Theorem ovoliunlem1 19403

Description: Lemma for ovoliun 19406. (Contributed by Mario Carneiro, 12-Jun-2014.)

Hypotheses

Ref	Expression
ovoliun.t	|- T = seq 1 ( + , G )
ovoliun.g	|- G = ( n e. NN |-> ( vol * ` A ) )
ovoliun.a	|- ( ( ph /\ n e. NN ) -> A C_ RR )
ovoliun.v	|- ( ( ph /\ n e. NN ) -> ( vol * ` A ) e. RR )
ovoliun.r	|- ( ph -> sup ( ran T , RR* , < ) e. RR )
ovoliun.b	|- ( ph -> B e. RR+ )
ovoliun.s	|- S = seq 1 ( + , ( ( abs o. - ) o. ( F ` n ) ) )
ovoliun.u	|- U = seq 1 ( + , ( ( abs o. - ) o. H ) )
ovoliun.h	|- H = ( k e. NN |-> ( ( F ` ( 1st ` ( J ` k ) ) ) ` ( 2nd ` ( J ` k ) ) ) )
ovoliun.j	|- ( ph -> J : NN -1-1-onto-> ( NN X. NN ) )
ovoliun.f	|- ( ph -> F : NN --> ( ( <_ i^i ( RR X. RR ) ) ^m NN ) )
ovoliun.x1	|- ( ( ph /\ n e. NN ) -> A C_ U. ran ( (,) o. ( F ` n ) ) )
ovoliun.x2	|- ( ( ph /\ n e. NN ) -> sup ( ran S , RR* , < ) <_ ( ( vol * ` A ) + ( B / ( 2 ^ n ) ) ) )
ovoliun.k	|- ( ph -> K e. NN )
ovoliun.l1	|- ( ph -> L e. ZZ )
ovoliun.l2	|- ( ph -> A. w e. ( 1 ... K ) ( 1st ` ( J ` w ) ) <_ L )

Assertion

Ref	Expression
ovoliunlem1	|- ( ph -> ( U ` K ) <_ ( sup ( ran T , RR* , < ) + B ) )

Distinct variable groups:   A, k   k, n, B   k, F, n   w, k, J, n   n, K, w   k, L, n, w   n, H   ph, k, n   S, k   k, G   T, k   n, G   T, n

Allowed substitution hints:   ph( w)   A( w, n)   B( w)   S( w, n)   T( w)   U( w, k, n)   F( w)   G( w)   H( w, k)   K( k)

Proof of Theorem ovoliunlem1

Dummy variables j m x y z i are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5731	. . . . . . . . 9 |- ( j = ( J ` m ) -> ( 1st ` j ) = ( 1st ` ( J ` m ) ) )
2	1	fveq2d 5735	. . . . . . . 8 |- ( j = ( J ` m ) -> ( F ` ( 1st ` j ) ) = ( F ` ( 1st ` ( J ` m ) ) ) )
3		fveq2 5731	. . . . . . . 8 |- ( j = ( J ` m ) -> ( 2nd ` j ) = ( 2nd ` ( J ` m ) ) )
4	2, 3	fveq12d 5737	. . . . . . 7 |- ( j = ( J ` m ) -> ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) = ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) )
5	4	fveq2d 5735	. . . . . 6 |- ( j = ( J ` m ) -> ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) = ( 2nd ` ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) ) )
6	4	fveq2d 5735	. . . . . 6 |- ( j = ( J ` m ) -> ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) = ( 1st ` ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) ) )
7	5, 6	oveq12d 6102	. . . . 5 |- ( j = ( J ` m ) -> ( ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) - ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) ) = ( ( 2nd ` ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) ) - ( 1st ` ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) ) ) )
8		fzfid 11317	. . . . 5 |- ( ph -> ( 1 ... K ) e. Fin )
9		ovoliun.j	. . . . . . 7 |- ( ph -> J : NN -1-1-onto-> ( NN X. NN ) )
10		f1of1 5676	. . . . . . 7 |- ( J : NN -1-1-onto-> ( NN X. NN ) -> J : NN -1-1-> ( NN X. NN ) )
