Theorem opnvonmbllem2 40847

Description: An open subset of the n-dimensional Real numbers is Lebesgue measurable. This is Proposition 115G (a) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 24-Dec-2020.)

Hypotheses

Ref	Expression
opnvonmbllem2.x	|- ( ph -> X e. Fin )
opnvonmbllem2.n	|- S = dom (voln ` X )
opnvonmbllem2.g	|- ( ph -> G e. ( TopOpen ` (ℝ^ ` X ) ) )
opnvonmbl.k	|- K = { h e. ( ( QQ X. QQ ) ^m X ) | X_ i e. X ( ( [,) o. h ) ` i ) C_ G }

Assertion

Ref	Expression
opnvonmbllem2	|- ( ph -> G e. S )

Distinct variable groups:   h, G, i   h, K, i   S, h, i   h, X, i   ph, h, i

Proof of Theorem opnvonmbllem2

Dummy variables x k c d e are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opnvonmbllem2.x	. . . . . . . . . . 11 |- ( ph -> X e. Fin )
2		eqid 2622	. . . . . . . . . . . 12 |- ( dist ` (ℝ^ ` X ) ) = ( dist ` (ℝ^ ` X ) )
3	2	rrxmetfi 40507	. . . . . . . . . . 11 |- ( X e. Fin -> ( dist ` (ℝ^ ` X ) ) e. ( Met ` ( RR ^m X ) ) )
4	1, 3	syl 17	. . . . . . . . . 10 |- ( ph -> ( dist ` (ℝ^ ` X ) ) e. ( Met ` ( RR ^m X ) ) )
5		metxmet 22139	. . . . . . . . . 10 |- ( ( dist ` (ℝ^ ` X ) ) e. ( Met ` ( RR ^m X ) ) -> ( dist ` (ℝ^ ` X ) ) e. ( *Met ` ( RR ^m X ) ) )
6	4, 5	syl 17	. . . . . . . . 9 |- ( ph -> ( dist ` (ℝ^ ` X ) ) e. ( *Met ` ( RR ^m X ) ) )
7	6	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. G ) -> ( dist ` (ℝ^ ` X ) ) e. ( *Met ` ( RR ^m X ) ) )
8		opnvonmbllem2.g	. . . . . . . . . 10 |- ( ph -> G e. ( TopOpen ` (ℝ^ ` X ) ) )
9		eqid 2622	. . . . . . . . . . . . . 14 |- (ℝ^ ` X ) = (ℝ^ ` X )
10	9	rrxval 23175	. . . . . . . . . . . . 13 |- ( X e. Fin -> (ℝ^ ` X ) = (toCHil ` (RRfld freeLMod X ) ) )
11	1, 10	syl 17	. . . . . . . . . . . 12 |- ( ph -> (ℝ^ ` X ) = (toCHil ` (RRfld freeLMod X ) ) )
12	11	fveq2d 6195	. . . . . . . . . . 11 |- ( ph -> ( TopOpen ` (ℝ^ ` X ) ) = ( TopOpen ` (toCHil ` (RRfld freeLMod X ) ) ) )
13		ovex 6678	. . . . . . . . . . . . 13 |- (RRfld freeLMod X ) e. _V
14		eqid 2622	. . . . . . . . . . . . . 14 |- (toCHil ` (RRfld freeLMod X ) ) = (toCHil ` (RRfld freeLMod X ) )
15		eqid 2622	. . . . . . . . . . . . . 14 |- ( dist ` (toCHil ` (RRfld freeLMod X ) ) ) = ( dist ` (toCHil ` (RRfld freeLMod X ) ) )
16		eqid 2622	. . . . . . . . . . . . . 14 |- ( TopOpen ` (toCHil ` (RRfld freeLMod X ) ) ) = ( TopOpen ` (toCHil ` (RRfld freeLMod X ) ) )
17	14, 15, 16	tchtopn 23025	. . . . . . . . . . . . 13 |- ( (RRfld freeLMod X ) e. _V -> ( TopOpen ` (toCHil ` (RRfld freeLMod X ) ) ) = ( MetOpen ` ( dist ` (toCHil ` (RRfld freeLMod X ) ) ) ) )
18	13, 17	ax-mp 5	. . . . . . . . . . . 12 |- ( TopOpen ` (toCHil ` (RRfld freeLMod X ) ) ) = ( MetOpen ` ( dist ` (toCHil ` (RRfld freeLMod X ) ) ) )
19	18	a1i 11	. . . . . . . . . . 11 |- ( ph -> ( TopOpen ` (toCHil ` (RRfld freeLMod X ) ) ) = ( MetOpen ` ( dist ` (toCHil ` (RRfld freeLMod X ) ) ) ) )
20	11	eqcomd 2628	. . . . . . . . . . . . 13 |- ( ph -> (toCHil ` (RRfld freeLMod X ) ) = (ℝ^ ` X ) )
21	20	fveq2d 6195	. . . . . . . . . . . 12 |- ( ph -> ( dist ` (toCHil ` (RRfld freeLMod X ) ) ) = ( dist ` (ℝ^ ` X ) ) )
