Theorem voliunlem2 22583

Description: Lemma for voliun 22586. (Contributed by Mario Carneiro, 20-Mar-2014.)

Hypotheses

Ref	Expression
voliunlem.3	|- ( ph -> F : NN --> dom vol )
voliunlem.5	|- ( ph -> Disj i e. NN ( F ` i ) )
voliunlem.6	|- H = ( n e. NN |-> ( vol* ` ( x i^i ( F ` n ) ) ) )

Assertion

Ref	Expression
voliunlem2	|- ( ph -> U. ran F e. dom vol )

Distinct variable groups:   i, n, x, F   ph, n, x

Allowed substitution hints:   ph( i)   H( x, i, n)

Proof of Theorem voliunlem2

Dummy variables k z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		voliunlem.3	. . . . 5 |- ( ph -> F : NN --> dom vol )
2		frn 5747	. . . . 5 |- ( F : NN --> dom vol -> ran F C_ dom vol )
3	1, 2	syl 17	. . . 4 |- ( ph -> ran F C_ dom vol )
4		mblss 22563	. . . . . 6 |- ( x e. dom vol -> x C_ RR )
5		selpw 3949	. . . . . 6 |- ( x e. ~P RR <-> x C_ RR )
6	4, 5	sylibr 217	. . . . 5 |- ( x e. dom vol -> x e. ~P RR )
7	6	ssriv 3422	. . . 4 |- dom vol C_ ~P RR
8	3, 7	syl6ss 3430	. . 3 |- ( ph -> ran F C_ ~P RR )
9		sspwuni 4360	. . 3 |- ( ran F C_ ~P RR <-> U. ran F C_ RR )
10	8, 9	sylib 201	. 2 |- ( ph -> U. ran F C_ RR )
11		elpwi 3951	. . . 4 |- ( x e. ~P RR -> x C_ RR )
12		inundif 3836	. . . . . . . 8 |- ( ( x i^i U. ran F ) u. ( x \ U. ran F ) ) = x
13	12	fveq2i 5882	. . . . . . 7 |- ( vol* ` ( ( x i^i U. ran F ) u. ( x \ U. ran F ) ) ) = ( vol* ` x )
14		inss1 3643	. . . . . . . . 9 |- ( x i^i U. ran F ) C_ x
15		simp2 1031	. . . . . . . . 9 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> x C_ RR )
16	14, 15	syl5ss 3429	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( x i^i U. ran F ) C_ RR )
17		ovolsscl 22517	. . . . . . . . . 10 |- ( ( ( x i^i U. ran F ) C_ x /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i U. ran F ) ) e. RR )
18	14, 17	mp3an1 1377	. . . . . . . . 9 |- ( ( x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i U. ran F ) ) e. RR )
19	18	3adant1 1048	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i U. ran F ) ) e. RR )
20		difss 3549	. . . . . . . . 9 |- ( x \ U. ran F ) C_ x
21	20, 15	syl5ss 3429	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( x \ U. ran F ) C_ RR )
22		ovolsscl 22517	. . . . . . . . . 10 |- ( ( ( x \ U. ran F ) C_ x /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x \ U. ran F ) ) e. RR )
23	20, 22	mp3an1 1377	. . . . . . . . 9 |- ( ( x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x \ U. ran F ) ) e. RR )
24	23	3adant1 1048	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x \ U. ran F ) ) e. RR )
25		ovolun 22530	. . . . . . . 8 |- ( ( ( ( x i^i U. ran F ) C_ RR /\ ( vol* ` ( x i^i U. ran F ) ) e. RR ) /\ ( ( x \ U. ran F ) C_ RR /\ ( vol* ` ( x \ U. ran F ) ) e. RR ) ) -> ( vol* ` ( ( x i^i U. ran F ) u. ( x \ U. ran F ) ) ) <_ ( ( vol* ` ( x i^i U. ran F ) ) + ( vol* ` ( x \ U. ran F ) ) ) )
26	16, 19, 21, 24, 25	syl22anc 1293	. . . . . . 7 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( ( x i^i U. ran F ) u. ( x \ U. ran F ) ) ) <_ ( ( vol* ` ( x i^i U. ran F ) ) + ( vol* ` ( x \ U. ran F ) ) ) )
27	13, 26	syl5eqbrr 4430	. . . . . 6 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` x ) <_ ( ( vol* ` ( x i^i U. ran F ) ) + ( vol* ` ( x \ U. ran F ) ) ) )
28	19	rexrd 9708	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i U. ran F ) ) e. RR* )
29		nnuz 11218	. . . . . . . . . . . 12 |- NN = ( ZZ>= ` 1 )
30		1zzd 10992	. . . . . . . . . . . 12 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> 1 e. ZZ )
31		fveq2 5879	. . . . . . . . . . . . . . . . 17 |- ( n = k -> ( F ` n ) = ( F ` k ) )
32	31	ineq2d 3625	. . . . . . . . . . . . . . . 16 |- ( n = k -> ( x i^i ( F ` n ) ) = ( x i^i ( F ` k ) ) )
33	32	fveq2d 5883	. . . . . . . . . . . . . . 15 |- ( n = k -> ( vol* ` ( x i^i ( F ` n ) ) ) = ( vol* ` ( x i^i ( F ` k ) ) ) )
34		voliunlem.6	. . . . . . . . . . . . . . 15 |- H = ( n e. NN |-> ( vol* ` ( x i^i ( F ` n ) ) ) )