11	9, 10	syl 16	. . . . . 6 |- ( ph -> J : NN -1-1-> ( NN X. NN ) )
12		elfznn 11085	. . . . . . 7 |- ( m e. ( 1 ... K ) -> m e. NN )
13	12	ssriv 3354	. . . . . 6 |- ( 1 ... K ) C_ NN
14		f1ores 5692	. . . . . 6 |- ( ( J : NN -1-1-> ( NN X. NN ) /\ ( 1 ... K ) C_ NN ) -> ( J |` ( 1 ... K ) ) : ( 1 ... K ) -1-1-onto-> ( J " ( 1 ... K ) ) )
15	11, 13, 14	sylancl 645	. . . . 5 |- ( ph -> ( J |` ( 1 ... K ) ) : ( 1 ... K ) -1-1-onto-> ( J " ( 1 ... K ) ) )
16		fvres 5748	. . . . . 6 |- ( m e. ( 1 ... K ) -> ( ( J |` ( 1 ... K ) ) ` m ) = ( J ` m ) )
17	16	adantl 454	. . . . 5 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( ( J |` ( 1 ... K ) ) ` m ) = ( J ` m ) )
18		inss2 3564	. . . . . . . . 9 |- ( <_ i^i ( RR X. RR ) ) C_ ( RR X. RR )
19		ovoliun.f	. . . . . . . . . . . . 13 |- ( ph -> F : NN --> ( ( <_ i^i ( RR X. RR ) ) ^m NN ) )
20	19	adantr 453	. . . . . . . . . . . 12 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> F : NN --> ( ( <_ i^i ( RR X. RR ) ) ^m NN ) )
21		imassrn 5219	. . . . . . . . . . . . . . 15 |- ( J " ( 1 ... K ) ) C_ ran J
22		f1of 5677	. . . . . . . . . . . . . . . . 17 |- ( J : NN -1-1-onto-> ( NN X. NN ) -> J : NN --> ( NN X. NN ) )
23	9, 22	syl 16	. . . . . . . . . . . . . . . 16 |- ( ph -> J : NN --> ( NN X. NN ) )
24		frn 5600	. . . . . . . . . . . . . . . 16 |- ( J : NN --> ( NN X. NN ) -> ran J C_ ( NN X. NN ) )
25	23, 24	syl 16	. . . . . . . . . . . . . . 15 |- ( ph -> ran J C_ ( NN X. NN ) )
26	21, 25	syl5ss 3361	. . . . . . . . . . . . . 14 |- ( ph -> ( J " ( 1 ... K ) ) C_ ( NN X. NN ) )
27	26	sselda 3350	. . . . . . . . . . . . 13 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> j e. ( NN X. NN ) )
28		xp1st 6379	. . . . . . . . . . . . 13 |- ( j e. ( NN X. NN ) -> ( 1st ` j ) e. NN )
29	27, 28	syl 16	. . . . . . . . . . . 12 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( 1st ` j ) e. NN )
30	20, 29	ffvelrnd 5874	. . . . . . . . . . 11 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( F ` ( 1st ` j ) ) e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) )
31		reex 9086	. . . . . . . . . . . . . 14 |- RR e. _V
32	31, 31	xpex 4993	. . . . . . . . . . . . 13 |- ( RR X. RR ) e. _V
33	32	inex2 4348	. . . . . . . . . . . 12 |- ( <_ i^i ( RR X. RR ) ) e. _V
34		nnex 10011	. . . . . . . . . . . 12 |- NN e. _V
35	33, 34	elmap 7045	. . . . . . . . . . 11 |- ( ( F ` ( 1st ` j ) ) e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) <-> ( F ` ( 1st ` j ) ) : NN --> ( <_ i^i ( RR X. RR ) ) )
36	30, 35	sylib 190	. . . . . . . . . 10 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( F ` ( 1st ` j ) ) : NN --> ( <_ i^i ( RR X. RR ) ) )
37		xp2nd 6380	. . . . . . . . . . 11 |- ( j e. ( NN X. NN ) -> ( 2nd ` j ) e. NN )
38	27, 37	syl 16	. . . . . . . . . 10 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( 2nd ` j ) e. NN )
