Theorem sxbrsigalem2 29181

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 29179	. . 3 |- U. ran R = ( RR X. RR )
5		sxbrsigalem0 29166	. . 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 2496	. 2 |- U. ran R = U. ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) )
7		vex 3034	. . . . . 6 |- u e. _V
8		vex 3034	. . . . . 6 |- v e. _V
9	7, 8	xpex 6614	. . . . 5 |- ( u X. v ) e. _V
10	3, 9	elrnmpt2 6428	. . . 4 |- ( d e. ran R <-> E. u e. ran I E. v e. ran I d = ( u X. v ) )
11		simpr 468	. . . . . . 7 |- ( ( ( u e. ran I /\ v e. ran I ) /\ d = ( u X. v ) ) -> d = ( u X. v ) )
12	1, 2	dya2icobrsiga 29171	. . . . . . . . . . . . 13 |- ran I C_ 𝔅ℝ
13		brsigasspwrn 29081	. . . . . . . . . . . . 13 |- 𝔅ℝ C_ ~P RR
14	12, 13	sstri 3427	. . . . . . . . . . . 12 |- ran I C_ ~P RR
15	14	sseli 3414	. . . . . . . . . . 11 |- ( u e. ran I -> u e. ~P RR )
16	15	elpwid 3952	. . . . . . . . . 10 |- ( u e. ran I -> u C_ RR )
17	14	sseli 3414	. . . . . . . . . . 11 |- ( v e. ran I -> v e. ~P RR )
18	17	elpwid 3952	. . . . . . . . . 10 |- ( v e. ran I -> v C_ RR )
19		xpinpreima2 28787	. . . . . . . . . 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 485	. . . . . . . . 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 9648	. . . . . . . . . . . . . . . . 17 |- RR e. _V
22	21	mptex 6152	. . . . . . . . . . . . . . . 16 |- ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) e. _V
23	22	rnex 6746	. . . . . . . . . . . . . . 15 |- ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) e. _V
24	21	mptex 6152	. . . . . . . . . . . . . . . 16 |- ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) e. _V
25	24	rnex 6746	. . . . . . . . . . . . . . 15 |- ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) e. _V
26	23, 25	unex 6608	. . . . . . . . . . . . . 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 29038	. . . . . . . . . . . 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 1461	. . . . . . . . . . 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 28362	. . . . . . . . . . . . 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 6336	. . . . . . . . . . . . . 14 |- ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) e. _V
34	2, 33	elrnmpt2 6428	. . . . . . . . . . . . 13 |- ( u e. ran I <-> E. x e. ZZ E. n e. ZZ u = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) )
35		simpr 468	. . . . . . . . . . . . . . . . 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 4862	. . . . . . . . . . . . . . . 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 5268	. . . . . . . . . . . . . . . . . . 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 464	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( x e. ZZ /\ n e. ZZ ) -> x e. ZZ )
39	38	zred 11063	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( x e. ZZ /\ n e. ZZ ) -> x e. RR )
40		2rp 11330	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- 2 e. RR+
41	40	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( x e. ZZ /\ n e. ZZ ) -> 2 e. RR+ )
42		simpr 468	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( x e. ZZ /\ n e. ZZ ) -> n e. ZZ )
43	41, 42	rpexpcld 12477	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( 2 ^ n ) e. RR+ )
44	39, 43	rerpdivcld 11392	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( x / ( 2 ^ n ) ) e. RR )
45	44	rexrd 9708	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( x / ( 2 ^ n ) ) e. RR* )
46		1red 9676	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( x e. ZZ /\ n e. ZZ ) -> 1 e. RR )
47	39, 46	readdcld 9688	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( x + 1 ) e. RR )
48	47, 43	rerpdivcld 11392	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( ( x + 1 ) / ( 2 ^ n ) ) e. RR )
49	48	rexrd 9708	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( ( x + 1 ) / ( 2 ^ n ) ) e. RR* )
50		pnfxr 11435	. . . . . . . . . . . . . . . . . . . . . 22 |- +oo e. RR*
51	50	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( x e. ZZ /\ n e. ZZ ) -> +oo e. RR* )
52	39	lep1d 10560	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( x e. ZZ /\ n e. ZZ ) -> x <_ ( x + 1 ) )
53	39, 47, 43, 52	lediv1dd 11419	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( x e. ZZ /\ n e. ZZ ) -> ( x / ( 2 ^ n ) ) <_ ( ( x + 1 ) / ( 2 ^ n ) ) )
54		pnfge 11455	. . . . . . . . . . . . . . . . . . . . . 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 28440	. . . . . . . . . . . . . . . . . . . . 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 1300	. . . . . . . . . . . . . . . . . . . 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 4862	. . . . . . . . . . . . . . . . . . 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 2520	. . . . . . . . . . . . . . . . . 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 3588	. . . . . . . . . . . . . . . . . . . . 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 2471	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) = ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR )
