Theorem sxbrsigalem2 26838

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 26836	. . 3 |- U. ran R = ( RR X. RR )
5		sxbrsigalem0 26823	. . 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 2483	. 2 |- U. ran R = U. ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) )
7		vex 3074	. . . . . 6 |- u e. _V
8		vex 3074	. . . . . 6 |- v e. _V
9	7, 8	xpex 6611	. . . . 5 |- ( u X. v ) e. _V
10	3, 9	elrnmpt2 6306	. . . 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 26828	. . . . . . . . . . . . 13 |- ran I C_ 𝔅ℝ
13		brsigasspwrn 26737	. . . . . . . . . . . . 13 |- 𝔅ℝ C_ ~P RR
14	12, 13	sstri 3466	. . . . . . . . . . . 12 |- ran I C_ ~P RR
15	14	sseli 3453	. . . . . . . . . . 11 |- ( u e. ran I -> u e. ~P RR )
16	15	elpwid 3971	. . . . . . . . . 10 |- ( u e. ran I -> u C_ RR )
17	14	sseli 3453	. . . . . . . . . . 11 |- ( v e. ran I -> v e. ~P RR )
18	17	elpwid 3971	. . . . . . . . . 10 |- ( v e. ran I -> v C_ RR )
19		xpinpreima2 26475	. . . . . . . . . 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 9477	. . . . . . . . . . . . . . . . 17 |- RR e. _V
22	21	mptex 6050	. . . . . . . . . . . . . . . 16 |- ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) e. _V
23	22	rnex 6615	. . . . . . . . . . . . . . 15 |- ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) e. _V
24	21	mptex 6050	. . . . . . . . . . . . . . . 16 |- ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) e. _V
25	24	rnex 6615	. . . . . . . . . . . . . . 15 |- ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) e. _V
26	23, 25	unex 6481	. . . . . . . . . . . . . 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 26723	. . . . . . . . . . . 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 1379	. . . . . . . . . . 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 26145	. . . . . . . . . . . . 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 6218	. . . . . . . . . . . . . 14 |- ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) e. _V
34	2, 33	elrnmpt2 6306	. . . . . . . . . . . . 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 4964	. . . . . . . . . . . . . . . 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 5364	. . . . . . . . . . . . . . . . . . 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 10851	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( x e. ZZ /\ n e. ZZ ) -> x e. RR )
40		2rp 11100	. . . . . . . . . . . . . . . . . . . . . . . . 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 12141	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( 2 ^ n ) e. RR+ )
44	39, 43	rerpdivcld 11158	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( x / ( 2 ^ n ) ) e. RR )
45	44	rexrd 9537	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( x / ( 2 ^ n ) ) e. RR* )
46		1re 9489	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- 1 e. RR
47	46	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( x e. ZZ /\ n e. ZZ ) -> 1 e. RR )
48	39, 47	readdcld 9517	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( x + 1 ) e. RR )
49	48, 43	rerpdivcld 11158	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( ( x + 1 ) / ( 2 ^ n ) ) e. RR )
50	49	rexrd 9537	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( ( x + 1 ) / ( 2 ^ n ) ) e. RR* )
51		pnfxr 11196	. . . . . . . . . . . . . . . . . . . . . 22 |- +oo e. RR*
52	51	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( x e. ZZ /\ n e. ZZ ) -> +oo e. RR* )
53	39	lep1d 10368	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( x e. ZZ /\ n e. ZZ ) -> x <_ ( x + 1 ) )
54	39, 48, 43, 53	lediv1dd 11185	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( x / ( 2 ^ n ) ) <_ ( ( x + 1 ) / ( 2 ^ n ) ) )
55		pnfge 11214	. . . . . . . . . . . . . . . . . . . . . 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 26211	. . . . . . . . . . . . . . . . . . . . 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 1227	. . . . . . . . . . . . . . . . . . . 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 4964	. . . . . . . . . . . . . . . . . . 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 2507	. . . . . . . . . . . . . . . . . 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 3620	. . . . . . . . . . . . . . . . . . . . 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 2451	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) = ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR )
64		oveq1 6200	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( e = ( x / ( 2 ^ n ) ) -> ( e [,) +oo ) = ( ( x / ( 2 ^ n ) ) [,) +oo ) )
65	64	xpeq1d 4964	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( e = ( x / ( 2 ^ n ) ) -> ( ( e [,) +oo ) X. RR ) = ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) )