39	36, 38	ffvelrnd 5874	. . . . . . . . 9 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) e. ( <_ i^i ( RR X. RR ) ) )
40	18, 39	sseldi 3348	. . . . . . . 8 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) e. ( RR X. RR ) )
41		xp2nd 6380	. . . . . . . 8 |- ( ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) e. ( RR X. RR ) -> ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) e. RR )
42	40, 41	syl 16	. . . . . . 7 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) e. RR )
43		xp1st 6379	. . . . . . . 8 |- ( ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) e. ( RR X. RR ) -> ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) e. RR )
44	40, 43	syl 16	. . . . . . 7 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) e. RR )
45	42, 44	resubcld 9470	. . . . . 6 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) - ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) ) e. RR )
46	45	recnd 9119	. . . . 5 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) - ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) ) e. CC )
47	7, 8, 15, 17, 46	fsumf1o 12522	. . . 4 |- ( ph -> sum_ j e. ( J " ( 1 ... K ) ) ( ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) - ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) ) = sum_ m e. ( 1 ... K ) ( ( 2nd ` ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) ) - ( 1st ` ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) ) ) )
48	19	adantr 453	. . . . . . . . . . 11 |- ( ( ph /\ k e. NN ) -> F : NN --> ( ( <_ i^i ( RR X. RR ) ) ^m NN ) )
49	23	ffvelrnda 5873	. . . . . . . . . . . 12 |- ( ( ph /\ k e. NN ) -> ( J ` k ) e. ( NN X. NN ) )
50		xp1st 6379	. . . . . . . . . . . 12 |- ( ( J ` k ) e. ( NN X. NN ) -> ( 1st ` ( J ` k ) ) e. NN )
51	49, 50	syl 16	. . . . . . . . . . 11 |- ( ( ph /\ k e. NN ) -> ( 1st ` ( J ` k ) ) e. NN )
52	48, 51	ffvelrnd 5874	. . . . . . . . . 10 |- ( ( ph /\ k e. NN ) -> ( F ` ( 1st ` ( J ` k ) ) ) e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) )
53	33, 34	elmap 7045	. . . . . . . . . 10 |- ( ( F ` ( 1st ` ( J ` k ) ) ) e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) <-> ( F ` ( 1st ` ( J ` k ) ) ) : NN --> ( <_ i^i ( RR X. RR ) ) )
54	52, 53	sylib 190	. . . . . . . . 9 |- ( ( ph /\ k e. NN ) -> ( F ` ( 1st ` ( J ` k ) ) ) : NN --> ( <_ i^i ( RR X. RR ) ) )
55		xp2nd 6380	. . . . . . . . . 10 |- ( ( J ` k ) e. ( NN X. NN ) -> ( 2nd ` ( J ` k ) ) e. NN )
56	49, 55	syl 16	. . . . . . . . 9 |- ( ( ph /\ k e. NN ) -> ( 2nd ` ( J ` k ) ) e. NN )
57	54, 56	ffvelrnd 5874	. . . . . . . 8 |- ( ( ph /\ k e. NN ) -> ( ( F ` ( 1st ` ( J ` k ) ) ) ` ( 2nd ` ( J ` k ) ) ) e. ( <_ i^i ( RR X. RR ) ) )
58		ovoliun.h	. . . . . . . 8 |- H = ( k e. NN |-> ( ( F ` ( 1st ` ( J ` k ) ) ) ` ( 2nd ` ( J ` k ) ) ) )
59	57, 58	fmptd 5896	. . . . . . 7 |- ( ph -> H : NN --> ( <_ i^i ( RR X. RR ) ) )
