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_ 𝔅ℝ

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.

Step	Hyp	Ref	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
3	1, 2	elrnmpt2 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*
6	5	a1i 11	. . . . . . . . 9 |- ( ( x e. ZZ /\ n e. ZZ ) -> -oo e. RR* )
7		simpl 457	. . . . . . . . . . . 12 |- ( ( x e. ZZ /\ n e. ZZ ) -> x e. ZZ )
8	7	zred 10990	. . . . . . . . . . 11 |- ( ( x e. ZZ /\ n e. ZZ ) -> x e. RR )
9		2rp 11250	. . . . . . . . . . . . 13 |- 2 e. RR+
10	9	a1i 11	. . . . . . . . . . . 12 |- ( ( x e. ZZ /\ n e. ZZ ) -> 2 e. RR+ )
11		simpr 461	. . . . . . . . . . . 12 |- ( ( x e. ZZ /\ n e. ZZ ) -> n e. ZZ )
12	10, 11	rpexpcld 12335	. . . . . . . . . . 11 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( 2 ^ n ) e. RR+ )
13	8, 12	rerpdivcld 11308	. . . . . . . . . 10 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( x / ( 2 ^ n ) ) e. RR )
14	13	rexrd 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 )
16	8, 15	readdcld 9640	. . . . . . . . . . 11 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( x + 1 ) e. RR )
17	16, 12	rerpdivcld 11308	. . . . . . . . . 10 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( ( x + 1 ) / ( 2 ^ n ) ) e. RR )
18	17	rexrd 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 ) ) )
20	13, 19	syl 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 ) ) ) )
22	6, 14, 18, 20, 21	syl31anc 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 |- 𝔅ℝ e. (sigAlgebra ` RR )
24		elrnsiga 28287	. . . . . . . . . 10 |- (𝔅ℝ e. (sigAlgebra ` RR ) -> 𝔅ℝ e. U. ran sigAlgebra )
25	23, 24	ax-mp 5	. . . . . . . . 9 |- 𝔅ℝ 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 (,) ) ) )
29	26, 27, 28	mp2an 672	. . . . . . . . . 10 |- ( -oo (,) ( ( x + 1 ) / ( 2 ^ n ) ) ) e. (sigaGen ` ( topGen ` ran (,) ) )
30		df-brsiga 28314	. . . . . . . . . 10 |- 𝔅ℝ = (sigaGen ` ( topGen ` ran (,) ) )
31	29, 30	eleqtrri 2544	. . . . . . . . 9 |- ( -oo (,) ( ( x + 1 ) / ( 2 ^ n ) ) ) e. 𝔅ℝ
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 (,) ) ) )
34	26, 32, 33	mp2an 672	. . . . . . . . . 10 |- ( -oo (,) ( x / ( 2 ^ n ) ) ) e. (sigaGen ` ( topGen ` ran (,) ) )
35	34, 30	eleqtrri 2544	. . . . . . . . 9 |- ( -oo (,) ( x / ( 2 ^ n ) ) ) e. 𝔅ℝ
36		difelsiga 28294	. . . . . . . . 9 |- ( (𝔅ℝ e. U. ran sigAlgebra /\ ( -oo (,) ( ( x + 1 ) / ( 2 ^ n ) ) ) e. 𝔅ℝ /\ ( -oo (,) ( x / ( 2 ^ n ) ) ) e. 𝔅ℝ ) -> ( ( -oo (,) ( ( x + 1 ) / ( 2 ^ n ) ) ) \ ( -oo (,) ( x / ( 2 ^ n ) ) ) ) e. 𝔅ℝ )
37	25, 31, 35, 36	mp3an 1324	. . . . . . . 8 |- ( ( -oo (,) ( ( x + 1 ) / ( 2 ^ n ) ) ) \ ( -oo (,) ( x / ( 2 ^ n ) ) ) ) e. 𝔅ℝ
38	22, 37	syl6eqelr 2554	. . . . . . 7 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) e. 𝔅ℝ )
39	38	adantr 465	. . . . . 6 |- ( ( ( x e. ZZ /\ n e. ZZ ) /\ d = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) ) -> ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) e. 𝔅ℝ )
40	4, 39	eqeltrd 2545	. . . . 5 |- ( ( ( x e. ZZ /\ n e. ZZ ) /\ d = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) ) -> d e. 𝔅ℝ )
41	40	ex 434	. . . 4 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( d = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) -> d e. 𝔅ℝ ) )
42	41	rexlimivv 2954	. . 3 |- ( E. x e. ZZ E. n e. ZZ d = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) -> d e. 𝔅ℝ )
43	3, 42	sylbi 195	. 2 |- ( d e. ran I -> d e. 𝔅ℝ )
44	43	ssriv 3503	1 |- ran I C_ 𝔅ℝ

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  𝔅_ℝcbrsiga 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