Theorem sxbrsigalem2 27897

Description: The sigma-algebra generated by the dyadic closed-below, open-above rectangular subsets of ( RR X. RR ) is a subset of the sigma algebra generated by the closed half-spaces of ( RR X. RR ). The proof goes by noting the fact that the dyadic rectangles are intersections of a 'vertical band' and an 'horizontal band', which themselves are differences of closed half-spaces. (Contributed by Thierry Arnoux, 17-Sep-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 ) ) ) )
dya2ioc.2	|- R = ( u e. ran I , v e. ran I |-> ( u X. v ) )

Assertion

Ref	Expression
sxbrsigalem2	|- (sigaGen ` ran R ) C_ (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) )

Distinct variable groups:   x, n   x, I   v, u, I, x   u, n, v   R, n, x   x, J   e, f, n, u, v, x

Allowed substitution hints:   R( v, u, e, f)   I( e, f, n)   J( v, u, e, f, n)

Proof of Theorem sxbrsigalem2

Dummy variable d is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sxbrsiga.0	. . . 4 |- J = ( topGen ` ran (,) )
2		dya2ioc.1	. . . 4 |- I = ( x e. ZZ , n e. ZZ |-> ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) )
3		dya2ioc.2	. . . 4 |- R = ( u e. ran I , v e. ran I |-> ( u X. v ) )
4	1, 2, 3	dya2iocucvr 27895	. . 3 |- U. ran R = ( RR X. RR )
5		sxbrsigalem0 27882	. . 3 |- U. ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) = ( RR X. RR )
6	4, 5	eqtr4i 2499	. 2 |- U. ran R = U. ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) )
7		vex 3116	. . . . . 6 |- u e. _V
8		vex 3116	. . . . . 6 |- v e. _V
9	7, 8	xpex 6711	. . . . 5 |- ( u X. v ) e. _V
10	3, 9	elrnmpt2 6397	. . . 4 |- ( d e. ran R <-> E. u e. ran I E. v e. ran I d = ( u X. v ) )
11		simpr 461	. . . . . . 7 |- ( ( ( u e. ran I /\ v e. ran I ) /\ d = ( u X. v ) ) -> d = ( u X. v ) )
12	1, 2	dya2icobrsiga 27887	. . . . . . . . . . . . 13 |- ran I C_ 𝔅ℝ
13		brsigasspwrn 27796	. . . . . . . . . . . . 13 |- 𝔅ℝ C_ ~P RR
14	12, 13	sstri 3513	. . . . . . . . . . . 12 |- ran I C_ ~P RR
15	14	sseli 3500	. . . . . . . . . . 11 |- ( u e. ran I -> u e. ~P RR )
16	15	elpwid 4020	. . . . . . . . . 10 |- ( u e. ran I -> u C_ RR )
17	14	sseli 3500	. . . . . . . . . . 11 |- ( v e. ran I -> v e. ~P RR )
18	17	elpwid 4020	. . . . . . . . . 10 |- ( v e. ran I -> v C_ RR )
19		xpinpreima2 27525	. . . . . . . . . 10 |- ( ( u C_ RR /\ v C_ RR ) -> ( u X. v ) = ( ( `' ( 1st |` ( RR X. RR ) ) " u ) i^i ( `' ( 2nd |` ( RR X. RR ) ) " v ) ) )
20	16, 18, 19	syl2an 477	. . . . . . . . 9 |- ( ( u e. ran I /\ v e. ran I ) -> ( u X. v ) = ( ( `' ( 1st |` ( RR X. RR ) ) " u ) i^i ( `' ( 2nd |` ( RR X. RR ) ) " v ) ) )
21		reex 9579	. . . . . . . . . . . . . . . . 17 |- RR e. _V
22	21	mptex 6129	. . . . . . . . . . . . . . . 16 |- ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) e. _V
23	22	rnex 6715	. . . . . . . . . . . . . . 15 |- ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) e. _V
24	21	mptex 6129	. . . . . . . . . . . . . . . 16 |- ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) e. _V
25	24	rnex 6715	. . . . . . . . . . . . . . 15 |- ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) e. _V
26	23, 25	unex 6580	. . . . . . . . . . . . . 14 |- ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) e. _V
27	26	a1i 11	. . . . . . . . . . . . 13 |- ( T. -> ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) e. _V )
28	27	sgsiga 27782	. . . . . . . . . . . 12 |- ( T. -> (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) e. U. ran sigAlgebra )
29	28	trud 1388	. . . . . . . . . . 11 |- (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) e. U. ran sigAlgebra
30	29	a1i 11	. . . . . . . . . 10 |- ( ( u e. ran I /\ v e. ran I ) -> (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) e. U. ran sigAlgebra )
31		1stpreima 27196	. . . . . . . . . . . . 13 |- ( u C_ RR -> ( `' ( 1st |` ( RR X. RR ) ) " u ) = ( u X. RR ) )
32	16, 31	syl 16	. . . . . . . . . . . 12 |- ( u e. ran I -> ( `' ( 1st |` ( RR X. RR ) ) " u ) = ( u X. RR ) )
33		ovex 6307	. . . . . . . . . . . . . 14 |- ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) e. _V
34	2, 33	elrnmpt2 6397	. . . . . . . . . . . . 13 |- ( u e. ran I <-> E. x e. ZZ E. n e. ZZ u = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) )
35		simpr 461	. . . . . . . . . . . . . . . . 17 |- ( ( ( x e. ZZ /\ n e. ZZ ) /\ u = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) ) -> u = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) )
36	35	xpeq1d 5022	. . . . . . . . . . . . . . . 16 |- ( ( ( x e. ZZ /\ n e. ZZ ) /\ u = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) ) -> ( u X. RR ) = ( ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) X. RR ) )