60		eqid 2438	. . . . . . . 8 |- ( ( abs o. - ) o. H ) = ( ( abs o. - ) o. H )
61	60	ovolfsval 19372	. . . . . . 7 |- ( ( H : NN --> ( <_ i^i ( RR X. RR ) ) /\ m e. NN ) -> ( ( ( abs o. - ) o. H ) ` m ) = ( ( 2nd ` ( H ` m ) ) - ( 1st ` ( H ` m ) ) ) )
62	59, 12, 61	syl2an 465	. . . . . 6 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( ( ( abs o. - ) o. H ) ` m ) = ( ( 2nd ` ( H ` m ) ) - ( 1st ` ( H ` m ) ) ) )
63	12	adantl 454	. . . . . . . . 9 |- ( ( ph /\ m e. ( 1 ... K ) ) -> m e. NN )
64		fveq2 5731	. . . . . . . . . . . . 13 |- ( k = m -> ( J ` k ) = ( J ` m ) )
65	64	fveq2d 5735	. . . . . . . . . . . 12 |- ( k = m -> ( 1st ` ( J ` k ) ) = ( 1st ` ( J ` m ) ) )
66	65	fveq2d 5735	. . . . . . . . . . 11 |- ( k = m -> ( F ` ( 1st ` ( J ` k ) ) ) = ( F ` ( 1st ` ( J ` m ) ) ) )
67	64	fveq2d 5735	. . . . . . . . . . 11 |- ( k = m -> ( 2nd ` ( J ` k ) ) = ( 2nd ` ( J ` m ) ) )
68	66, 67	fveq12d 5737	. . . . . . . . . 10 |- ( k = m -> ( ( F ` ( 1st ` ( J ` k ) ) ) ` ( 2nd ` ( J ` k ) ) ) = ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) )
69		fvex 5745	. . . . . . . . . 10 |- ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) e. _V
70	68, 58, 69	fvmpt 5809	. . . . . . . . 9 |- ( m e. NN -> ( H ` m ) = ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) )
71	63, 70	syl 16	. . . . . . . 8 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( H ` m ) = ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) )
72	71	fveq2d 5735	. . . . . . 7 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( 2nd ` ( H ` m ) ) = ( 2nd ` ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) ) )
73	71	fveq2d 5735	. . . . . . 7 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( 1st ` ( H ` m ) ) = ( 1st ` ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) ) )
74	72, 73	oveq12d 6102	. . . . . 6 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( ( 2nd ` ( H ` m ) ) - ( 1st ` ( H ` m ) ) ) = ( ( 2nd ` ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) ) - ( 1st ` ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) ) ) )
75	62, 74	eqtrd 2470	. . . . 5 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( ( ( abs o. - ) o. H ) ` m ) = ( ( 2nd ` ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) ) - ( 1st ` ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) ) ) )
76		ovoliun.k	. . . . . 6 |- ( ph -> K e. NN )
77		nnuz 10526	. . . . . 6 |- NN = ( ZZ>= ` 1 )
78	76, 77	syl6eleq 2528	. . . . 5 |- ( ph -> K e. ( ZZ>= ` 1 ) )
79		ffvelrn 5871	. . . . . . . . . . 11 |- ( ( H : NN --> ( <_ i^i ( RR X. RR ) ) /\ m e. NN ) -> ( H ` m ) e. ( <_ i^i ( RR X. RR ) ) )
80	59, 12, 79	syl2an 465	. . . . . . . . . 10 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( H ` m ) e. ( <_ i^i ( RR X. RR ) ) )
81	18, 80	sseldi 3348	. . . . . . . . 9 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( H ` m ) e. ( RR X. RR ) )
