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_ 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 6215	. . . 4 |- ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) e. _V
3	1, 2	elrnmpt2 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*
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 10848	. . . . . . . . . . 11 |- ( ( x e. ZZ /\ n e. ZZ ) -> x e. RR )
9		2rp 11097	. . . . . . . . . . . . 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 12132	. . . . . . . . . . 11 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( 2 ^ n ) e. RR+ )
13	8, 12	rerpdivcld 11155	. . . . . . . . . 10 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( x / ( 2 ^ n ) ) e. RR )
14	13	rexrd 9534	. . . . . . . . 9 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( x / ( 2 ^ n ) ) e. RR* )
15		1re 9486	. . . . . . . . . . . . 13 |- 1 e. RR
16	15	a1i 11	. . . . . . . . . . . 12 |- ( ( x e. ZZ /\ n e. ZZ ) -> 1 e. RR )
17	8, 16	readdcld 9514	. . . . . . . . . . 11 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( x + 1 ) e. RR )
18	17, 12	rerpdivcld 11155	. . . . . . . . . 10 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( ( x + 1 ) / ( 2 ^ n ) ) e. RR )
19	18	rexrd 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 ) ) )
21	13, 20	syl 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 ) ) ) )
23	6, 14, 19, 21, 22	syl31anc 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 |- 𝔅ℝ e. (sigAlgebra ` RR )
25		elrnsiga 26703	. . . . . . . . . 10 |- (𝔅ℝ e. (sigAlgebra ` RR ) -> 𝔅ℝ e. U. ran sigAlgebra )
26	24, 25	ax-mp 5	. . . . . . . . 9 |- 𝔅ℝ 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 (,) ) ) )
30	27, 28, 29	mp2an 672	. . . . . . . . . 10 |- ( -oo (,) ( ( x + 1 ) / ( 2 ^ n ) ) ) e. (sigaGen ` ( topGen ` ran (,) ) )
31		df-brsiga 26730	. . . . . . . . . 10 |- 𝔅ℝ = (sigaGen ` ( topGen ` ran (,) ) )
32	30, 31	eleqtrri 2538	. . . . . . . . 9 |- ( -oo (,) ( ( x + 1 ) / ( 2 ^ n ) ) ) e. 𝔅ℝ
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 (,) ) ) )
35	27, 33, 34	mp2an 672	. . . . . . . . . 10 |- ( -oo (,) ( x / ( 2 ^ n ) ) ) e. (sigaGen ` ( topGen ` ran (,) ) )
36	35, 31	eleqtrri 2538	. . . . . . . . 9 |- ( -oo (,) ( x / ( 2 ^ n ) ) ) e. 𝔅ℝ
37		difelsiga 26710	. . . . . . . . 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 )
38	26, 32, 36, 37	mp3an 1315	. . . . . . . 8 |- ( ( -oo (,) ( ( x + 1 ) / ( 2 ^ n ) ) ) \ ( -oo (,) ( x / ( 2 ^ n ) ) ) ) e. 𝔅ℝ
39	23, 38	syl6eqelr 2548	. . . . . . 7 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) e. 𝔅ℝ )
40	39	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 )
41	4, 40	eqeltrd 2539	. . . . 5 |- ( ( ( x e. ZZ /\ n e. ZZ ) /\ d = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) ) -> d e. 𝔅ℝ )
42	41	ex 434	. . . 4 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( d = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) -> d e. 𝔅ℝ ) )
43	42	rexlimivv 2942	. . 3 |- ( E. x e. ZZ E. n e. ZZ d = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) -> d e. 𝔅ℝ )
44	3, 43	sylbi 195	. 2 |- ( d e. ran I -> d e. 𝔅ℝ )
45	44	ssriv 3458	1 |- ran I C_ 𝔅ℝ

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  𝔅_ℝcbrsiga 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