Theorem voliunlem2 23319

Description: Lemma for voliun 23322. (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 6053	. . . . 5 |- ( F : NN --> dom vol -> ran F C_ dom vol )
3	1, 2	syl 17	. . . 4 |- ( ph -> ran F C_ dom vol )
4		mblss 23299	. . . . . 6 |- ( x e. dom vol -> x C_ RR )
5		selpw 4165	. . . . . 6 |- ( x e. ~P RR <-> x C_ RR )
6	4, 5	sylibr 224	. . . . 5 |- ( x e. dom vol -> x e. ~P RR )
7	6	ssriv 3607	. . . 4 |- dom vol C_ ~P RR
8	3, 7	syl6ss 3615	. . 3 |- ( ph -> ran F C_ ~P RR )
9		sspwuni 4611	. . 3 |- ( ran F C_ ~P RR <-> U. ran F C_ RR )
10	8, 9	sylib 208	. 2 |- ( ph -> U. ran F C_ RR )
11		elpwi 4168	. . . 4 |- ( x e. ~P RR -> x C_ RR )
12		inundif 4046	. . . . . . . 8 |- ( ( x i^i U. ran F ) u. ( x \ U. ran F ) ) = x
13	12	fveq2i 6194	. . . . . . 7 |- ( vol* ` ( ( x i^i U. ran F ) u. ( x \ U. ran F ) ) ) = ( vol* ` x )
14		inss1 3833	. . . . . . . . 9 |- ( x i^i U. ran F ) C_ x
15		simp2 1062	. . . . . . . . 9 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> x C_ RR )
16	14, 15	syl5ss 3614	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( x i^i U. ran F ) C_ RR )
17		ovolsscl 23254	. . . . . . . . . 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 1411	. . . . . . . . 9 |- ( ( x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i U. ran F ) ) e. RR )
19	18	3adant1 1079	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i U. ran F ) ) e. RR )
20		difss 3737	. . . . . . . . 9 |- ( x \ U. ran F ) C_ x
21	20, 15	syl5ss 3614	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( x \ U. ran F ) C_ RR )
22		ovolsscl 23254	. . . . . . . . . 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 1411	. . . . . . . . 9 |- ( ( x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x \ U. ran F ) ) e. RR )
24	23	3adant1 1079	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x \ U. ran F ) ) e. RR )
25		ovolun 23267	. . . . . . . 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 1327	. . . . . . 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 4689	. . . . . 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 10089	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i U. ran F ) ) e. RR* )
29		nnuz 11723	. . . . . . . . . . . 12 |- NN = ( ZZ>= ` 1 )
30		1zzd 11408	. . . . . . . . . . . 12 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> 1 e. ZZ )
31		fveq2 6191	. . . . . . . . . . . . . . . . 17 |- ( n = k -> ( F ` n ) = ( F ` k ) )
32	31	ineq2d 3814	. . . . . . . . . . . . . . . 16 |- ( n = k -> ( x i^i ( F ` n ) ) = ( x i^i ( F ` k ) ) )
33	32	fveq2d 6195	. . . . . . . . . . . . . . 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 6201	. . . . . . . . . . . . . . 15 |- ( vol* ` ( x i^i ( F ` k ) ) ) e. _V
36	33, 34, 35	fvmpt 6282	. . . . . . . . . . . . . 14 |- ( k e. NN -> ( H ` k ) = ( vol* ` ( x i^i ( F ` k ) ) ) )
37	36	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) /\ k e. NN ) -> ( H ` k ) = ( vol* ` ( x i^i ( F ` k ) ) ) )
38		inss1 3833	. . . . . . . . . . . . . . . 16 |- ( x i^i ( F ` k ) ) C_ x
39		ovolsscl 23254	. . . . . . . . . . . . . . . 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 1411	. . . . . . . . . . . . . . 15 |- ( ( x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i ( F ` k ) ) ) e. RR )
41	40	3adant1 1079	. . . . . . . . . . . . . 14 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i ( F ` k ) ) ) e. RR )
42	41	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) /\ k e. NN ) -> ( vol* ` ( x i^i ( F ` k ) ) ) e. RR )
43	37, 42	eqeltrd 2701	. . . . . . . . . . . 12 |- ( ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) /\ k e. NN ) -> ( H ` k ) e. RR )
44	29, 30, 43	serfre 12830	. . . . . . . . . . 11 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> seq 1 ( + , H ) : NN --> RR )
45		frn 6053	. . . . . . . . . . 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 10083	. . . . . . . . . 10 |- RR C_ RR*
48	46, 47	syl6ss 3615	. . . . . . . . 9 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ran seq 1 ( + , H ) C_ RR* )
49		supxrcl 12145	. . . . . . . . 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 1063	. . . . . . . . . 10 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` x ) e. RR )
52	51, 24	resubcld 10458	. . . . . . . . 9 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) e. RR )
53	52	rexrd 10089	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) e. RR* )
54		iunin2 4584	. . . . . . . . . . 11 |- U_ n e. NN ( x i^i ( F ` n ) ) = ( x i^i U_ n e. NN ( F ` n ) )
55		ffn 6045	. . . . . . . . . . . . . 14 |- ( F : NN --> dom vol -> F Fn NN )