35		fvex 5889	. . . . . . . . . . . . . . 15 |- ( vol* ` ( x i^i ( F ` k ) ) ) e. _V
36	33, 34, 35	fvmpt 5963	. . . . . . . . . . . . . 14 |- ( k e. NN -> ( H ` k ) = ( vol* ` ( x i^i ( F ` k ) ) ) )
37	36	adantl 473	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) /\ k e. NN ) -> ( H ` k ) = ( vol* ` ( x i^i ( F ` k ) ) ) )
38		inss1 3643	. . . . . . . . . . . . . . . 16 |- ( x i^i ( F ` k ) ) C_ x
39		ovolsscl 22517	. . . . . . . . . . . . . . . 16 |- ( ( ( x i^i ( F ` k ) ) C_ x /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i ( F ` k ) ) ) e. RR )
40	38, 39	mp3an1 1377	. . . . . . . . . . . . . . 15 |- ( ( x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i ( F ` k ) ) ) e. RR )
41	40	3adant1 1048	. . . . . . . . . . . . . 14 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i ( F ` k ) ) ) e. RR )
42	41	adantr 472	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) /\ k e. NN ) -> ( vol* ` ( x i^i ( F ` k ) ) ) e. RR )
43	37, 42	eqeltrd 2549	. . . . . . . . . . . 12 |- ( ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) /\ k e. NN ) -> ( H ` k ) e. RR )
44	29, 30, 43	serfre 12280	. . . . . . . . . . 11 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> seq 1 ( + , H ) : NN --> RR )
45		frn 5747	. . . . . . . . . . 11 |- ( seq 1 ( + , H ) : NN --> RR -> ran seq 1 ( + , H ) C_ RR )
46	44, 45	syl 17	. . . . . . . . . 10 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ran seq 1 ( + , H ) C_ RR )
47		ressxr 9702	. . . . . . . . . 10 |- RR C_ RR*
48	46, 47	syl6ss 3430	. . . . . . . . 9 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ran seq 1 ( + , H ) C_ RR* )
49		supxrcl 11625	. . . . . . . . 9 |- ( ran seq 1 ( + , H ) C_ RR* -> sup ( ran seq 1 ( + , H ) , RR* , < ) e. RR* )
50	48, 49	syl 17	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> sup ( ran seq 1 ( + , H ) , RR* , < ) e. RR* )
51		simp3 1032	. . . . . . . . . 10 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` x ) e. RR )
52	51, 24	resubcld 10068	. . . . . . . . 9 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) e. RR )
53	52	rexrd 9708	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) e. RR* )
54		iunin2 4333	. . . . . . . . . . 11 |- U_ n e. NN ( x i^i ( F ` n ) ) = ( x i^i U_ n e. NN ( F ` n ) )
55		ffn 5739	. . . . . . . . . . . . . 14 |- ( F : NN --> dom vol -> F Fn NN )
56		fniunfv 6170	. . . . . . . . . . . . . 14 |- ( F Fn NN -> U_ n e. NN ( F ` n ) = U. ran F )
57	1, 55, 56	3syl 18	. . . . . . . . . . . . 13 |- ( ph -> U_ n e. NN ( F ` n ) = U. ran F )
58	57	3ad2ant1 1051	. . . . . . . . . . . 12 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> U_ n e. NN ( F ` n ) = U. ran F )
59	58	ineq2d 3625	. . . . . . . . . . 11 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( x i^i U_ n e. NN ( F ` n ) ) = ( x i^i U. ran F ) )
60	54, 59	syl5eq 2517	. . . . . . . . . 10 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> U_ n e. NN ( x i^i ( F ` n ) ) = ( x i^i U. ran F ) )
61	60	fveq2d 5883	. . . . . . . . 9 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` U_ n e. NN ( x i^i ( F ` n ) ) ) = ( vol* ` ( x i^i U. ran F ) ) )
62		eqid 2471	. . . . . . . . . 10 |- seq 1 ( + , H ) = seq 1 ( + , H )
63		inss1 3643	. . . . . . . . . . . 12 |- ( x i^i ( F ` n ) ) C_ x
64	63, 15	syl5ss 3429	. . . . . . . . . . 11 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( x i^i ( F ` n ) ) C_ RR )
65	64	adantr 472	. . . . . . . . . 10 |- ( ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) /\ n e. NN ) -> ( x i^i ( F ` n ) ) C_ RR )
66		ovolsscl 22517	. . . . . . . . . . . . 13 |- ( ( ( x i^i ( F ` n ) ) C_ x /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i ( F ` n ) ) ) e. RR )
67	63, 66	mp3an1 1377	. . . . . . . . . . . 12 |- ( ( x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i ( F ` n ) ) ) e. RR )
