Theorem sxbrsigalem2 30348

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 30346	. . 3 |- U. ran R = ( RR X. RR )
5		sxbrsigalem0 30333	. . 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 2647	. 2 |- U. ran R = U. ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) )
7		vex 3203	. . . . . 6 |- u e. _V
8		vex 3203	. . . . . 6 |- v e. _V
9	7, 8	xpex 6962	. . . . 5 |- ( u X. v ) e. _V
10	3, 9	elrnmpt2 6773	. . . 4 |- ( d e. ran R <-> E. u e. ran I E. v e. ran I d = ( u X. v ) )
11		simpr 477	. . . . . . 7 |- ( ( ( u e. ran I /\ v e. ran I ) /\ d = ( u X. v ) ) -> d = ( u X. v ) )
12	1, 2	dya2icobrsiga 30338	. . . . . . . . . . . . 13 |- ran I C_ 𝔅ℝ
13		brsigasspwrn 30248	. . . . . . . . . . . . 13 |- 𝔅ℝ C_ ~P RR
14	12, 13	sstri 3612	. . . . . . . . . . . 12 |- ran I C_ ~P RR
15	14	sseli 3599	. . . . . . . . . . 11 |- ( u e. ran I -> u e. ~P RR )
16	15	elpwid 4170	. . . . . . . . . 10 |- ( u e. ran I -> u C_ RR )
17	14	sseli 3599	. . . . . . . . . . 11 |- ( v e. ran I -> v e. ~P RR )
18	17	elpwid 4170	. . . . . . . . . 10 |- ( v e. ran I -> v C_ RR )
19		xpinpreima2 29953	. . . . . . . . . 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 494	. . . . . . . . 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 10027	. . . . . . . . . . . . . . . . 17 |- RR e. _V
22	21	mptex 6486	. . . . . . . . . . . . . . . 16 |- ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) e. _V
23	22	rnex 7100	. . . . . . . . . . . . . . 15 |- ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) e. _V
24	21	mptex 6486	. . . . . . . . . . . . . . . 16 |- ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) e. _V
25	24	rnex 7100	. . . . . . . . . . . . . . 15 |- ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) e. _V
26	23, 25	unex 6956	. . . . . . . . . . . . . 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 30205	. . . . . . . . . . . 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 1493	. . . . . . . . . . 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 29484	. . . . . . . . . . . . 13 |- ( u C_ RR -> ( `' ( 1st |` ( RR X. RR ) ) " u ) = ( u X. RR ) )
32	16, 31	syl 17	. . . . . . . . . . . 12 |- ( u e. ran I -> ( `' ( 1st |` ( RR X. RR ) ) " u ) = ( u X. RR ) )
33		ovex 6678	. . . . . . . . . . . . . 14 |- ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) e. _V
34	2, 33	elrnmpt2 6773	. . . . . . . . . . . . 13 |- ( u e. ran I <-> E. x e. ZZ E. n e. ZZ u = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) )
35		simpr 477	. . . . . . . . . . . . . . . . 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 5138	. . . . . . . . . . . . . . . 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 5559	. . . . . . . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( x e. ZZ /\ n e. ZZ ) -> x e. ZZ )
39	38	zred 11482	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( x e. ZZ /\ n e. ZZ ) -> x e. RR )
40		2rp 11837	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- 2 e. RR+
41	40	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( x e. ZZ /\ n e. ZZ ) -> 2 e. RR+ )
42		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( x e. ZZ /\ n e. ZZ ) -> n e. ZZ )
43	41, 42	rpexpcld 13032	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( 2 ^ n ) e. RR+ )
44	39, 43	rerpdivcld 11903	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( x / ( 2 ^ n ) ) e. RR )
45	44	rexrd 10089	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( x / ( 2 ^ n ) ) e. RR* )
46		1red 10055	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( x e. ZZ /\ n e. ZZ ) -> 1 e. RR )
47	39, 46	readdcld 10069	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( x + 1 ) e. RR )
48	47, 43	rerpdivcld 11903	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( ( x + 1 ) / ( 2 ^ n ) ) e. RR )
49	48	rexrd 10089	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( ( x + 1 ) / ( 2 ^ n ) ) e. RR* )
50		pnfxr 10092	. . . . . . . . . . . . . . . . . . . . . 22 |- +oo e. RR*
51	50	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( x e. ZZ /\ n e. ZZ ) -> +oo e. RR* )
52	39	lep1d 10955	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( x e. ZZ /\ n e. ZZ ) -> x <_ ( x + 1 ) )
53	39, 47, 43, 52	lediv1dd 11930	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( x / ( 2 ^ n ) ) <_ ( ( x + 1 ) / ( 2 ^ n ) ) )
54		pnfge 11964	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( x + 1 ) / ( 2 ^ n ) ) e. RR* -> ( ( x + 1 ) / ( 2 ^ n ) ) <_ +oo )
55	49, 54	syl 17	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( ( x + 1 ) / ( 2 ^ n ) ) <_ +oo )
56		difico 29545	. . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
57	45, 49, 51, 53, 55, 56	syl32anc 1334	. . . . . . . . . . . . . . . . . . . 20 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( ( ( x / ( 2 ^ n ) ) [,) +oo ) \ ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) )
58	57	xpeq1d 5138	. . . . . . . . . . . . . . . . . . 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 ) )
59	37, 58	syl5reqr 2671	. . . . . . . . . . . . . . . . . 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 ) ) )
60	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 )
61		ssun1 3776	. . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
62		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) = ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR )
63		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( e = ( x / ( 2 ^ n ) ) -> ( e [,) +oo ) = ( ( x / ( 2 ^ n ) ) [,) +oo ) )