56		fniunfv 6505	. . . . . . . . . . . . . 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 1082	. . . . . . . . . . . 12 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> U_ n e. NN ( F ` n ) = U. ran F )
59	58	ineq2d 3814	. . . . . . . . . . 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 2668	. . . . . . . . . 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 6195	. . . . . . . . 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 2622	. . . . . . . . . 10 |- seq 1 ( + , H ) = seq 1 ( + , H )
63		inss1 3833	. . . . . . . . . . . 12 |- ( x i^i ( F ` n ) ) C_ x
64	63, 15	syl5ss 3614	. . . . . . . . . . 11 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( x i^i ( F ` n ) ) C_ RR )
65	64	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) /\ n e. NN ) -> ( x i^i ( F ` n ) ) C_ RR )
66		ovolsscl 23254	. . . . . . . . . . . . 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 1411	. . . . . . . . . . . 12 |- ( ( x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i ( F ` n ) ) ) e. RR )
68	67	3adant1 1079	. . . . . . . . . . 11 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i ( F ` n ) ) ) e. RR )
69	68	adantr 481	. . . . . . . . . 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 23273	. . . . . . . . 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 4677	. . . . . . . 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 1082	. . . . . . . . . . . . 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 1082	. . . . . . . . . . . . 13 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> Disj i e. NN ( F ` i ) )
75	72, 74, 34, 15, 51	voliunlem1 23318	. . . . . . . . . . . 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 6359	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) /\ k e. NN ) -> ( seq 1 ( + , H ) ` k ) e. RR )
77	24	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) /\ k e. NN ) -> ( vol* ` ( x \ U. ran F ) ) e. RR )
78		simpl3 1066	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) /\ k e. NN ) -> ( vol* ` x ) e. RR )
79		leaddsub 10504	. . . . . . . . . . . . 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 1326	. . . . . . . . . . . 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 222	. . . . . . . . . . 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 2966	. . . . . . . . . 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 6045	. . . . . . . . . . 11 |- ( seq 1 ( + , H ) : NN --> RR -> seq 1 ( + , H ) Fn NN )
84		breq1 4656	. . . . . . . . . . . 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 6362	. . . . . . . . . . 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 247	. . . . . . . . 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 12156	. . . . . . . . . 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 693	. . . . . . . . 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 247	. . . . . . . 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 11993	. . . . . . 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 10504	. . . . . . . 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 1326	. . . . . . 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 247	. . . . . 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 10069	. . . . . . 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 10179	. . . . . 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 957	. . . . 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 1267	. . . 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 491	. . 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 2966	. 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 23294	. 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 698	1 |- ( ph -> U. ran F e. dom vol )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   U_ciun 4520  Disj wdisj 4620   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   supcsup 8346   RRcr 9935   1c1 9937   + caddc 9939   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   seqcseq 12801   vol*covol 23231   volcvol 23232

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-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

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-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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  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-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  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-ovol 23233  df-vol 23234

This theorem is referenced by:  voliunlem3  23320  iunmbl  23321