68	67	3adant1 1048	. . . . . . . . . . 11 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i ( F ` n ) ) ) e. RR )
69	68	adantr 472	. . . . . . . . . 10 |- ( ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) /\ n e. NN ) -> ( vol* ` ( x i^i ( F ` n ) ) ) e. RR )
70	62, 34, 65, 69	ovoliun 22536	. . . . . . . . 9 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` U_ n e. NN ( x i^i ( F ` n ) ) ) <_ sup ( ran seq 1 ( + , H ) , RR* , < ) )
71	61, 70	eqbrtrrd 4418	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i U. ran F ) ) <_ sup ( ran seq 1 ( + , H ) , RR* , < ) )
72	1	3ad2ant1 1051	. . . . . . . . . . . . 13 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> F : NN --> dom vol )
73		voliunlem.5	. . . . . . . . . . . . . 14 |- ( ph -> Disj i e. NN ( F ` i ) )
74	73	3ad2ant1 1051	. . . . . . . . . . . . 13 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> Disj i e. NN ( F ` i ) )
75	72, 74, 34, 15, 51	voliunlem1 22582	. . . . . . . . . . . 12 |- ( ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) /\ k e. NN ) -> ( ( seq 1 ( + , H ) ` k ) + ( vol* ` ( x \ U. ran F ) ) ) <_ ( vol* ` x ) )
76	44	ffvelrnda 6037	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) /\ k e. NN ) -> ( seq 1 ( + , H ) ` k ) e. RR )
77	24	adantr 472	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) /\ k e. NN ) -> ( vol* ` ( x \ U. ran F ) ) e. RR )
78		simpl3 1035	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) /\ k e. NN ) -> ( vol* ` x ) e. RR )
79		leaddsub 10111	. . . . . . . . . . . . 13 |- ( ( ( seq 1 ( + , H ) ` k ) e. RR /\ ( vol* ` ( x \ U. ran F ) ) e. RR /\ ( vol* ` x ) e. RR ) -> ( ( ( seq 1 ( + , H ) ` k ) + ( vol* ` ( x \ U. ran F ) ) ) <_ ( vol* ` x ) <-> ( seq 1 ( + , H ) ` k ) <_ ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) ) )
80	76, 77, 78, 79	syl3anc 1292	. . . . . . . . . . . 12 |- ( ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) /\ k e. NN ) -> ( ( ( seq 1 ( + , H ) ` k ) + ( vol* ` ( x \ U. ran F ) ) ) <_ ( vol* ` x ) <-> ( seq 1 ( + , H ) ` k ) <_ ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) ) )
81	75, 80	mpbid 215	. . . . . . . . . . 11 |- ( ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) /\ k e. NN ) -> ( seq 1 ( + , H ) ` k ) <_ ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) )
82	81	ralrimiva 2809	. . . . . . . . . 10 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> A. k e. NN ( seq 1 ( + , H ) ` k ) <_ ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) )
83		ffn 5739	. . . . . . . . . . 11 |- ( seq 1 ( + , H ) : NN --> RR -> seq 1 ( + , H ) Fn NN )
84		breq1 4398	. . . . . . . . . . . 12 |- ( z = ( seq 1 ( + , H ) ` k ) -> ( z <_ ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) <-> ( seq 1 ( + , H ) ` k ) <_ ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) ) )
85	84	ralrn 6040	. . . . . . . . . . 11 |- ( seq 1 ( + , H ) Fn NN -> ( A. z e. ran seq 1 ( + , H ) z <_ ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) <-> A. k e. NN ( seq 1 ( + , H ) ` k ) <_ ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) ) )
86	44, 83, 85	3syl 18	. . . . . . . . . 10 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( A. z e. ran seq 1 ( + , H ) z <_ ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) <-> A. k e. NN ( seq 1 ( + , H ) ` k ) <_ ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) ) )
87	82, 86	mpbird 240	. . . . . . . . 9 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> A. z e. ran seq 1 ( + , H ) z <_ ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) )
88		supxrleub 11637	. . . . . . . . . 10 |- ( ( ran seq 1 ( + , H ) C_ RR* /\ ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) e. RR* ) -> ( sup ( ran seq 1 ( + , H ) , RR* , < ) <_ ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) <-> A. z e. ran seq 1 ( + , H ) z <_ ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) ) )