82		xp2nd 6380	. . . . . . . . 9 |- ( ( H ` m ) e. ( RR X. RR ) -> ( 2nd ` ( H ` m ) ) e. RR )
83	81, 82	syl 16	. . . . . . . 8 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( 2nd ` ( H ` m ) ) e. RR )
84		xp1st 6379	. . . . . . . . 9 |- ( ( H ` m ) e. ( RR X. RR ) -> ( 1st ` ( H ` m ) ) e. RR )
85	81, 84	syl 16	. . . . . . . 8 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( 1st ` ( H ` m ) ) e. RR )
86	83, 85	resubcld 9470	. . . . . . 7 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( ( 2nd ` ( H ` m ) ) - ( 1st ` ( H ` m ) ) ) e. RR )
87	86	recnd 9119	. . . . . 6 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( ( 2nd ` ( H ` m ) ) - ( 1st ` ( H ` m ) ) ) e. CC )
88	74, 87	eqeltrrd 2513	. . . . 5 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( ( 2nd ` ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) ) - ( 1st ` ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) ) ) e. CC )
89	75, 78, 88	fsumser 12529	. . . 4 |- ( ph -> sum_ m e. ( 1 ... K ) ( ( 2nd ` ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) ) - ( 1st ` ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) ) ) = ( seq 1 ( + , ( ( abs o. - ) o. H ) ) ` K ) )
90	47, 89	eqtrd 2470	. . 3 |- ( ph -> sum_ j e. ( J " ( 1 ... K ) ) ( ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) - ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) ) = ( seq 1 ( + , ( ( abs o. - ) o. H ) ) ` K ) )
91		ovoliun.u	. . . 4 |- U = seq 1 ( + , ( ( abs o. - ) o. H ) )
92	91	fveq1i 5732	. . 3 |- ( U ` K ) = ( seq 1 ( + , ( ( abs o. - ) o. H ) ) ` K )
93	90, 92	syl6eqr 2488	. 2 |- ( ph -> sum_ j e. ( J " ( 1 ... K ) ) ( ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) - ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) ) = ( U ` K ) )
94		f1oeng 7129	. . . . . . 7 |- ( ( ( 1 ... K ) e. Fin /\ ( J |` ( 1 ... K ) ) : ( 1 ... K ) -1-1-onto-> ( J " ( 1 ... K ) ) ) -> ( 1 ... K ) ~~ ( J " ( 1 ... K ) ) )
95	8, 15, 94	syl2anc 644	. . . . . 6 |- ( ph -> ( 1 ... K ) ~~ ( J " ( 1 ... K ) ) )
96	95	ensymd 7161	. . . . 5 |- ( ph -> ( J " ( 1 ... K ) ) ~~ ( 1 ... K ) )
97		enfii 7329	. . . . 5 |- ( ( ( 1 ... K ) e. Fin /\ ( J " ( 1 ... K ) ) ~~ ( 1 ... K ) ) -> ( J " ( 1 ... K ) ) e. Fin )
98	8, 96, 97	syl2anc 644	. . . 4 |- ( ph -> ( J " ( 1 ... K ) ) e. Fin )
99	98, 45	fsumrecl 12533	. . 3 |- ( ph -> sum_ j e. ( J " ( 1 ... K ) ) ( ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) - ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) ) e. RR )
100		fzfid 11317	. . . . 5 |- ( ph -> ( 1 ... L ) e. Fin )
101		elfznn 11085	. . . . . 6 |- ( n e. ( 1 ... L ) -> n e. NN )
102		ovoliun.v	. . . . . 6 |- ( ( ph /\ n e. NN ) -> ( vol * ` A ) e. RR )
103	101, 102	sylan2 462	. . . . 5 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( vol * ` A ) e. RR )
104	100, 103	fsumrecl 12533	. . . 4 |- ( ph -> sum_ n e. ( 1 ... L ) ( vol * ` A ) e. RR )