66	65	eqeq2d 2465	. . . . . . . . . . . . . . . . . . . . . . . 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 3172	. . . . . . . . . . . . . . . . . . . . . . 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 2451	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) = ( e e. RR |-> ( ( e [,) +oo ) X. RR ) )
70		ovex 6218	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( e [,) +oo ) e. _V
71	70, 21	xpex 6611	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( e [,) +oo ) X. RR ) e. _V
72	69, 71	elrnmpti 5191	. . . . . . . . . . . . . . . . . . . . . 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 3455	. . . . . . . . . . . . . . . . . . . 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 26728	. . . . . . . . . . . . . . . . . . . 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 2451	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR ) = ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR )
78		oveq1 6200	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( e = ( ( x + 1 ) / ( 2 ^ n ) ) -> ( e [,) +oo ) = ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) )
79	78	xpeq1d 4964	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( e = ( ( x + 1 ) / ( 2 ^ n ) ) -> ( ( e [,) +oo ) X. RR ) = ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR ) )
80	79	eqeq2d 2465	. . . . . . . . . . . . . . . . . . . . . . . 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 3172	. . . . . . . . . . . . . . . . . . . . . . 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 5191	. . . . . . . . . . . . . . . . . . . . . 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 3455	. . . . . . . . . . . . . . . . . . . 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 26728	. . . . . . . . . . . . . . . . . . . 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 26714	. . . . . . . . . . . . . . . . . . 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 1219	. . . . . . . . . . . . . . . . . 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 2539	. . . . . . . . . . . . . . . . 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 2539	. . . . . . . . . . . . . . 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 2945	. . . . . . . . . . . . 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 2539	. . . . . . . . . . 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 26146	. . . . . . . . . . . . 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 6306	. . . . . . . . . . . . 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 4965	. . . . . . . . . . . . . . . 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 5365	. . . . . . . . . . . . . . . . . . 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 4965	. . . . . . . . . . . . . . . . . . 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 2507	. . . . . . . . . . . . . . . . . 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 3621	. . . . . . . . . . . . . . . . . . . . 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 2451	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) = ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) )
108		oveq1 6200	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( f = ( x / ( 2 ^ n ) ) -> ( f [,) +oo ) = ( ( x / ( 2 ^ n ) ) [,) +oo ) )
109	108	xpeq2d 4965	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( f = ( x / ( 2 ^ n ) ) -> ( RR X. ( f [,) +oo ) ) = ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) )
110	109	eqeq2d 2465	. . . . . . . . . . . . . . . . . . . . . . . 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 3172	. . . . . . . . . . . . . . . . . . . . . . 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 2451	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) = ( f e. RR |-> ( RR X. ( f [,) +oo ) ) )
114		ovex 6218	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( f [,) +oo ) e. _V
115	21, 114	xpex 6611	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( RR X. ( f [,) +oo ) ) e. _V
116	113, 115	elrnmpti 5191	. . . . . . . . . . . . . . . . . . . . . 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 3455	. . . . . . . . . . . . . . . . . . . 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 26728	. . . . . . . . . . . . . . . . . . . 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 2451	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) = ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) )
122		oveq1 6200	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( f = ( ( x + 1 ) / ( 2 ^ n ) ) -> ( f [,) +oo ) = ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) )
