /  (
2 ^ n ) ) )  e.  (sigaGen `  ( topGen `  ran  (,) )
)
30 df-brsiga 28314 . . . . . . . . . 10  |- 𝔅  =  (sigaGen `  ( topGen `
 ran  (,) )
)
3129, 30eleqtrri 2544 . . . . . . . . 9  |-  ( -oo (,) ( ( x  + 
1 )  /  (
2 ^ n ) ) )  e. 𝔅
32 iooretop 21398 . . . . . . . . . . 11  |-  ( -oo (,) ( x  /  (
2 ^ n ) ) )  e.  (
topGen `  ran  (,) )
33 elsigagen 28308 . . . . . . . . . . 11  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( -oo (,) ( x  /  (
2 ^ n ) ) )  e.  (
topGen `  ran  (,) )
)  ->  ( -oo (,) ( x  /  (
2 ^ n ) ) )  e.  (sigaGen `  ( topGen `  ran  (,) )
) )
3426, 32, 33mp2an 672 . . . . . . . . . 10  |-  ( -oo (,) ( x  /  (
2 ^ n ) ) )  e.  (sigaGen `  ( topGen `  ran  (,) )
)
3534, 30eleqtrri 2544 . . . . . . . . 9  |-  ( -oo (,) ( x  /  (
2 ^ n ) ) )  e. 𝔅
36 difelsiga 28294 . . . . . . . . 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 )
3725, 31, 35, 36mp3an 1324 . . . . . . . 8  |-  ( ( -oo (,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) 
\  ( -oo (,) ( x  /  (
2 ^ n ) ) ) )  e. 𝔅
3822, 37syl6eqelr 2554 . . . . . . 7  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) )  e. 𝔅 )
3938adantr 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 )
404, 39eqeltrd 2545 . . . . 5  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  d  e. 𝔅 )
4140ex 434 . . . 4  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) )  ->  d  e. 𝔅 ) )
4241rexlimivv 2954 . . 3  |-  ( E. x  e.  ZZ  E. n  e.  ZZ  d  =  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  + 
1 )  /  (
2 ^ n ) ) )  ->  d  e. 𝔅 )
433, 42sylbi 195 . 2  |-  ( d  e.  ran  I  -> 
d  e. 𝔅 )
4443ssriv 3503 1  |-  ran  I  C_ 𝔅
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1395    e. wcel 1819   E.wrex 2808    \ cdif 3468    C_ wss 3471   U.cuni 4251   class class class wbr 4456   ran crn 5009   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   RRcr 9508   1c1 9510    + caddc 9512   -oocmnf 9643   RR*cxr 9644    < clt 9645    / cdiv 10227   2c2 10606   ZZcz 10885   RR+crp 11245   (,)cioo 11554   [,)cico 11556   ^cexp 12168   topGenctg 14854   Topctop 19520  sigAlgebracsiga 28268  sigaGencsigagen 28299  𝔅cbrsiga 28313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-ac2 8860  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-acn 8340  df-ac 8514  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-ioo 11558  df-ico 11560  df-seq 12110  df-exp 12169  df-topgen 14860  df-top 19525  df-bases 19527  df-siga 28269  df-sigagen 28300  df-brsiga 28314
This theorem is referenced by:  sxbrsigalem2  28418  sxbrsigalem5  28420
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