105		ovoliun.b	. . . . . . 7 |- ( ph -> B e. RR+ )
106	105	rpred 10653	. . . . . 6 |- ( ph -> B e. RR )
107		2nn 10138	. . . . . . . 8 |- 2 e. NN
108		nnnn0 10233	. . . . . . . 8 |- ( n e. NN -> n e. NN0 )
109		nnexpcl 11399	. . . . . . . 8 |- ( ( 2 e. NN /\ n e. NN0 ) -> ( 2 ^ n ) e. NN )
110	107, 108, 109	sylancr 646	. . . . . . 7 |- ( n e. NN -> ( 2 ^ n ) e. NN )
111	101, 110	syl 16	. . . . . 6 |- ( n e. ( 1 ... L ) -> ( 2 ^ n ) e. NN )
112		nndivre 10040	. . . . . 6 |- ( ( B e. RR /\ ( 2 ^ n ) e. NN ) -> ( B / ( 2 ^ n ) ) e. RR )
113	106, 111, 112	syl2an 465	. . . . 5 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( B / ( 2 ^ n ) ) e. RR )
114	100, 113	fsumrecl 12533	. . . 4 |- ( ph -> sum_ n e. ( 1 ... L ) ( B / ( 2 ^ n ) ) e. RR )
115	104, 114	readdcld 9120	. . 3 |- ( ph -> ( sum_ n e. ( 1 ... L ) ( vol * ` A ) + sum_ n e. ( 1 ... L ) ( B / ( 2 ^ n ) ) ) e. RR )
116		ovoliun.r	. . . 4 |- ( ph -> sup ( ran T , RR* , < ) e. RR )
117	116, 106	readdcld 9120	. . 3 |- ( ph -> ( sup ( ran T , RR* , < ) + B ) e. RR )
118		relxp 4986	. . . . . . . . . . . . . . 15 |- Rel ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) )
119		relres 5177	. . . . . . . . . . . . . . 15 |- Rel ( ( J " ( 1 ... K ) ) |` { n } )
120		opelxp 4911	. . . . . . . . . . . . . . . 16 |- ( <. x , y >. e. ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) <-> ( x e. { n } /\ y e. ( ( J " ( 1 ... K ) ) " { n } ) ) )
121		vex 2961	. . . . . . . . . . . . . . . . . 18 |- y e. _V
122	121	opelres 5154	. . . . . . . . . . . . . . . . 17 |- ( <. x , y >. e. ( ( J " ( 1 ... K ) ) |` { n } ) <-> ( <. x , y >. e. ( J " ( 1 ... K ) ) /\ x e. { n } ) )
123		ancom 439	. . . . . . . . . . . . . . . . 17 |- ( ( x e. { n } /\ <. x , y >. e. ( J " ( 1 ... K ) ) ) <-> ( <. x , y >. e. ( J " ( 1 ... K ) ) /\ x e. { n } ) )
124		elsni 3840	. . . . . . . . . . . . . . . . . . . . 21 |- ( x e. { n } -> x = n )
125	124	opeq1d 3992	. . . . . . . . . . . . . . . . . . . 20 |- ( x e. { n } -> <. x , y >. = <. n , y >. )
126	125	eleq1d 2504	. . . . . . . . . . . . . . . . . . 19 |- ( x e. { n } -> ( <. x , y >. e. ( J " ( 1 ... K ) ) <-> <. n , y >. e. ( J " ( 1 ... K ) ) ) )
127		vex 2961	. . . . . . . . . . . . . . . . . . . 20 |- n e. _V
128	127, 121	elimasn 5232	. . . . . . . . . . . . . . . . . . 19 |- ( y e. ( ( J " ( 1 ... K ) ) " { n } ) <-> <. n , y >. e. ( J " ( 1 ... K ) ) )
129	126, 128	syl6bbr 256	. . . . . . . . . . . . . . . . . 18 |- ( x e. { n } -> ( <. x , y >. e. ( J " ( 1 ... K ) ) <-> y e. ( ( J " ( 1 ... K ) ) " { n } ) ) )
130	129	pm5.32i 620	. . . . . . . . . . . . . . . . 17 |- ( ( x e. { n } /\ <. x , y >. e. ( J " ( 1 ... K ) ) ) <-> ( x e. { n } /\ y e. ( ( J " ( 1 ... K ) ) " { n } ) ) )
131	122, 123, 130	3bitr2ri 267	. . . . . . . . . . . . . . . 16 |- ( ( x e. { n } /\ y e. ( ( J " ( 1 ... K ) ) " { n } ) ) <-> <. x , y >. e. ( ( J " ( 1 ... K ) ) |` { n } ) )