123	122	xpeq2d 4965	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( f = ( ( x + 1 ) / ( 2 ^ n ) ) -> ( RR X. ( f [,) +oo ) ) = ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) )
124	123	eqeq2d 2465	. . . . . . . . . . . . . . . . . . . . . . . 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 3172	. . . . . . . . . . . . . . . . . . . . . . 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 5191	. . . . . . . . . . . . . . . . . . . . . 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 3455	. . . . . . . . . . . . . . . . . . . 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 26728	. . . . . . . . . . . . . . . . . . . 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 26714	. . . . . . . . . . . . . . . . . . 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 1219	. . . . . . . . . . . . . . . . . 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 2539	. . . . . . . . . . . . . . . . 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 2539	. . . . . . . . . . . . . . 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 2945	. . . . . . . . . . . . 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 2539	. . . . . . . . . . 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 26716	. . . . . . . . . 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 1219	. . . . . . . . 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 2539	. . . . . . . 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 2539	. . . . . 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 2945	. . . 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 3461	. 2 |- ran R C_ (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) )
151		sigagenss2 26731	. 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 1315	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 1370   T. wtru 1371   e. wcel 1758   E.wrex 2796   _Vcvv 3071   \ cdif 3426   u. cun 3427   i^i cin 3428   C_ wss 3429   ~Pcpw 3961   U.cuni 4192   class class class wbr 4393   |-> cmpt 4451   X. cxp 4939   `'ccnv 4940   ran crn 4942   |` cres 4943   "cima 4944   ` cfv 5519  (class class class)co 6193   |-> cmpt2 6195   1stc1st 6678   2ndc2nd 6679   RRcr 9385   1c1 9387   + caddc 9389   +oocpnf 9519   RR*cxr 9521   <_ cle 9523   / cdiv 10097   2c2 10475   ZZcz 10750   RR+crp 11095   (,)cioo 11404   [,)cico 11406   ^cexp 11975   topGenctg 14487  sigAlgebracsiga 26688  sigaGencsigagen 26719  𝔅_ℝcbrsiga 26733

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 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-ac2 8736  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464  ax-addf 9465  ax-mulf 9466

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 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-of 6423  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-er 7204  df-map 7319  df-pm 7320  df-ixp 7367  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-fsupp 7725  df-fi 7765  df-sup 7795  df-oi 7828  df-card 8213  df-acn 8216  df-ac 8390  df-cda 8441  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-10 10492  df-n0 10684  df-z 10751  df-dec 10860  df-uz 10966  df-q 11058  df-rp 11096  df-xneg 11193  df-xadd 11194  df-xmul 11195  df-ioo 11408  df-ioc 11409  df-ico 11410  df-icc 11411  df-fz 11548  df-fzo 11659  df-fl 11752  df-mod 11819  df-seq 11917  df-exp 11976  df-fac 12162  df-bc 12189  df-hash 12214  df-shft 12667  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-limsup 13060  df-clim 13077  df-rlim 13078  df-sum 13275  df-ef 13464  df-sin 13466  df-cos 13467  df-pi 13469  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-starv 14364  df-sca 14365  df-vsca 14366  df-ip 14367  df-tset 14368  df-ple 14369  df-ds 14371  df-unif 14372  df-hom 14373  df-cco 14374  df-rest 14472  df-topn 14473  df-0g 14491  df-gsum 14492  df-topgen 14493  df-pt 14494  df-prds 14497  df-xrs 14551  df-qtop 14556  df-imas 14557  df-xps 14559  df-mre 14635  df-mrc 14636  df-acs 14638  df-mnd 15526  df-submnd 15576  df-mulg 15659  df-cntz 15946  df-cmn 16392  df-psmet 17927  df-xmet 17928  df-met 17929  df-bl 17930  df-mopn 17931  df-fbas 17932  df-fg 17933  df-cnfld 17937  df-refld 18153  df-top 18628  df-bases 18630  df-topon 18631  df-topsp 18632  df-cld 18748  df-ntr 18749  df-cls 18750  df-nei 18827  df-lp 18865  df-perf 18866  df-cn 18956  df-cnp 18957  df-haus 19044  df-cmp 19115  df-tx 19260  df-hmeo 19453  df-fil 19544  df-fm 19636  df-flim 19637  df-flf 19638  df-fcls 19639  df-xms 20020  df-ms 20021  df-tms 20022  df-cncf 20579  df-cfil 20891  df-cmet 20893  df-cms 20971  df-limc 21467  df-dv 21468  df-log 22134  df-cxp 22135  df-logb 26588  df-siga 26689  df-sigagen 26720  df-brsiga 26734

This theorem is referenced by:  sxbrsigalem4  26839