64	63	xpeq1d 5138	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( e = ( x / ( 2 ^ n ) ) -> ( ( e [,) +oo ) X. RR ) = ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) )
65	64	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
66	65	rspcev 3309	. . . . . . . . . . . . . . . . . . . . . . 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 ) )
67	44, 62, 66	sylancl 694	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( x e. ZZ /\ n e. ZZ ) -> E. e e. RR ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) = ( ( e [,) +oo ) X. RR ) )
68		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) = ( e e. RR |-> ( ( e [,) +oo ) X. RR ) )
69		ovex 6678	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( e [,) +oo ) e. _V
70	69, 21	xpex 6962	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( e [,) +oo ) X. RR ) e. _V
71	68, 70	elrnmpti 5376	. . . . . . . . . . . . . . . . . . . . . 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 ) )
72	67, 71	sylibr 224	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) e. ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) )
73	61, 72	sseldi 3601	. . . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
74		elsigagen 30210	. . . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
75	26, 73, 74	sylancr 695	. . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
76		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR ) = ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR )
77		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( e = ( ( x + 1 ) / ( 2 ^ n ) ) -> ( e [,) +oo ) = ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) )
78	77	xpeq1d 5138	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( e = ( ( x + 1 ) / ( 2 ^ n ) ) -> ( ( e [,) +oo ) X. RR ) = ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR ) )
79	78	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
80	79	rspcev 3309	. . . . . . . . . . . . . . . . . . . . . . 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 ) )
81	48, 76, 80	sylancl 694	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( x e. ZZ /\ n e. ZZ ) -> E. e e. RR ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR ) = ( ( e [,) +oo ) X. RR ) )
82	68, 70	elrnmpti 5376	. . . . . . . . . . . . . . . . . . . . . 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 ) )
83	81, 82	sylibr 224	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR ) e. ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) )
84	61, 83	sseldi 3601	. . . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
85		elsigagen 30210	. . . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
86	26, 84, 85	sylancr 695	. . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
87		difelsiga 30196	. . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
88	60, 75, 86, 87	syl3anc 1326	. . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
89	59, 88	eqeltrd 2701	. . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
90	89	adantr 481	. . . . . . . . . . . . . . . 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 ) ) ) ) ) )
91	36, 90	eqeltrd 2701	. . . . . . . . . . . . . . 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 ) ) ) ) ) )
92	91	ex 450	. . . . . . . . . . . . . 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 ) ) ) ) ) ) )
93	92	rexlimivv 3036	. . . . . . . . . . . . 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 ) ) ) ) ) )
94	34, 93	sylbi 207	. . . . . . . . . . . 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 ) ) ) ) ) )
95	32, 94	eqeltrd 2701	. . . . . . . . . . 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 ) ) ) ) ) )
96	95	adantr 481	. . . . . . . . . 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 ) ) ) ) ) )
97		2ndpreima 29485	. . . . . . . . . . . . 13 |- ( v C_ RR -> ( `' ( 2nd |` ( RR X. RR ) ) " v ) = ( RR X. v ) )
98	18, 97	syl 17	. . . . . . . . . . . 12 |- ( v e. ran I -> ( `' ( 2nd |` ( RR X. RR ) ) " v ) = ( RR X. v ) )
99	2, 33	elrnmpt2 6773	. . . . . . . . . . . . 13 |- ( v e. ran I <-> E. x e. ZZ E. n e. ZZ v = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) )
100		simpr 477	. . . . . . . . . . . . . . . . 17 |- ( ( ( x e. ZZ /\ n e. ZZ ) /\ v = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) ) -> v = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) )
101	100	xpeq2d 5139	. . . . . . . . . . . . . . . 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 ) ) ) ) )
102		difxp2 5560	. . . . . . . . . . . . . . . . . . 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 ) ) )
103	57	xpeq2d 5139	. . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
104	102, 103	syl5reqr 2671	. . . . . . . . . . . . . . . . . 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 ) ) ) )
105		ssun2 3777	. . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
106		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) = ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) )
107		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( f = ( x / ( 2 ^ n ) ) -> ( f [,) +oo ) = ( ( x / ( 2 ^ n ) ) [,) +oo ) )
108	107	xpeq2d 5139	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( f = ( x / ( 2 ^ n ) ) -> ( RR X. ( f [,) +oo ) ) = ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) )
109	108	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
110	109	rspcev 3309	. . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