37		difxp1 5430	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( x / ( 2 ^ n ) ) [,) +oo ) \ ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) X. RR ) = ( ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) \ ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR ) )
38		simpl 457	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( x e. ZZ /\ n e. ZZ ) -> x e. ZZ )
39	38	zred 10962	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( x e. ZZ /\ n e. ZZ ) -> x e. RR )
40		2rp 11221	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- 2 e. RR+
41	40	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( x e. ZZ /\ n e. ZZ ) -> 2 e. RR+ )
42		simpr 461	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( x e. ZZ /\ n e. ZZ ) -> n e. ZZ )
43	41, 42	rpexpcld 12297	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( 2 ^ n ) e. RR+ )
44	39, 43	rerpdivcld 11279	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( x / ( 2 ^ n ) ) e. RR )
45	44	rexrd 9639	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( x / ( 2 ^ n ) ) e. RR* )
46		1re 9591	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- 1 e. RR
47	46	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( x e. ZZ /\ n e. ZZ ) -> 1 e. RR )
48	39, 47	readdcld 9619	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( x + 1 ) e. RR )
49	48, 43	rerpdivcld 11279	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( ( x + 1 ) / ( 2 ^ n ) ) e. RR )
50	49	rexrd 9639	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( ( x + 1 ) / ( 2 ^ n ) ) e. RR* )
51		pnfxr 11317	. . . . . . . . . . . . . . . . . . . . . 22 |- +oo e. RR*
52	51	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( x e. ZZ /\ n e. ZZ ) -> +oo e. RR* )
53	39	lep1d 10473	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( x e. ZZ /\ n e. ZZ ) -> x <_ ( x + 1 ) )
54	39, 48, 43, 53	lediv1dd 11306	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( x / ( 2 ^ n ) ) <_ ( ( x + 1 ) / ( 2 ^ n ) ) )
55		pnfge 11335	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( x + 1 ) / ( 2 ^ n ) ) e. RR* -> ( ( x + 1 ) / ( 2 ^ n ) ) <_ +oo )
56	50, 55	syl 16	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( ( x + 1 ) / ( 2 ^ n ) ) <_ +oo )
57		difico 27262	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( x / ( 2 ^ n ) ) e. RR* /\ ( ( x + 1 ) / ( 2 ^ n ) ) e. RR* /\ +oo e. RR* ) /\ ( ( x / ( 2 ^ n ) ) <_ ( ( x + 1 ) / ( 2 ^ n ) ) /\ ( ( x + 1 ) / ( 2 ^ n ) ) <_ +oo ) ) -> ( ( ( x / ( 2 ^ n ) ) [,) +oo ) \ ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) )
58	45, 50, 52, 54, 56, 57	syl32anc 1236	. . . . . . . . . . . . . . . . . . . 20 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( ( ( x / ( 2 ^ n ) ) [,) +oo ) \ ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) )
59	58	xpeq1d 5022	. . . . . . . . . . . . . . . . . . 19 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( ( ( ( x / ( 2 ^ n ) ) [,) +oo ) \ ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) X. RR ) = ( ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) X. RR ) )
60	37, 59	syl5reqr 2523	. . . . . . . . . . . . . . . . . 18 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) X. RR ) = ( ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) \ ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR ) ) )
61	29	a1i 11	. . . . . . . . . . . . . . . . . . 19 |- ( ( x e. ZZ /\ n e. ZZ ) -> (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) e. U. ran sigAlgebra )
62		ssun1 3667	. . . . . . . . . . . . . . . . . . . . 21 |- ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) C_ ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) )
63		eqid 2467	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) = ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR )
64		oveq1 6289	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( e = ( x / ( 2 ^ n ) ) -> ( e [,) +oo ) = ( ( x / ( 2 ^ n ) ) [,) +oo ) )
65	64	xpeq1d 5022	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( e = ( x / ( 2 ^ n ) ) -> ( ( e [,) +oo ) X. RR ) = ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) )
66	65	eqeq2d 2481	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( e = ( x / ( 2 ^ n ) ) -> ( ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) = ( ( e [,) +oo ) X. RR ) <-> ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) = ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) ) )
67	66	rspcev 3214	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( x / ( 2 ^ n ) ) e. RR /\ ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) = ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) ) -> E. e e. RR ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) = ( ( e [,) +oo ) X. RR ) )