22	21	fveq2d 6195	. . . . . . . . . . 11 |- ( ph -> ( MetOpen ` ( dist ` (toCHil ` (RRfld freeLMod X ) ) ) ) = ( MetOpen ` ( dist ` (ℝ^ ` X ) ) ) )
23	12, 19, 22	3eqtrd 2660	. . . . . . . . . 10 |- ( ph -> ( TopOpen ` (ℝ^ ` X ) ) = ( MetOpen ` ( dist ` (ℝ^ ` X ) ) ) )
24	8, 23	eleqtrd 2703	. . . . . . . . 9 |- ( ph -> G e. ( MetOpen ` ( dist ` (ℝ^ ` X ) ) ) )
25	24	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. G ) -> G e. ( MetOpen ` ( dist ` (ℝ^ ` X ) ) ) )
26		simpr 477	. . . . . . . 8 |- ( ( ph /\ x e. G ) -> x e. G )
27		eqid 2622	. . . . . . . . 9 |- ( MetOpen ` ( dist ` (ℝ^ ` X ) ) ) = ( MetOpen ` ( dist ` (ℝ^ ` X ) ) )
28	27	mopni2 22298	. . . . . . . 8 |- ( ( ( dist ` (ℝ^ ` X ) ) e. ( *Met ` ( RR ^m X ) ) /\ G e. ( MetOpen ` ( dist ` (ℝ^ ` X ) ) ) /\ x e. G ) -> E. e e. RR+ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) C_ G )
29	7, 25, 26, 28	syl3anc 1326	. . . . . . 7 |- ( ( ph /\ x e. G ) -> E. e e. RR+ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) C_ G )
30	1	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. G ) /\ e e. RR+ ) -> X e. Fin )
31		eqid 2622	. . . . . . . . . . . . . . . . . 18 |- ( TopOpen ` (ℝ^ ` X ) ) = ( TopOpen ` (ℝ^ ` X ) )
32	31	rrxtoponfi 40511	. . . . . . . . . . . . . . . . 17 |- ( X e. Fin -> ( TopOpen ` (ℝ^ ` X ) ) e. (TopOn ` ( RR ^m X ) ) )
33	1, 32	syl 17	. . . . . . . . . . . . . . . 16 |- ( ph -> ( TopOpen ` (ℝ^ ` X ) ) e. (TopOn ` ( RR ^m X ) ) )
34		toponss 20731	. . . . . . . . . . . . . . . 16 |- ( ( ( TopOpen ` (ℝ^ ` X ) ) e. (TopOn ` ( RR ^m X ) ) /\ G e. ( TopOpen ` (ℝ^ ` X ) ) ) -> G C_ ( RR ^m X ) )
35	33, 8, 34	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ph -> G C_ ( RR ^m X ) )
36	35	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. G ) -> G C_ ( RR ^m X ) )
37	36, 26	sseldd 3604	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. G ) -> x e. ( RR ^m X ) )
38	37	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. G ) /\ e e. RR+ ) -> x e. ( RR ^m X ) )
39		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. G ) /\ e e. RR+ ) -> e e. RR+ )
40	30, 38, 39	hoiqssbl 40839	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. G ) /\ e e. RR+ ) -> E. c e. ( QQ ^m X ) E. d e. ( QQ ^m X ) ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) ) )
41	40	3adant3 1081	. . . . . . . . . 10 |- ( ( ( ph /\ x e. G ) /\ e e. RR+ /\ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) C_ G ) -> E. c e. ( QQ ^m X ) E. d e. ( QQ ^m X ) ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) ) )
42		nfv 1843	. . . . . . . . . . . . . . . 16 |- F/ i ( ph /\ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) C_ G )
43		nfv 1843	. . . . . . . . . . . . . . . 16 |- F/ i ( c e. ( QQ ^m X ) /\ d e. ( QQ ^m X ) )
44		nfcv 2764	. . . . . . . . . . . . . . . . . 18 |- F/_ i x
45		nfixp1 7928	. . . . . . . . . . . . . . . . . 18 |- F/_ i X_ i e. X ( ( c ` i ) [,) ( d ` i ) )
46	44, 45	nfel 2777	. . . . . . . . . . . . . . . . 17 |- F/ i x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) )