132	120, 131	bitri 242	. . . . . . . . . . . . . . 15 |- ( <. x , y >. e. ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) <-> <. x , y >. e. ( ( J " ( 1 ... K ) ) |` { n } ) )
133	118, 119, 132	eqrelriiv 4973	. . . . . . . . . . . . . 14 |- ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) = ( ( J " ( 1 ... K ) ) |` { n } )
134		df-res 4893	. . . . . . . . . . . . . 14 |- ( ( J " ( 1 ... K ) ) |` { n } ) = ( ( J " ( 1 ... K ) ) i^i ( { n } X. _V ) )
135	133, 134	eqtri 2458	. . . . . . . . . . . . 13 |- ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) = ( ( J " ( 1 ... K ) ) i^i ( { n } X. _V ) )
136	135	a1i 11	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) = ( ( J " ( 1 ... K ) ) i^i ( { n } X. _V ) ) )
137	136	iuneq2dv 4116	. . . . . . . . . . 11 |- ( ph -> U_ n e. ( 1 ... L ) ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) = U_ n e. ( 1 ... L ) ( ( J " ( 1 ... K ) ) i^i ( { n } X. _V ) ) )
138		iunin2 4157	. . . . . . . . . . 11 |- U_ n e. ( 1 ... L ) ( ( J " ( 1 ... K ) ) i^i ( { n } X. _V ) ) = ( ( J " ( 1 ... K ) ) i^i U_ n e. ( 1 ... L ) ( { n } X. _V ) )
139	137, 138	syl6eq 2486	. . . . . . . . . 10 |- ( ph -> U_ n e. ( 1 ... L ) ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) = ( ( J " ( 1 ... K ) ) i^i U_ n e. ( 1 ... L ) ( { n } X. _V ) ) )
140		relxp 4986	. . . . . . . . . . . . . 14 |- Rel ( NN X. NN )
141		relss 4966	. . . . . . . . . . . . . 14 |- ( ( J " ( 1 ... K ) ) C_ ( NN X. NN ) -> ( Rel ( NN X. NN ) -> Rel ( J " ( 1 ... K ) ) ) )
142	26, 140, 141	ee10 1386	. . . . . . . . . . . . 13 |- ( ph -> Rel ( J " ( 1 ... K ) ) )
143		ovoliun.l2	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> A. w e. ( 1 ... K ) ( 1st ` ( J ` w ) ) <_ L )
144		ffn 5594	. . . . . . . . . . . . . . . . . . . . 21 |- ( J : NN --> ( NN X. NN ) -> J Fn NN )
145	23, 144	syl 16	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> J Fn NN )
146		fveq2 5731	. . . . . . . . . . . . . . . . . . . . . 22 |- ( j = ( J ` w ) -> ( 1st ` j ) = ( 1st ` ( J ` w ) ) )
147	146	breq1d 4225	. . . . . . . . . . . . . . . . . . . . 21 |- ( j = ( J ` w ) -> ( ( 1st ` j ) <_ L <-> ( 1st ` ( J ` w ) ) <_ L ) )
148	147	ralima 5981	. . . . . . . . . . . . . . . . . . . 20 |- ( ( J Fn NN /\ ( 1 ... K ) C_ NN ) -> ( A. j e. ( J " ( 1 ... K ) ) ( 1st ` j ) <_ L <-> A. w e. ( 1 ... K ) ( 1st ` ( J ` w ) ) <_ L ) )
149	145, 13, 148	sylancl 645	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( A. j e. ( J " ( 1 ... K ) ) ( 1st ` j ) <_ L <-> A. w e. ( 1 ... K ) ( 1st ` ( J ` w ) ) <_ L ) )
150	143, 149	mpbird 225	. . . . . . . . . . . . . . . . . 18 |- ( ph -> A. j e. ( J " ( 1 ... K ) ) ( 1st ` j ) <_ L )
151	150	r19.21bi 2806	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( 1st ` j ) <_ L )