68	44, 63, 67	sylancl 662	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( x e. ZZ /\ n e. ZZ ) -> E. e e. RR ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) = ( ( e [,) +oo ) X. RR ) )
69		eqid 2467	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) = ( e e. RR |-> ( ( e [,) +oo ) X. RR ) )
70		ovex 6307	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( e [,) +oo ) e. _V
71	70, 21	xpex 6711	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( e [,) +oo ) X. RR ) e. _V
72	69, 71	elrnmpti 5251	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) e. ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) <-> E. e e. RR ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) = ( ( e [,) +oo ) X. RR ) )
73	68, 72	sylibr 212	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) e. ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) )
74	62, 73	sseldi 3502	. . . . . . . . . . . . . . . . . . . 20 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) e. ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) )
75		elsigagen 27787	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) e. _V /\ ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) e. ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) -> ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
76	26, 74, 75	sylancr 663	. . . . . . . . . . . . . . . . . . 19 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
77		eqid 2467	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR ) = ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR )
78		oveq1 6289	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( e = ( ( x + 1 ) / ( 2 ^ n ) ) -> ( e [,) +oo ) = ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) )
79	78	xpeq1d 5022	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( e = ( ( x + 1 ) / ( 2 ^ n ) ) -> ( ( e [,) +oo ) X. RR ) = ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR ) )
80	79	eqeq2d 2481	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( e = ( ( x + 1 ) / ( 2 ^ n ) ) -> ( ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR ) = ( ( e [,) +oo ) X. RR ) <-> ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR ) = ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR ) ) )
81	80	rspcev 3214	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( x + 1 ) / ( 2 ^ n ) ) e. RR /\ ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR ) = ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR ) ) -> E. e e. RR ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR ) = ( ( e [,) +oo ) X. RR ) )
82	49, 77, 81	sylancl 662	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( x e. ZZ /\ n e. ZZ ) -> E. e e. RR ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR ) = ( ( e [,) +oo ) X. RR ) )
83	69, 71	elrnmpti 5251	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR ) e. ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) <-> E. e e. RR ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR ) = ( ( e [,) +oo ) X. RR ) )
84	82, 83	sylibr 212	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR ) e. ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) )
85	62, 84	sseldi 3502	. . . . . . . . . . . . . . . . . . . 20 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR ) e. ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) )
86		elsigagen 27787	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) e. _V /\ ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR ) e. ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) -> ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
87	26, 85, 86	sylancr 663	. . . . . . . . . . . . . . . . . . 19 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
88		difelsiga 27773	. . . . . . . . . . . . . . . . . . 19 |- ( ( (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) e. U. ran sigAlgebra /\ ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) /\ ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) ) -> ( ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) \ ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR ) ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
89	61, 76, 87, 88	syl3anc 1228	. . . . . . . . . . . . . . . . . 18 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) \ ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR ) ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
90	60, 89	eqeltrd 2555	. . . . . . . . . . . . . . . . 17 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) X. RR ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
91	90	adantr 465	. . . . . . . . . . . . . . . 16 |- ( ( ( x e. ZZ /\ n e. ZZ ) /\ u = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) ) -> ( ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) X. RR ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