47		nfcv 2764	. . . . . . . . . . . . . . . . . 18 |- F/_ i ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e )
48	45, 47	nfss 3596	. . . . . . . . . . . . . . . . 17 |- F/ i X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e )
49	46, 48	nfan 1828	. . . . . . . . . . . . . . . 16 |- F/ i ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) )
50	42, 43, 49	nf3an 1831	. . . . . . . . . . . . . . 15 |- F/ i ( ( ph /\ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) C_ G ) /\ ( c e. ( QQ ^m X ) /\ d e. ( QQ ^m X ) ) /\ ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) ) )
51	1	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) C_ G ) -> X e. Fin )
52	51	3ad2ant1 1082	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) C_ G ) /\ ( c e. ( QQ ^m X ) /\ d e. ( QQ ^m X ) ) /\ ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) ) ) -> X e. Fin )
53		elmapi 7879	. . . . . . . . . . . . . . . . 17 |- ( c e. ( QQ ^m X ) -> c : X --> QQ )
54	53	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( c e. ( QQ ^m X ) /\ d e. ( QQ ^m X ) ) -> c : X --> QQ )
55	54	3ad2ant2 1083	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) C_ G ) /\ ( c e. ( QQ ^m X ) /\ d e. ( QQ ^m X ) ) /\ ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) ) ) -> c : X --> QQ )
56		elmapi 7879	. . . . . . . . . . . . . . . . 17 |- ( d e. ( QQ ^m X ) -> d : X --> QQ )
57	56	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( c e. ( QQ ^m X ) /\ d e. ( QQ ^m X ) ) -> d : X --> QQ )
58	57	3ad2ant2 1083	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) C_ G ) /\ ( c e. ( QQ ^m X ) /\ d e. ( QQ ^m X ) ) /\ ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) ) ) -> d : X --> QQ )
59		simp3r 1090	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) C_ G ) /\ ( c e. ( QQ ^m X ) /\ d e. ( QQ ^m X ) ) /\ ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) ) ) -> X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) )
60		simp1r 1086	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) C_ G ) /\ ( c e. ( QQ ^m X ) /\ d e. ( QQ ^m X ) ) /\ ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) ) ) -> ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) C_ G )
61		simp3l 1089	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) C_ G ) /\ ( c e. ( QQ ^m X ) /\ d e. ( QQ ^m X ) ) /\ ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) ) ) -> x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) )
62		opnvonmbl.k	. . . . . . . . . . . . . . 15 |- K = { h e. ( ( QQ X. QQ ) ^m X ) | X_ i e. X ( ( [,) o. h ) ` i ) C_ G }
63		eqid 2622	. . . . . . . . . . . . . . 15 |- ( i e. X |-> <. ( c ` i ) , ( d ` i ) >. ) = ( i e. X |-> <. ( c ` i ) , ( d ` i ) >. )
64	50, 52, 55, 58, 59, 60, 61, 62, 63	opnvonmbllem1 40846	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) C_ G ) /\ ( c e. ( QQ ^m X ) /\ d e. ( QQ ^m X ) ) /\ ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) ) ) -> E. h e. K x e. X_ i e. X ( ( [,) o. h ) ` i ) )
