Theorem voliunlem2 21158

Description: Lemma for voliun 21161. (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 5666	. . . . 5 |- ( F : NN --> dom vol -> ran F C_ dom vol )
3	1, 2	syl 16	. . . 4 |- ( ph -> ran F C_ dom vol )
4		mblss 21139	. . . . . 6 |- ( x e. dom vol -> x C_ RR )
5		selpw 3968	. . . . . 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 3461	. . . 4 |- dom vol C_ ~P RR
8	3, 7	syl6ss 3469	. . 3 |- ( ph -> ran F C_ ~P RR )
9		sspwuni 4357	. . 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 3970	. . . 4 |- ( x e. ~P RR -> x C_ RR )
12		inundif 3858	. . . . . . . 8 |- ( ( x i^i U. ran F ) u. ( x \ U. ran F ) ) = x
13	12	fveq2i 5795	. . . . . . 7 |- ( vol* ` ( ( x i^i U. ran F ) u. ( x \ U. ran F ) ) ) = ( vol* ` x )
14		inss1 3671	. . . . . . . . 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 3468	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( x i^i U. ran F ) C_ RR )
17		ovolsscl 21094	. . . . . . . . . 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 1302	. . . . . . . . 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 3584	. . . . . . . . 9 |- ( x \ U. ran F ) C_ x
21	20, 15	syl5ss 3468	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( x \ U. ran F ) C_ RR )
22		ovolsscl 21094	. . . . . . . . . 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 1302	. . . . . . . . 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 21107	. . . . . . . 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 1220	. . . . . . 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 4427	. . . . . 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 9537	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i U. ran F ) ) e. RR* )
29		nnuz 11000	. . . . . . . . . . . 12 |- NN = ( ZZ>= ` 1 )
30		1zzd 10781	. . . . . . . . . . . 12 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> 1 e. ZZ )
31		fveq2 5792	. . . . . . . . . . . . . . . . 17 |- ( n = k -> ( F ` n ) = ( F ` k ) )
32	31	ineq2d 3653	. . . . . . . . . . . . . . . 16 |- ( n = k -> ( x i^i ( F ` n ) ) = ( x i^i ( F ` k ) ) )
33	32	fveq2d 5796	. . . . . . . . . . . . . . 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 5802	. . . . . . . . . . . . . . 15 |- ( vol* ` ( x i^i ( F ` k ) ) ) e. _V
36	33, 34, 35	fvmpt 5876	. . . . . . . . . . . . . 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 3671	. . . . . . . . . . . . . . . 16 |- ( x i^i ( F ` k ) ) C_ x
39		ovolsscl 21094	. . . . . . . . . . . . . . . 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 1302	. . . . . . . . . . . . . . 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 2539	. . . . . . . . . . . 12 |- ( ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) /\ k e. NN ) -> ( H ` k ) e. RR )
44	29, 30, 43	serfre 11945	. . . . . . . . . . 11 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> seq 1 ( + , H ) : NN --> RR )
45		frn 5666	. . . . . . . . . . 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 9531	. . . . . . . . . 10 |- RR C_ RR*
48	46, 47	syl6ss 3469	. . . . . . . . 9 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ran seq 1 ( + , H ) C_ RR* )
49		supxrcl 11381	. . . . . . . . 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 9880	. . . . . . . . 9 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) e. RR )
53	52	rexrd 9537	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) e. RR* )
54		iunin2 4335	. . . . . . . . . . 11 |- U_ n e. NN ( x i^i ( F ` n ) ) = ( x i^i U_ n e. NN ( F ` n ) )
55		ffn 5660	. . . . . . . . . . . . . 14 |- ( F : NN --> dom vol -> F Fn NN )
56		fniunfv 6066	. . . . . . . . . . . . . 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 3653	. . . . . . . . . . 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 2504	. . . . . . . . . 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 5796	. . . . . . . . 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 2451	. . . . . . . . . 10 |- seq 1 ( + , H ) = seq 1 ( + , H )
63		inss1 3671	. . . . . . . . . . . 12 |- ( x i^i ( F ` n ) ) C_ x
64	63, 15	syl5ss 3468	. . . . . . . . . . 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 21094	. . . . . . . . . . . . 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 1302	. . . . . . . . . . . 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 21113	. . . . . . . . 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 4415	. . . . . . . 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 21157	. . . . . . . . . . . 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 5945	. . . . . . . . . . . . 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 9919	. . . . . . . . . . . . 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 1219	. . . . . . . . . . . 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 2825	. . . . . . . . . 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 5660	. . . . . . . . . . 11 |- ( seq 1 ( + , H ) : NN --> RR -> seq 1 ( + , H ) Fn NN )
84		breq1 4396	. . . . . . . . . . . 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 5948	. . . . . . . . . . 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 11393	. . . . . . . . . 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 11240	. . . . . . 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 9919	. . . . . . . 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 1219	. . . . . . 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 9517	. . . . . . 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 9620	. . . . . 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 1190	. . . 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 2825	. 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 21134	. 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 1370   e. wcel 1758   A.wral 2795   \ cdif 3426   u. cun 3427   i^i cin 3428   C_ wss 3429   ~Pcpw 3961   U.cuni 4192   U_ciun 4272  Disj wdisj 4363   class class class wbr 4393   |-> cmpt 4451   dom cdm 4941   ran crn 4942   Fn wfn 5514   -->wf 5515   ` cfv 5519  (class class class)co 6193   supcsup 7794   RRcr 9385   1c1 9387   + caddc 9389   RR*cxr 9521   < clt 9522   <_ cle 9523   - cmin 9699   NNcn 10426   seqcseq 11916   vol*covol 21071   volcvol 21072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cc 8708  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-disj 4364  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-map 7319  df-pm 7320  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-sup 7795  df-oi 7828  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-n0 10684  df-z 10751  df-uz 10966  df-q 11058  df-rp 11096  df-ioo 11408  df-ico 11410  df-icc 11411  df-fz 11548  df-fzo 11659  df-fl 11752  df-seq 11917  df-exp 11976  df-hash 12214  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-clim 13077  df-rlim 13078  df-sum 13275  df-ovol 21073  df-vol 21074

This theorem is referenced by:  voliunlem3  21159  iunmbl  21160