92	36, 91	eqeltrd 2555	. . . . . . . . . . . . . . 15 |- ( ( ( x e. ZZ /\ n e. ZZ ) /\ u = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) ) -> ( u X. RR ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
93	92	ex 434	. . . . . . . . . . . . . 14 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( u = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) -> ( u X. RR ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) ) )
94	93	rexlimivv 2960	. . . . . . . . . . . . 13 |- ( E. x e. ZZ E. n e. ZZ u = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) -> ( u X. RR ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
95	34, 94	sylbi 195	. . . . . . . . . . . 12 |- ( u e. ran I -> ( u X. RR ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
96	32, 95	eqeltrd 2555	. . . . . . . . . . 11 |- ( u e. ran I -> ( `' ( 1st |` ( RR X. RR ) ) " u ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
97	96	adantr 465	. . . . . . . . . 10 |- ( ( u e. ran I /\ v e. ran I ) -> ( `' ( 1st |` ( RR X. RR ) ) " u ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
98		2ndpreima 27197	. . . . . . . . . . . . 13 |- ( v C_ RR -> ( `' ( 2nd |` ( RR X. RR ) ) " v ) = ( RR X. v ) )
99	18, 98	syl 16	. . . . . . . . . . . 12 |- ( v e. ran I -> ( `' ( 2nd |` ( RR X. RR ) ) " v ) = ( RR X. v ) )
100	2, 33	elrnmpt2 6397	. . . . . . . . . . . . 13 |- ( v e. ran I <-> E. x e. ZZ E. n e. ZZ v = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) )
101		simpr 461	. . . . . . . . . . . . . . . . 17 |- ( ( ( x e. ZZ /\ n e. ZZ ) /\ v = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) ) -> v = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) )
102	101	xpeq2d 5023	. . . . . . . . . . . . . . . 16 |- ( ( ( x e. ZZ /\ n e. ZZ ) /\ v = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) ) -> ( RR X. v ) = ( RR X. ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) ) )
103		difxp2 5431	. . . . . . . . . . . . . . . . . . 19 |- ( RR X. ( ( ( x / ( 2 ^ n ) ) [,) +oo ) \ ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) ) = ( ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) \ ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) )
104	58	xpeq2d 5023	. . . . . . . . . . . . . . . . . . 19 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( RR X. ( ( ( x / ( 2 ^ n ) ) [,) +oo ) \ ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) ) = ( RR X. ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) ) )
105	103, 104	syl5reqr 2523	. . . . . . . . . . . . . . . . . 18 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( RR X. ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) ) = ( ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) \ ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) ) )
106		ssun2 3668	. . . . . . . . . . . . . . . . . . . . 21 |- ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) C_ ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) )
107		eqid 2467	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) = ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) )
108		oveq1 6289	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( f = ( x / ( 2 ^ n ) ) -> ( f [,) +oo ) = ( ( x / ( 2 ^ n ) ) [,) +oo ) )
109	108	xpeq2d 5023	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( f = ( x / ( 2 ^ n ) ) -> ( RR X. ( f [,) +oo ) ) = ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) )
110	109	eqeq2d 2481	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( f = ( x / ( 2 ^ n ) ) -> ( ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) = ( RR X. ( f [,) +oo ) ) <-> ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) = ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) ) )
111	110	rspcev 3214	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( x / ( 2 ^ n ) ) e. RR /\ ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) = ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) ) -> E. f e. RR ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) = ( RR X. ( f [,) +oo ) ) )
112	44, 107, 111	sylancl 662	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( x e. ZZ /\ n e. ZZ ) -> E. f e. RR ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) = ( RR X. ( f [,) +oo ) ) )
113		eqid 2467	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) = ( f e. RR |-> ( RR X. ( f [,) +oo ) ) )
114		ovex 6307	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( f [,) +oo ) e. _V
115	21, 114	xpex 6711	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( RR X. ( f [,) +oo ) ) e. _V