65	64	3exp 1264	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) C_ G ) -> ( ( c e. ( QQ ^m X ) /\ d e. ( QQ ^m X ) ) -> ( ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) ) -> E. h e. K x e. X_ i e. X ( ( [,) o. h ) ` i ) ) ) )
66	65	adantlr 751	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. G ) /\ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) C_ G ) -> ( ( c e. ( QQ ^m X ) /\ d e. ( QQ ^m X ) ) -> ( ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) ) -> E. h e. K x e. X_ i e. X ( ( [,) o. h ) ` i ) ) ) )
67	66	3adant2 1080	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. G ) /\ e e. RR+ /\ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) C_ G ) -> ( ( c e. ( QQ ^m X ) /\ d e. ( QQ ^m X ) ) -> ( ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) ) -> E. h e. K x e. X_ i e. X ( ( [,) o. h ) ` i ) ) ) )
68	67	rexlimdvv 3037	. . . . . . . . . 10 |- ( ( ( ph /\ x e. G ) /\ e e. RR+ /\ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) C_ G ) -> ( E. c e. ( QQ ^m X ) E. d e. ( QQ ^m X ) ( x e. X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) /\ X_ i e. X ( ( c ` i ) [,) ( d ` i ) ) C_ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) ) -> E. h e. K x e. X_ i e. X ( ( [,) o. h ) ` i ) ) )
69	41, 68	mpd 15	. . . . . . . . 9 |- ( ( ( ph /\ x e. G ) /\ e e. RR+ /\ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) C_ G ) -> E. h e. K x e. X_ i e. X ( ( [,) o. h ) ` i ) )
70	69	3exp 1264	. . . . . . . 8 |- ( ( ph /\ x e. G ) -> ( e e. RR+ -> ( ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) C_ G -> E. h e. K x e. X_ i e. X ( ( [,) o. h ) ` i ) ) ) )
71	70	rexlimdv 3030	. . . . . . 7 |- ( ( ph /\ x e. G ) -> ( E. e e. RR+ ( x ( ball ` ( dist ` (ℝ^ ` X ) ) ) e ) C_ G -> E. h e. K x e. X_ i e. X ( ( [,) o. h ) ` i ) ) )
72	29, 71	mpd 15	. . . . . 6 |- ( ( ph /\ x e. G ) -> E. h e. K x e. X_ i e. X ( ( [,) o. h ) ` i ) )
73		eliun 4524	. . . . . 6 |- ( x e. U_ h e. K X_ i e. X ( ( [,) o. h ) ` i ) <-> E. h e. K x e. X_ i e. X ( ( [,) o. h ) ` i ) )
74	72, 73	sylibr 224	. . . . 5 |- ( ( ph /\ x e. G ) -> x e. U_ h e. K X_ i e. X ( ( [,) o. h ) ` i ) )
75	74	ralrimiva 2966	. . . 4 |- ( ph -> A. x e. G x e. U_ h e. K X_ i e. X ( ( [,) o. h ) ` i ) )
76		dfss3 3592	. . . 4 |- ( G C_ U_ h e. K X_ i e. X ( ( [,) o. h ) ` i ) <-> A. x e. G x e. U_ h e. K X_ i e. X ( ( [,) o. h ) ` i ) )
77	75, 76	sylibr 224	. . 3 |- ( ph -> G C_ U_ h e. K X_ i e. X ( ( [,) o. h ) ` i ) )
78	62	eleq2i 2693	. . . . . . . . 9 |- ( h e. K <-> h e. { h e. ( ( QQ X. QQ ) ^m X ) | X_ i e. X ( ( [,) o. h ) ` i ) C_ G } )
79	78	biimpi 206	. . . . . . . 8 |- ( h e. K -> h e. { h e. ( ( QQ X. QQ ) ^m X ) | X_ i e. X ( ( [,) o. h ) ` i ) C_ G } )
80	79	adantl 482	. . . . . . 7 |- ( ( ph /\ h e. K ) -> h e. { h e. ( ( QQ X. QQ ) ^m X ) | X_ i e. X ( ( [,) o. h ) ` i ) C_ G } )
81		rabid 3116	. . . . . . 7 |- ( h e. { h e. ( ( QQ X. QQ ) ^m X ) | X_ i e. X ( ( [,) o. h ) ` i ) C_ G } <-> ( h e. ( ( QQ X. QQ ) ^m X ) /\ X_ i e. X ( ( [,) o. h ) ` i ) C_ G ) )
82	80, 81	sylib 208	. . . . . 6 |- ( ( ph /\ h e. K ) -> ( h e. ( ( QQ X. QQ ) ^m X ) /\ X_ i e. X ( ( [,) o. h ) ` i ) C_ G ) )
