Theorem voliunlem2 21007

Description: Lemma for voliun 21010. (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 5560	. . . . 5 |- ( F : NN --> dom vol -> ran F C_ dom vol )
3	1, 2	syl 16	. . . 4 |- ( ph -> ran F C_ dom vol )
4		mblss 20989	. . . . . 6 |- ( x e. dom vol -> x C_ RR )
5		selpw 3862	. . . . . 6 |- ( x e. ~P RR <-> x C_ RR )
6	4, 5	sylibr 212	. . . . 5 |- ( x e. dom vol -> x e. ~P RR )
7	6	ssriv 3355	. . . 4 |- dom vol C_ ~P RR
8	3, 7	syl6ss 3363	. . 3 |- ( ph -> ran F C_ ~P RR )
9		sspwuni 4251	. . 3 |- ( ran F C_ ~P RR <-> U. ran F C_ RR )
10	8, 9	sylib 196	. 2 |- ( ph -> U. ran F C_ RR )
11		elpwi 3864	. . . 4 |- ( x e. ~P RR -> x C_ RR )
12		inundif 3752	. . . . . . . 8 |- ( ( x i^i U. ran F ) u. ( x \ U. ran F ) ) = x
13	12	fveq2i 5689	. . . . . . 7 |- ( vol* ` ( ( x i^i U. ran F ) u. ( x \ U. ran F ) ) ) = ( vol* ` x )
14		inss1 3565	. . . . . . . . 9 |- ( x i^i U. ran F ) C_ x
15		simp2 989	. . . . . . . . 9 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> x C_ RR )
16	14, 15	syl5ss 3362	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( x i^i U. ran F ) C_ RR )
17		ovolsscl 20944	. . . . . . . . . 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 1301	. . . . . . . . 9 |- ( ( x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i U. ran F ) ) e. RR )
19	18	3adant1 1006	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i U. ran F ) ) e. RR )
20		difss 3478	. . . . . . . . 9 |- ( x \ U. ran F ) C_ x
21	20, 15	syl5ss 3362	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( x \ U. ran F ) C_ RR )
22		ovolsscl 20944	. . . . . . . . . 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 1301	. . . . . . . . 9 |- ( ( x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x \ U. ran F ) ) e. RR )
24	23	3adant1 1006	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x \ U. ran F ) ) e. RR )
25		ovolun 20957	. . . . . . . 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 1219	. . . . . . 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 4321	. . . . . 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 9425	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i U. ran F ) ) e. RR* )
29		nnuz 10888	. . . . . . . . . . . 12 |- NN = ( ZZ>= ` 1 )
30		1zzd 10669	. . . . . . . . . . . 12 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> 1 e. ZZ )
31		fveq2 5686	. . . . . . . . . . . . . . . . 17 |- ( n = k -> ( F ` n ) = ( F ` k ) )
32	31	ineq2d 3547	. . . . . . . . . . . . . . . 16 |- ( n = k -> ( x i^i ( F ` n ) ) = ( x i^i ( F ` k ) ) )
33	32	fveq2d 5690	. . . . . . . . . . . . . . 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 5696	. . . . . . . . . . . . . . 15 |- ( vol* ` ( x i^i ( F ` k ) ) ) e. _V
36	33, 34, 35	fvmpt 5769	. . . . . . . . . . . . . 14 |- ( k e. NN -> ( H ` k ) = ( vol* ` ( x i^i ( F ` k ) ) ) )
37	36	adantl 466	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) /\ k e. NN ) -> ( H ` k ) = ( vol* ` ( x i^i ( F ` k ) ) ) )
38		inss1 3565	. . . . . . . . . . . . . . . 16 |- ( x i^i ( F ` k ) ) C_ x
39		ovolsscl 20944	. . . . . . . . . . . . . . . 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 1301	. . . . . . . . . . . . . . 15 |- ( ( x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i ( F ` k ) ) ) e. RR )
41	40	3adant1 1006	. . . . . . . . . . . . . 14 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i ( F ` k ) ) ) e. RR )
42	41	adantr 465	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) /\ k e. NN ) -> ( vol* ` ( x i^i ( F ` k ) ) ) e. RR )
43	37, 42	eqeltrd 2512	. . . . . . . . . . . 12 |- ( ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) /\ k e. NN ) -> ( H ` k ) e. RR )
44	29, 30, 43	serfre 11827	. . . . . . . . . . 11 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> seq 1 ( + , H ) : NN --> RR )
45		frn 5560	. . . . . . . . . . 11 |- ( seq 1 ( + , H ) : NN --> RR -> ran seq 1 ( + , H ) C_ RR )
46	44, 45	syl 16	. . . . . . . . . 10 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ran seq 1 ( + , H ) C_ RR )
47		ressxr 9419	. . . . . . . . . 10 |- RR C_ RR*
48	46, 47	syl6ss 3363	. . . . . . . . 9 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ran seq 1 ( + , H ) C_ RR* )
49		supxrcl 11269	. . . . . . . . 9 |- ( ran seq 1 ( + , H ) C_ RR* -> sup ( ran seq 1 ( + , H ) , RR* , < ) e. RR* )
50	48, 49	syl 16	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> sup ( ran seq 1 ( + , H ) , RR* , < ) e. RR* )
51		simp3 990	. . . . . . . . . 10 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` x ) e. RR )
52	51, 24	resubcld 9768	. . . . . . . . 9 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) e. RR )
53	52	rexrd 9425	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) e. RR* )
54		iunin2 4229	. . . . . . . . . . 11 |- U_ n e. NN ( x i^i ( F ` n ) ) = ( x i^i U_ n e. NN ( F ` n ) )