116	113, 115	elrnmpti 5251	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) e. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) <-> E. f e. RR ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) = ( RR X. ( f [,) +oo ) ) )
117	112, 116	sylibr 212	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) e. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) )
118	106, 117	sseldi 3502	. . . . . . . . . . . . . . . . . . . 20 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) e. ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) )
119		elsigagen 27787	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) e. _V /\ ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) e. ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) -> ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
120	26, 118, 119	sylancr 663	. . . . . . . . . . . . . . . . . . 19 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
121		eqid 2467	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) = ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) )
122		oveq1 6289	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( f = ( ( x + 1 ) / ( 2 ^ n ) ) -> ( f [,) +oo ) = ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) )
123	122	xpeq2d 5023	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( f = ( ( x + 1 ) / ( 2 ^ n ) ) -> ( RR X. ( f [,) +oo ) ) = ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) )
124	123	eqeq2d 2481	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( f = ( ( x + 1 ) / ( 2 ^ n ) ) -> ( ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) = ( RR X. ( f [,) +oo ) ) <-> ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) = ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) ) )
125	124	rspcev 3214	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( x + 1 ) / ( 2 ^ n ) ) e. RR /\ ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) = ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) ) -> E. f e. RR ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) = ( RR X. ( f [,) +oo ) ) )
126	49, 121, 125	sylancl 662	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( x e. ZZ /\ n e. ZZ ) -> E. f e. RR ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) = ( RR X. ( f [,) +oo ) ) )
127	113, 115	elrnmpti 5251	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) e. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) <-> E. f e. RR ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) = ( RR X. ( f [,) +oo ) ) )
128	126, 127	sylibr 212	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) e. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) )
129	106, 128	sseldi 3502	. . . . . . . . . . . . . . . . . . . 20 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) e. ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) )
130		elsigagen 27787	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) e. _V /\ ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) e. ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) -> ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
131	26, 129, 130	sylancr 663	. . . . . . . . . . . . . . . . . . 19 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
132		difelsiga 27773	. . . . . . . . . . . . . . . . . . 19 |- ( ( (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) e. U. ran sigAlgebra /\ ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) /\ ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) ) -> ( ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) \ ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
133	61, 120, 131, 132	syl3anc 1228	. . . . . . . . . . . . . . . . . 18 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) \ ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
134	105, 133	eqeltrd 2555	. . . . . . . . . . . . . . . . 17 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( RR X. ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
135	134	adantr 465	. . . . . . . . . . . . . . . 16 |- ( ( ( x e. ZZ /\ n e. ZZ ) /\ v = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) ) -> ( RR X. ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
136	102, 135	eqeltrd 2555	. . . . . . . . . . . . . . 15 |- ( ( ( x e. ZZ /\ n e. ZZ ) /\ v = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) ) -> ( RR X. v ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
137	136	ex 434	. . . . . . . . . . . . . 14 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( v = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) -> ( RR X. v ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) ) )
138	137	rexlimivv 2960	. . . . . . . . . . . . 13 |- ( E. x e. ZZ E. n e. ZZ v = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) -> ( RR X. v ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