83	82	simprd 479	. . . . 5 |- ( ( ph /\ h e. K ) -> X_ i e. X ( ( [,) o. h ) ` i ) C_ G )
84	83	ralrimiva 2966	. . . 4 |- ( ph -> A. h e. K X_ i e. X ( ( [,) o. h ) ` i ) C_ G )
85		iunss 4561	. . . 4 |- ( U_ h e. K X_ i e. X ( ( [,) o. h ) ` i ) C_ G <-> A. h e. K X_ i e. X ( ( [,) o. h ) ` i ) C_ G )
86	84, 85	sylibr 224	. . 3 |- ( ph -> U_ h e. K X_ i e. X ( ( [,) o. h ) ` i ) C_ G )
87	77, 86	eqssd 3620	. 2 |- ( ph -> G = U_ h e. K X_ i e. X ( ( [,) o. h ) ` i ) )
88		opnvonmbllem2.n	. . . 4 |- S = dom (voln ` X )
89	1, 88	dmovnsal 40826	. . 3 |- ( ph -> S e. SAlg )
90		ssrab2 3687	. . . . . 6 |- { h e. ( ( QQ X. QQ ) ^m X ) | X_ i e. X ( ( [,) o. h ) ` i ) C_ G } C_ ( ( QQ X. QQ ) ^m X )
91	62, 90	eqsstri 3635	. . . . 5 |- K C_ ( ( QQ X. QQ ) ^m X )
92	91	a1i 11	. . . 4 |- ( ph -> K C_ ( ( QQ X. QQ ) ^m X ) )
93		qct 39578	. . . . . . 7 |- QQ ~<_ om
94	93	a1i 11	. . . . . 6 |- ( ph -> QQ ~<_ om )
95		xpct 8839	. . . . . 6 |- ( ( QQ ~<_ om /\ QQ ~<_ om ) -> ( QQ X. QQ ) ~<_ om )
96	94, 94, 95	syl2anc 693	. . . . 5 |- ( ph -> ( QQ X. QQ ) ~<_ om )
97	96, 1	mpct 39393	. . . 4 |- ( ph -> ( ( QQ X. QQ ) ^m X ) ~<_ om )
98		ssct 8041	. . . 4 |- ( ( K C_ ( ( QQ X. QQ ) ^m X ) /\ ( ( QQ X. QQ ) ^m X ) ~<_ om ) -> K ~<_ om )
99	92, 97, 98	syl2anc 693	. . 3 |- ( ph -> K ~<_ om )
100		reex 10027	. . . . . . . . . 10 |- RR e. _V
101	100, 100	xpex 6962	. . . . . . . . 9 |- ( RR X. RR ) e. _V
102		qssre 11798	. . . . . . . . . 10 |- QQ C_ RR
103		xpss12 5225	. . . . . . . . . 10 |- ( ( QQ C_ RR /\ QQ C_ RR ) -> ( QQ X. QQ ) C_ ( RR X. RR ) )
104	102, 102, 103	mp2an 708	. . . . . . . . 9 |- ( QQ X. QQ ) C_ ( RR X. RR )
105		mapss 7900	. . . . . . . . 9 |- ( ( ( RR X. RR ) e. _V /\ ( QQ X. QQ ) C_ ( RR X. RR ) ) -> ( ( QQ X. QQ ) ^m X ) C_ ( ( RR X. RR ) ^m X ) )
106	101, 104, 105	mp2an 708	. . . . . . . 8 |- ( ( QQ X. QQ ) ^m X ) C_ ( ( RR X. RR ) ^m X )
107	91	sseli 3599	. . . . . . . 8 |- ( h e. K -> h e. ( ( QQ X. QQ ) ^m X ) )
108	106, 107	sseldi 3601	. . . . . . 7 |- ( h e. K -> h e. ( ( RR X. RR ) ^m X ) )
109		elmapi 7879	. . . . . . 7 |- ( h e. ( ( RR X. RR ) ^m X ) -> h : X --> ( RR X. RR ) )
110	108, 109	syl 17	. . . . . 6 |- ( h e. K -> h : X --> ( RR X. RR ) )
111	110	adantl 482	. . . . 5 |- ( ( ph /\ h e. K ) -> h : X --> ( RR X. RR ) )
112		fveq2 6191	. . . . . . 7 |- ( k = i -> ( h ` k ) = ( h ` i ) )
113	112	fveq2d 6195	. . . . . 6 |- ( k = i -> ( 1st ` ( h ` k ) ) = ( 1st ` ( h ` i ) ) )
114	113	cbvmptv 4750	. . . . 5 |- ( k e. X |-> ( 1st ` ( h ` k ) ) ) = ( i e. X |-> ( 1st ` ( h ` i ) ) )
115	112	fveq2d 6195	. . . . . 6 |- ( k = i -> ( 2nd ` ( h ` k ) ) = ( 2nd ` ( h ` i ) ) )
116	115	cbvmptv 4750	. . . . 5 |- ( k e. X |-> ( 2nd ` ( h ` k ) ) ) = ( i e. X |-> ( 2nd ` ( h ` i ) ) )
117	111, 114, 116	hoicoto2 40819	. . . 4 |- ( ( ph /\ h e. K ) -> X_ i e. X ( ( [,) o. h ) ` i ) = X_ i e. X ( ( ( k e. X |-> ( 1st ` ( h ` k ) ) ) ` i ) [,) ( ( k e. X |-> ( 2nd ` ( h ` k ) ) ) ` i ) ) )
118	1	adantr 481	. . . . 5 |- ( ( ph /\ h e. K ) -> X e. Fin )
