Theorem voliunlem2 21696

Description: Lemma for voliun 21699. (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 5735	. . . . 5 |- ( F : NN --> dom vol -> ran F C_ dom vol )
3	1, 2	syl 16	. . . 4 |- ( ph -> ran F C_ dom vol )
4		mblss 21677	. . . . . 6 |- ( x e. dom vol -> x C_ RR )
5		selpw 4017	. . . . . 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 3508	. . . 4 |- dom vol C_ ~P RR
8	3, 7	syl6ss 3516	. . 3 |- ( ph -> ran F C_ ~P RR )
9		sspwuni 4411	. . 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 4019	. . . 4 |- ( x e. ~P RR -> x C_ RR )
12		inundif 3905	. . . . . . . 8 |- ( ( x i^i U. ran F ) u. ( x \ U. ran F ) ) = x
13	12	fveq2i 5867	. . . . . . 7 |- ( vol* ` ( ( x i^i U. ran F ) u. ( x \ U. ran F ) ) ) = ( vol* ` x )
14		inss1 3718	. . . . . . . . 9 |- ( x i^i U. ran F ) C_ x
15		simp2 997	. . . . . . . . 9 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> x C_ RR )
16	14, 15	syl5ss 3515	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( x i^i U. ran F ) C_ RR )
17		ovolsscl 21632	. . . . . . . . . 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 1311	. . . . . . . . 9 |- ( ( x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i U. ran F ) ) e. RR )
19	18	3adant1 1014	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i U. ran F ) ) e. RR )
20		difss 3631	. . . . . . . . 9 |- ( x \ U. ran F ) C_ x
21	20, 15	syl5ss 3515	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( x \ U. ran F ) C_ RR )
22		ovolsscl 21632	. . . . . . . . . 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 1311	. . . . . . . . 9 |- ( ( x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x \ U. ran F ) ) e. RR )
24	23	3adant1 1014	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x \ U. ran F ) ) e. RR )
25		ovolun 21645	. . . . . . . 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 1229	. . . . . . 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 4481	. . . . . 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 9639	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i U. ran F ) ) e. RR* )
29		nnuz 11113	. . . . . . . . . . . 12 |- NN = ( ZZ>= ` 1 )
30		1zzd 10891	. . . . . . . . . . . 12 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> 1 e. ZZ )
31		fveq2 5864	. . . . . . . . . . . . . . . . 17 |- ( n = k -> ( F ` n ) = ( F ` k ) )
32	31	ineq2d 3700	. . . . . . . . . . . . . . . 16 |- ( n = k -> ( x i^i ( F ` n ) ) = ( x i^i ( F ` k ) ) )
33	32	fveq2d 5868	. . . . . . . . . . . . . . 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 5874	. . . . . . . . . . . . . . 15 |- ( vol* ` ( x i^i ( F ` k ) ) ) e. _V
36	33, 34, 35	fvmpt 5948	. . . . . . . . . . . . . 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 3718	. . . . . . . . . . . . . . . 16 |- ( x i^i ( F ` k ) ) C_ x
39		ovolsscl 21632	. . . . . . . . . . . . . . . 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 1311	. . . . . . . . . . . . . . 15 |- ( ( x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i ( F ` k ) ) ) e. RR )
41	40	3adant1 1014	. . . . . . . . . . . . . 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 2555	. . . . . . . . . . . 12 |- ( ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) /\ k e. NN ) -> ( H ` k ) e. RR )
44	29, 30, 43	serfre 12100	. . . . . . . . . . 11 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> seq 1 ( + , H ) : NN --> RR )
45		frn 5735	. . . . . . . . . . 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 9633	. . . . . . . . . 10 |- RR C_ RR*
48	46, 47	syl6ss 3516	. . . . . . . . 9 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ran seq 1 ( + , H ) C_ RR* )
49		supxrcl 11502	. . . . . . . . 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 998	. . . . . . . . . 10 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` x ) e. RR )
52	51, 24	resubcld 9983	. . . . . . . . 9 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) e. RR )
53	52	rexrd 9639	. . . . . . . 8 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( ( vol* ` x ) - ( vol* ` ( x \ U. ran F ) ) ) e. RR* )
54		iunin2 4389	. . . . . . . . . . 11 |- U_ n e. NN ( x i^i ( F ` n ) ) = ( x i^i U_ n e. NN ( F ` n ) )
55		ffn 5729	. . . . . . . . . . . . . 14 |- ( F : NN --> dom vol -> F Fn NN )
56		fniunfv 6145	. . . . . . . . . . . . . 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 1017	. . . . . . . . . . . 12 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> U_ n e. NN ( F ` n ) = U. ran F )
59	58	ineq2d 3700	. . . . . . . . . . 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 2520	. . . . . . . . . 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 5868	. . . . . . . . 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 2467	. . . . . . . . . 10 |- seq 1 ( + , H ) = seq 1 ( + , H )
63		inss1 3718	. . . . . . . . . . . 12 |- ( x i^i ( F ` n ) ) C_ x
64	63, 15	syl5ss 3515	. . . . . . . . . . 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 21632	. . . . . . . . . . . . 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 1311	. . . . . . . . . . . 12 |- ( ( x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i ( F ` n ) ) ) e. RR )
68	67	3adant1 1014	. . . . . . . . . . 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 21651	. . . . . . . . 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 4469	. . . . . . . 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 1017	. . . . . . . . . . . . 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 1017	. . . . . . . . . . . . 13 |- ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> Disj i e. NN ( F ` i ) )
75	72, 74, 34, 15, 51	voliunlem1 21695	. . . . . . . . . . . 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 6019	. . . . . . . . . . . . 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 1001	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x C_ RR /\ ( vol* ` x ) e. RR ) /\ k e. NN ) -> ( vol* ` x ) e. RR )
79		leaddsub 10024	. . . . . . . . . . . . 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 1228	. . . . . . . . . . . 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 2878	. . . . . . . . . 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 5729	. . . . . . . . . . 11 |- ( seq 1 ( + , H ) : NN --> RR -> seq 1 ( + , H ) Fn NN )
84		breq1 4450	. . . . . . . . . . . 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 6022	. . . . . . . . . . 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 11514	. . . . . . . . . 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 11361	. . . . . . 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 10024	. . . . . . . 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 1228	. . . . . . 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 9619	. . . . . . 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 9722	. . . . . 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 920	. . . . 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 1198	. . . 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 2878	. 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 21672	. 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 973   = wceq 1379   e. wcel 1767   A.wral 2814   \ cdif 3473   u. cun 3474   i^i cin 3475   C_ wss 3476   ~Pcpw 4010   U.cuni 4245   U_ciun 4325  Disj wdisj 4417   class class class wbr 4447   |-> cmpt 4505   dom cdm 4999   ran crn 5000   Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   supcsup 7896   RRcr 9487   1c1 9489   + caddc 9491   RR*cxr 9623   < clt 9624   <_ cle 9625   - cmin 9801   NNcn 10532   seqcseq 12071   vol*covol 21609   volcvol 21610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cc 8811  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-disj 4418  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-q 11179  df-rp 11217  df-ioo 11529  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-fl 11893  df-seq 12072  df-exp 12131  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-rlim 13271  df-sum 13468  df-ovol 21611  df-vol 21612

This theorem is referenced by:  voliunlem3  21697  iunmbl  21698