111	44, 106, 110	sylancl 694	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( x e. ZZ /\ n e. ZZ ) -> E. f e. RR ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) = ( RR X. ( f [,) +oo ) ) )
112		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) = ( f e. RR |-> ( RR X. ( f [,) +oo ) ) )
113		ovex 6678	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( f [,) +oo ) e. _V
114	21, 113	xpex 6962	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( RR X. ( f [,) +oo ) ) e. _V
115	112, 114	elrnmpti 5376	. . . . . . . . . . . . . . . . . . . . . 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 ) ) )
116	111, 115	sylibr 224	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) e. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) )
117	105, 116	sseldi 3601	. . . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
118		elsigagen 30210	. . . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
119	26, 117, 118	sylancr 695	. . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
120		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) = ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) )
121		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( f = ( ( x + 1 ) / ( 2 ^ n ) ) -> ( f [,) +oo ) = ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) )
122	121	xpeq2d 5139	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( f = ( ( x + 1 ) / ( 2 ^ n ) ) -> ( RR X. ( f [,) +oo ) ) = ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) )
123	122	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
124	123	rspcev 3309	. . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
125	48, 120, 124	sylancl 694	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( x e. ZZ /\ n e. ZZ ) -> E. f e. RR ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) = ( RR X. ( f [,) +oo ) ) )
126	112, 114	elrnmpti 5376	. . . . . . . . . . . . . . . . . . . . . 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 ) ) )
127	125, 126	sylibr 224	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) e. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) )
128	105, 127	sseldi 3601	. . . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
129		elsigagen 30210	. . . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
130	26, 128, 129	sylancr 695	. . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
131		difelsiga 30196	. . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
132	60, 119, 130, 131	syl3anc 1326	. . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
133	104, 132	eqeltrd 2701	. . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
134	133	adantr 481	. . . . . . . . . . . . . . . 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 ) ) ) ) ) )
135	101, 134	eqeltrd 2701	. . . . . . . . . . . . . . 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 ) ) ) ) ) )
136	135	ex 450	. . . . . . . . . . . . . 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 ) ) ) ) ) ) )
137	136	rexlimivv 3036	. . . . . . . . . . . . 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 ) ) ) ) ) )
138	99, 137	sylbi 207	. . . . . . . . . . . 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 ) ) ) ) ) )
139	98, 138	eqeltrd 2701	. . . . . . . . . . 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 ) ) ) ) ) )
140	139	adantl 482	. . . . . . . . . 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 ) ) ) ) ) )
141		inelsiga 30198	. . . . . . . . . 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 ) ) ) ) ) )
142	30, 96, 140, 141	syl3anc 1326	. . . . . . . . 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 ) ) ) ) ) )
143	20, 142	eqeltrd 2701	. . . . . . . 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 ) ) ) ) ) )
144	143	adantr 481	. . . . . . 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 ) ) ) ) ) )
145	11, 144	eqeltrd 2701	. . . . . 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 ) ) ) ) ) )
146	145	ex 450	. . . . 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 ) ) ) ) ) ) )
147	146	rexlimivv 3036	. . . 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 ) ) ) ) ) )
148	10, 147	sylbi 207	. . 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 ) ) ) ) ) )
149	148	ssriv 3607	. 2 |- ran R C_ (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) )
150		sigagenss2 30213	. 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 ) ) ) ) ) )
151	6, 149, 26, 150	mp3an 1424	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 384   = wceq 1483   T. wtru 1484   e. wcel 1990   E.wrex 2913   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   ran crn 5115   |` cres 5116   "cima 5117   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   RRcr 9935   1c1 9937   + caddc 9939   +oocpnf 10071   RR*cxr 10073   <_ cle 10075   / cdiv 10684   2c2 11070   ZZcz 11377   RR+crp 11832   (,)cioo 12175   [,)cico 12177   ^cexp 12860   topGenctg 16098  sigAlgebracsiga 30170  sigaGencsigagen 30201  𝔅_ℝcbrsiga 30244

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-ac2 9285  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-ac 8939  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-refld 19951  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-fcls 21745  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-cfil 23053  df-cmet 23055  df-cms 23132  df-limc 23630  df-dv 23631  df-log 24303  df-cxp 24304  df-logb 24503  df-siga 30171  df-sigagen 30202  df-brsiga 30245

This theorem is referenced by:  sxbrsigalem4  30349