119	111	ffvelrnda 6359	. . . . . . 7 |- ( ( ( ph /\ h e. K ) /\ k e. X ) -> ( h ` k ) e. ( RR X. RR ) )
120		xp1st 7198	. . . . . . 7 |- ( ( h ` k ) e. ( RR X. RR ) -> ( 1st ` ( h ` k ) ) e. RR )
121	119, 120	syl 17	. . . . . 6 |- ( ( ( ph /\ h e. K ) /\ k e. X ) -> ( 1st ` ( h ` k ) ) e. RR )
122		eqid 2622	. . . . . 6 |- ( k e. X |-> ( 1st ` ( h ` k ) ) ) = ( k e. X |-> ( 1st ` ( h ` k ) ) )
123	121, 122	fmptd 6385	. . . . 5 |- ( ( ph /\ h e. K ) -> ( k e. X |-> ( 1st ` ( h ` k ) ) ) : X --> RR )
124		xp2nd 7199	. . . . . . 7 |- ( ( h ` k ) e. ( RR X. RR ) -> ( 2nd ` ( h ` k ) ) e. RR )
125	119, 124	syl 17	. . . . . 6 |- ( ( ( ph /\ h e. K ) /\ k e. X ) -> ( 2nd ` ( h ` k ) ) e. RR )
126		eqid 2622	. . . . . 6 |- ( k e. X |-> ( 2nd ` ( h ` k ) ) ) = ( k e. X |-> ( 2nd ` ( h ` k ) ) )
127	125, 126	fmptd 6385	. . . . 5 |- ( ( ph /\ h e. K ) -> ( k e. X |-> ( 2nd ` ( h ` k ) ) ) : X --> RR )
128	118, 88, 123, 127	hoimbl 40845	. . . 4 |- ( ( ph /\ h e. K ) -> X_ i e. X ( ( ( k e. X |-> ( 1st ` ( h ` k ) ) ) ` i ) [,) ( ( k e. X |-> ( 2nd ` ( h ` k ) ) ) ` i ) ) e. S )
129	117, 128	eqeltrd 2701	. . 3 |- ( ( ph /\ h e. K ) -> X_ i e. X ( ( [,) o. h ) ` i ) e. S )
130	89, 99, 129	saliuncl 40542	. 2 |- ( ph -> U_ h e. K X_ i e. X ( ( [,) o. h ) ` i ) e. S )
131	87, 130	eqeltrd 2701	1 |- ( ph -> G e. S )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   <.cop 4183   U_ciun 4520   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   omcom 7065   1stc1st 7166   2ndc2nd 7167   ^m cmap 7857   X_cixp 7908   ~<_ cdom 7953   Fincfn 7955   RRcr 9935   QQcq 11788   RR+crp 11832   [,)cico 12177   distcds 15950   TopOpenctopn 16082   *Metcxmt 19731   Metcme 19732   ballcbl 19733   MetOpencmopn 19736  RRfldcrefld 19950   freeLMod cfrlm 20090  TopOnctopon 20715  toCHilctch 22967  ℝ^crrx 23171  volncvoln 40752

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-cc 9257  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-disj 4621  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-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  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-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-prod 14636  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-prds 16108  df-pws 16110  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-rnghom 18715  df-drng 18749  df-field 18750  df-subrg 18778  df-abv 18817  df-staf 18845  df-srng 18846  df-lmod 18865  df-lss 18933  df-lmhm 19022  df-lvec 19103  df-sra 19172  df-rgmod 19173  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-refld 19951  df-phl 19971  df-dsmm 20076  df-frlm 20091  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cmp 21190  df-xms 22125  df-ms 22126  df-nm 22387  df-ngp 22388  df-tng 22389  df-nrg 22390  df-nlm 22391  df-clm 22863  df-cph 22968  df-tch 22969  df-rrx 23173  df-ovol 23233  df-vol 23234  df-salg 40529  df-sumge0 40580  df-mea 40667  df-ome 40704  df-caragen 40706  df-ovoln 40751  df-voln 40753

This theorem is referenced by:  opnvonmbl  40848