63		oveq1 6315	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( e = ( x / ( 2 ^ n ) ) -> ( e [,) +oo ) = ( ( x / ( 2 ^ n ) ) [,) +oo ) )
64	63	xpeq1d 4862	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( e = ( x / ( 2 ^ n ) ) -> ( ( e [,) +oo ) X. RR ) = ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) )
65	64	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 ) ) )
66	65	rspcev 3136	. . . . . . . . . . . . . . . . . . . . . . 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 675	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( x e. ZZ /\ n e. ZZ ) -> E. e e. RR ( ( ( x / ( 2 ^ n ) ) [,) +oo ) X. RR ) = ( ( e [,) +oo ) X. RR ) )
68		eqid 2471	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) = ( e e. RR |-> ( ( e [,) +oo ) X. RR ) )
69		ovex 6336	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( e [,) +oo ) e. _V
70	69, 21	xpex 6614	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( e [,) +oo ) X. RR ) e. _V
71	68, 70	elrnmpti 5091	. . . . . . . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . . . . . . . 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 3416	. . . . . . . . . . . . . . . . . . . 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 29043	. . . . . . . . . . . . . . . . . . . 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 676	. . . . . . . . . . . . . . . . . . 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 2471	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR ) = ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR )
77		oveq1 6315	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( e = ( ( x + 1 ) / ( 2 ^ n ) ) -> ( e [,) +oo ) = ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) )
78	77	xpeq1d 4862	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( e = ( ( x + 1 ) / ( 2 ^ n ) ) -> ( ( e [,) +oo ) X. RR ) = ( ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) X. RR ) )
79	78	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 ) ) )
80	79	rspcev 3136	. . . . . . . . . . . . . . . . . . . . . . 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 675	. . . . . . . . . . . . . . . . . . . . . 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 5091	. . . . . . . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . . . . . . . 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 3416	. . . . . . . . . . . . . . . . . . . 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 29043	. . . . . . . . . . . . . . . . . . . 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 676	. . . . . . . . . . . . . . . . . . 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 29029	. . . . . . . . . . . . . . . . . . 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 1292	. . . . . . . . . . . . . . . . . 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 2549	. . . . . . . . . . . . . . . . 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 472	. . . . . . . . . . . . . . . 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 2549	. . . . . . . . . . . . . . 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 441	. . . . . . . . . . . . . 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 2876	. . . . . . . . . . . . 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 200	. . . . . . . . . . . 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 2549	. . . . . . . . . . 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 472	. . . . . . . . . 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 28363	. . . . . . . . . . . . 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 6428	. . . . . . . . . . . . 13 |- ( v e. ran I <-> E. x e. ZZ E. n e. ZZ v = ( ( x / ( 2 ^ n ) ) [,) ( ( x + 1 ) / ( 2 ^ n ) ) ) )
100		simpr 468	. . . . . . . . . . . . . . . . 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 4863	. . . . . . . . . . . . . . . 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 5269	. . . . . . . . . . . . . . . . . . 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 4863	. . . . . . . . . . . . . . . . . . 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 2520	. . . . . . . . . . . . . . . . . 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 3589	. . . . . . . . . . . . . . . . . . . . 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 2471	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) = ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) )
107		oveq1 6315	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( f = ( x / ( 2 ^ n ) ) -> ( f [,) +oo ) = ( ( x / ( 2 ^ n ) ) [,) +oo ) )
108	107	xpeq2d 4863	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( f = ( x / ( 2 ^ n ) ) -> ( RR X. ( f [,) +oo ) ) = ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) )
109	108	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 ) ) ) )
110	109	rspcev 3136	. . . . . . . . . . . . . . . . . . . . . . 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 675	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( x e. ZZ /\ n e. ZZ ) -> E. f e. RR ( RR X. ( ( x / ( 2 ^ n ) ) [,) +oo ) ) = ( RR X. ( f [,) +oo ) ) )