55		ffn 5554	. . . . . . . . . . . . . 14 |- ( F : NN --> dom vol -> F Fn NN )
56		fniunfv 5959	. . . . . . . . . . . . . 14 |- ( F Fn NN -> U_ n e. NN ( F ` n ) = U. ran F )
57	1, 55, 56	3syl 20	. . . . . . . . . . . . 13 |- ( ph -> U_ n e. NN ( F ` n ) = U. ran F )
58	57	3ad2ant1 1009	. . . . . . . . . . . 12 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> U_ n e. NN ( F ` n ) = U. ran F )
59	58	ineq2d 3547	. . . . . . . . . . 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 2482	. . . . . . . . . 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 5690	. . . . . . . . 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 2438	. . . . . . . . . 10 |- seq 1 ( + , H ) = seq 1 ( + , H )
63		inss1 3565	. . . . . . . . . . . 12 |- ( x i^i ( F ` n ) ) C_ x
64	63, 15	syl5ss 3362	. . . . . . . . . . 11 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( x i^i ( F ` n ) ) C_ RR )
65	64	adantr 465	. . . . . . . . . 10 |- ( ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) /\ n e. NN ) -> ( x i^i ( F ` n ) ) C_ RR )
66		ovolsscl 20944	. . . . . . . . . . . . 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 1301	. . . . . . . . . . . 12 |- ( ( x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i ( F ` n ) ) ) e. RR )
68	67	3adant1 1006	. . . . . . . . . . 11 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i ( F ` n ) ) ) e. RR )
69	68	adantr 465	. . . . . . . . . 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 20963	. . . . . . . . 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 4309	. . . . . . . 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 1009	. . . . . . . . . . . . 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 1009	. . . . . . . . . . . . 13 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> Disj i e. NN ( F ` i ) )
75	72, 74, 34, 15, 51	voliunlem1 21006	. . . . . . . . . . . 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 5838	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) /\ k e. NN ) -> ( seq 1 ( + , H ) ` k ) e. RR )
77	24	adantr 465	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) /\ k e. NN ) -> ( vol* ` ( x \ U. ran F ) ) e. RR )
78		simpl3 993	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) /\ k e. NN ) -> ( vol* ` x ) e. RR )
79		leaddsub 9807	. . . . . . . . . . . . 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 1218	. . . . . . . . . . . 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 210	. . . . . . . . . . 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 2794	. . . . . . . . . 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 5554	. . . . . . . . . . 11 |- ( seq 1 ( + , H ) : NN --> RR -> seq 1 ( + , H ) Fn NN )
84		breq1 4290	. . . . . . . . . . . 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 5841	. . . . . . . . . . 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 20	. . . . . . . . . 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 232	. . . . . . . . 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 11281	. . . . . . . . . 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 661	. . . . . . . . 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 232	. . . . . . . 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 11128	. . . . . . 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 9807	. . . . . . . 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 1218	. . . . . . 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 232	. . . . . 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 9405	. . . . . . 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 9508	. . . . . 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 913	. . . . 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 1189	. . . 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 474	. . 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 2794	. 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 20984	. 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 664	1 |- ( ph -> U. ran F e. dom vol )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   A.wral 2710   \ cdif 3320   u. cun 3321   i^i cin 3322   C_ wss 3323   ~Pcpw 3855   U.cuni 4086   U_ciun 4166  Disj wdisj 4257   class class class wbr 4287   e. cmpt 4345   dom cdm 4835   ran crn 4836   Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086   supcsup 7682   RRcr 9273   1c1 9275   + caddc 9277   RR*cxr 9409   < clt 9410   <_ cle 9411   - cmin 9587   NNcn 10314   seqcseq 11798   vol*covol 20921   volcvol 20922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cc 8596  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-disj 4258  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-q 10946  df-rp 10984  df-ioo 11296  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-rlim 12959  df-sum 13156  df-ovol 20923  df-vol 20924

This theorem is referenced by:  voliunlem3  21008  iunmbl  21009