139	100, 138	sylbi 195	. . . . . . . . . . . 12 |- ( v e. ran I -> ( RR X. v ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
140	99, 139	eqeltrd 2555	. . . . . . . . . . 11 |- ( v e. ran I -> ( `' ( 2nd |` ( RR X. RR ) ) " v ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
141	140	adantl 466	. . . . . . . . . 10 |- ( ( u e. ran I /\ v e. ran I ) -> ( `' ( 2nd |` ( RR X. RR ) ) " v ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
142		inelsiga 27775	. . . . . . . . . 10 |- ( ( (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) e. U. ran sigAlgebra /\ ( `' ( 1st |` ( RR X. RR ) ) " u ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) /\ ( `' ( 2nd |` ( RR X. RR ) ) " v ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) ) -> ( ( `' ( 1st |` ( RR X. RR ) ) " u ) i^i ( `' ( 2nd |` ( RR X. RR ) ) " v ) ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
143	30, 97, 141, 142	syl3anc 1228	. . . . . . . . 9 |- ( ( u e. ran I /\ v e. ran I ) -> ( ( `' ( 1st |` ( RR X. RR ) ) " u ) i^i ( `' ( 2nd |` ( RR X. RR ) ) " v ) ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
144	20, 143	eqeltrd 2555	. . . . . . . 8 |- ( ( u e. ran I /\ v e. ran I ) -> ( u X. v ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
145	144	adantr 465	. . . . . . 7 |- ( ( ( u e. ran I /\ v e. ran I ) /\ d = ( u X. v ) ) -> ( u X. v ) e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
146	11, 145	eqeltrd 2555	. . . . . 6 |- ( ( ( u e. ran I /\ v e. ran I ) /\ d = ( u X. v ) ) -> d e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
147	146	ex 434	. . . . 5 |- ( ( u e. ran I /\ v e. ran I ) -> ( d = ( u X. v ) -> d e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) ) )
148	147	rexlimivv 2960	. . . 4 |- ( E. u e. ran I E. v e. ran I d = ( u X. v ) -> d e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
149	10, 148	sylbi 195	. . 3 |- ( d e. ran R -> d e. (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
150	149	ssriv 3508	. 2 |- ran R C_ (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) )
151		sigagenss2 27790	. 2 |- ( ( U. ran R = U. ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) /\ ran R C_ (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) /\ ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) e. _V ) -> (sigaGen ` ran R ) C_ (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) )
152	6, 150, 26, 151	mp3an 1324	1 |- (sigaGen ` ran R ) C_ (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) )

Syntax hints:   /\ wa 369   = wceq 1379   T. wtru 1380   e. wcel 1767   E.wrex 2815   _Vcvv 3113   \ cdif 3473   u. cun 3474   i^i cin 3475   C_ wss 3476   ~Pcpw 4010   U.cuni 4245   class class class wbr 4447   |-> cmpt 4505   X. cxp 4997   `'ccnv 4998   ran crn 5000   |` cres 5001   "cima 5002   ` cfv 5586  (class class class)co 6282   |-> cmpt2 6284   1stc1st 6779   2ndc2nd 6780   RRcr 9487   1c1 9489   + caddc 9491   +oocpnf 9621   RR*cxr 9623   <_ cle 9625   / cdiv 10202   2c2 10581   ZZcz 10860   RR+crp 11216   (,)cioo 11525   [,)cico 11527   ^cexp 12130   topGenctg 14689  sigAlgebracsiga 27747  sigaGencsigagen 27778  𝔅_ℝcbrsiga 27792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-ac2 8839  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-acn 8319  df-ac 8493  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-ioc 11530  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-fl 11893  df-mod 11961  df-seq 12072  df-exp 12131  df-fac 12318  df-bc 12345  df-hash 12370  df-shft 12859  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-limsup 13253  df-clim 13270  df-rlim 13271  df-sum 13468  df-ef 13661  df-sin 13663  df-cos 13664  df-pi 13666  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-hom 14575  df-cco 14576  df-rest 14674  df-topn 14675  df-0g 14693  df-gsum 14694  df-topgen 14695  df-pt 14696  df-prds 14699  df-xrs 14753  df-qtop 14758  df-imas 14759  df-xps 14761  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-submnd 15778  df-mulg 15861  df-cntz 16150  df-cmn 16596  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-mopn 18186  df-fbas 18187  df-fg 18188  df-cnfld 18192  df-refld 18408  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cld 19286  df-ntr 19287  df-cls 19288  df-nei 19365  df-lp 19403  df-perf 19404  df-cn 19494  df-cnp 19495  df-haus 19582  df-cmp 19653  df-tx 19798  df-hmeo 19991  df-fil 20082  df-fm 20174  df-flim 20175  df-flf 20176  df-fcls 20177  df-xms 20558  df-ms 20559  df-tms 20560  df-cncf 21117  df-cfil 21429  df-cmet 21431  df-cms 21509  df-limc 22005  df-dv 22006  df-log 22672  df-cxp 22673  df-logb 27647  df-siga 27748  df-sigagen 27779  df-brsiga 27793

This theorem is referenced by:  sxbrsigalem4  27898