112		eqid 2471	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) = ( f e. RR |-> ( RR X. ( f [,) +oo ) ) )
113		ovex 6336	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( f [,) +oo ) e. _V
114	21, 113	xpex 6614	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( RR X. ( f [,) +oo ) ) e. _V
115	112, 114	elrnmpti 5091	. . . . . . . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . . . . . . . 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 3416	. . . . . . . . . . . . . . . . . . . 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 29043	. . . . . . . . . . . . . . . . . . . 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 676	. . . . . . . . . . . . . . . . . . 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 2471	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) = ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) )
121		oveq1 6315	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( f = ( ( x + 1 ) / ( 2 ^ n ) ) -> ( f [,) +oo ) = ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) )
122	121	xpeq2d 4863	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( f = ( ( x + 1 ) / ( 2 ^ n ) ) -> ( RR X. ( f [,) +oo ) ) = ( RR X. ( ( ( x + 1 ) / ( 2 ^ n ) ) [,) +oo ) ) )
123	122	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 ) ) ) )
124	123	rspcev 3136	. . . . . . . . . . . . . . . . . . . . . . 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 675	. . . . . . . . . . . . . . . . . . . . . 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 5091	. . . . . . . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . . . . . . . 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 3416	. . . . . . . . . . . . . . . . . . . 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 29043	. . . . . . . . . . . . . . . . . . . 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 676	. . . . . . . . . . . . . . . . . . 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 29029	. . . . . . . . . . . . . . . . . . 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 1292	. . . . . . . . . . . . . . . . . 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 2549	. . . . . . . . . . . . . . . . 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 472	. . . . . . . . . . . . . . . 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 2549	. . . . . . . . . . . . . . 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 441	. . . . . . . . . . . . . 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 2876	. . . . . . . . . . . . 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 200	. . . . . . . . . . . 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 2549	. . . . . . . . . . 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 473	. . . . . . . . . 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 29031	. . . . . . . . . 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 1292	. . . . . . . . 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 2549	. . . . . . . 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 472	. . . . . . 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 2549	. . . . . 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 441	. . . . 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 2876	. . . 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 200	. . 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 3422	. 2 |- ran R C_ (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) )
150		sigagenss2 29046	. 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 1390	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 376   = wceq 1452   T. wtru 1453   e. wcel 1904   E.wrex 2757   _Vcvv 3031   \ cdif 3387   u. cun 3388   i^i cin 3389   C_ wss 3390   ~Pcpw 3942   U.cuni 4190   class class class wbr 4395   |-> cmpt 4454   X. cxp 4837   `'ccnv 4838   ran crn 4840   |` cres 4841   "cima 4842   ` cfv 5589  (class class class)co 6308   |-> cmpt2 6310   1stc1st 6810   2ndc2nd 6811   RRcr 9556   1c1 9558   + caddc 9560   +oocpnf 9690   RR*cxr 9692   <_ cle 9694   / cdiv 10291   2c2 10681   ZZcz 10961   RR+crp 11325   (,)cioo 11660   [,)cico 11662   ^cexp 12310   topGenctg 15414  sigAlgebracsiga 29003  sigaGencsigagen 29034  𝔅_ℝcbrsiga 29077

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-ac2 8911  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-acn 8394  df-ac 8565  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-sin 14200  df-cos 14201  df-pi 14203  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-refld 19250  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-fcls 21034  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-cfil 22303  df-cmet 22305  df-cms 22381  df-limc 22900  df-dv 22901  df-log 23585  df-cxp 23586  df-logb 23781  df-siga 29004  df-sigagen 29035  df-brsiga 29078

This theorem is referenced by:  sxbrsigalem4  29182