89	48, 53, 88	syl2anc 673	. . . . . . . . 9 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( sup ( ran seq 1 ( + , H ) , RR* , < ) <_ ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) <-> A. z e. ran seq 1 ( + , H ) z <_ ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) ) )
90	87, 89	mpbird 240	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> sup ( ran seq 1 ( + , H ) , RR* , < ) <_ ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) )
91	28, 50, 53, 71, 90	xrletrd 11482	. . . . . . 7 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i U. ran F ) ) <_ ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) )
92		leaddsub 10111	. . . . . . . 8 |- ( ( ( vol* ` ( x i^i U. ran F ) ) e. RR /\ ( vol* ` ( x \ U. ran F ) ) e. RR /\ ( vol* ` x ) e. RR ) -> ( ( ( vol* ` ( x i^i U. ran F ) ) + ( vol* ` ( x \ U. ran F ) ) ) <_ ( vol* ` x ) <-> ( vol* ` ( x i^i U. ran F ) ) <_ ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) ) )
93	19, 24, 51, 92	syl3anc 1292	. . . . . . 7 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( ( ( vol* ` ( x i^i U. ran F ) ) + ( vol* ` ( x \ U. ran F ) ) ) <_ ( vol* ` x ) <-> ( vol* ` ( x i^i U. ran F ) ) <_ ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) ) )
94	91, 93	mpbird 240	. . . . . 6 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( ( vol* ` ( x i^i U. ran F ) ) + ( vol* ` ( x \ U. ran F ) ) ) <_ ( vol* ` x ) )
95	19, 24	readdcld 9688	. . . . . . 7 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( ( vol* ` ( x i^i U. ran F ) ) + ( vol* ` ( x \ U. ran F ) ) ) e. RR )
96	51, 95	letri3d 9794	. . . . . 6 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( ( vol* ` x ) = ( ( vol* ` ( x i^i U. ran F ) ) + ( vol* ` ( x \ U. ran F ) ) ) <-> ( ( vol* ` x ) <_ ( ( vol* ` ( x i^i U. ran F ) ) + ( vol* ` ( x \ U. ran F ) ) ) /\ ( ( vol* ` ( x i^i U. ran F ) ) + ( vol* ` ( x \ U. ran F ) ) ) <_ ( vol* ` x ) ) ) )
97	27, 94, 96	mpbir2and 936	. . . . 5 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` x ) = ( ( vol* ` ( x i^i U. ran F ) ) + ( vol* ` ( x \ U. ran F ) ) ) )
98	97	3expia 1233	. . . 4 |- ( ( ph /\ x C_ RR ) -> ( ( vol* ` x ) e. RR -> ( vol* ` x ) = ( ( vol* ` ( x i^i U. ran F ) ) + ( vol* ` ( x \ U. ran F ) ) ) ) )
99	11, 98	sylan2 482	. . 3 |- ( ( ph /\ x e. ~P RR ) -> ( ( vol* ` x ) e. RR -> ( vol* ` x ) = ( ( vol* ` ( x i^i U. ran F ) ) + ( vol* ` ( x \ U. ran F ) ) ) ) )
100	99	ralrimiva 2809	. 2 |- ( ph -> A. x e. ~P RR ( ( vol* ` x ) e. RR -> ( vol* ` x ) = ( ( vol* ` ( x i^i U. ran F ) ) + ( vol* ` ( x \ U. ran F ) ) ) ) )
101		ismbl 22558	. 2 |- ( U. ran F e. dom vol <-> ( U. ran F C_ RR /\ A. x e. ~P RR ( ( vol* ` x ) e. RR -> ( vol* ` x ) = ( ( vol* ` ( x i^i U. ran F ) ) + ( vol* ` ( x \ U. ran F ) ) ) ) ) )
102	10, 100, 101	sylanbrc 677	1 |- ( ph -> U. ran F e. dom vol )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   A.wral 2756   \ cdif 3387   u. cun 3388   i^i cin 3389   C_ wss 3390   ~Pcpw 3942   U.cuni 4190   U_ciun 4269  Disj wdisj 4366   class class class wbr 4395   |-> cmpt 4454   dom cdm 4839   ran crn 4840   Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   supcsup 7972   RRcr 9556   1c1 9558   + caddc 9560   RR*cxr 9692   < clt 9693   <_ cle 9694   - cmin 9880   NNcn 10631   seqcseq 12251   vol*covol 22491   volcvol 22493

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-cc 8883  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

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-disj 4367  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-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  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-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-ioo 11664  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-rlim 13630  df-sum 13830  df-ovol 22494  df-vol 22496

This theorem is referenced by:  voliunlem3  22584  iunmbl  22585