152	29, 77	syl6eleq 2528	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( 1st ` j ) e. ( ZZ>= ` 1 ) )
153		ovoliun.l1	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> L e. ZZ )
154	153	adantr 453	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> L e. ZZ )
155		elfz5 11056	. . . . . . . . . . . . . . . . . 18 |- ( ( ( 1st ` j ) e. ( ZZ>= ` 1 ) /\ L e. ZZ ) -> ( ( 1st ` j ) e. ( 1 ... L ) <-> ( 1st ` j ) <_ L ) )
156	152, 154, 155	syl2anc 644	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( ( 1st ` j ) e. ( 1 ... L ) <-> ( 1st ` j ) <_ L ) )
157	151, 156	mpbird 225	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( 1st ` j ) e. ( 1 ... L ) )
158	157	ralrimiva 2791	. . . . . . . . . . . . . . 15 |- ( ph -> A. j e. ( J " ( 1 ... K ) ) ( 1st ` j ) e. ( 1 ... L ) )
159		vex 2961	. . . . . . . . . . . . . . . . . 18 |- x e. _V
160	159, 121	op1std 6360	. . . . . . . . . . . . . . . . 17 |- ( j = <. x , y >. -> ( 1st ` j ) = x )
161	160	eleq1d 2504	. . . . . . . . . . . . . . . 16 |- ( j = <. x , y >. -> ( ( 1st ` j ) e. ( 1 ... L ) <-> x e. ( 1 ... L ) ) )
162	161	rspccv 3051	. . . . . . . . . . . . . . 15 |- ( A. j e. ( J " ( 1 ... K ) ) ( 1st ` j ) e. ( 1 ... L ) -> ( <. x , y >. e. ( J " ( 1 ... K ) ) -> x e. ( 1 ... L ) ) )
163	158, 162	syl 16	. . . . . . . . . . . . . 14 |- ( ph -> ( <. x , y >. e. ( J " ( 1 ... K ) ) -> x e. ( 1 ... L ) ) )
164		opelxp 4911	. . . . . . . . . . . . . . 15 |- ( <. x , y >. e. ( U_ n e. ( 1 ... L ) { n } X. _V ) <-> ( x e. U_ n e. ( 1 ... L ) { n } /\ y e. _V ) )
165	121	biantru 493	. . . . . . . . . . . . . . 15 |- ( x e. U_ n e. ( 1 ... L ) { n } <-> ( x e. U_ n e. ( 1 ... L ) { n } /\ y e. _V ) )
166		iunid 4148	. . . . . . . . . . . . . . . 16 |- U_ n e. ( 1 ... L ) { n } = ( 1 ... L )
167	166	eleq2i 2502	. . . . . . . . . . . . . . 15 |- ( x e. U_ n e. ( 1 ... L ) { n } <-> x e. ( 1 ... L ) )
168	164, 165, 167	3bitr2i 266	. . . . . . . . . . . . . 14 |- ( <. x , y >. e. ( U_ n e. ( 1 ... L ) { n } X. _V ) <-> x e. ( 1 ... L ) )
169	163, 168	syl6ibr 220	. . . . . . . . . . . . 13 |- ( ph -> ( <. x , y >. e. ( J " ( 1 ... K ) ) -> <. x , y >. e. ( U_ n e. ( 1 ... L ) { n } X. _V ) ) )
170	142, 169	relssdv 4971	. . . . . . . . . . . 12 |- ( ph -> ( J " ( 1 ... K ) ) C_ ( U_ n e. ( 1 ... L ) { n } X. _V ) )
171		xpiundir 4936	. . . . . . . . . . . 12 |- ( U_ n e. ( 1 ... L ) { n } X. _V ) = U_ n e. ( 1 ... L ) ( { n } X. _V )
172	170, 171	syl6sseq 3396	. . . . . . . . . . 11 |- ( ph -> ( J " ( 1 ... K ) ) C_ U_ n e. ( 1 ... L ) ( { n } X. _V ) )
173		df-ss 3336	. . . . . . . . . . 11 |- ( ( J " ( 1 ... K ) ) C_ U_ n e. ( 1 ... L ) ( { n } X. _V ) <-> ( ( J " ( 1 ... K ) ) i^i U_ n e. ( 1 ... L ) ( { n } X. _V ) ) = ( J " ( 1 ... K ) ) )
174	172, 173	sylib 190	. . . . . . . . . 10 |- ( ph -> ( ( J " ( 1 ... K ) ) i^i U_ n e. ( 1 ... L ) ( { n } X. _V ) ) = ( J " ( 1 ... K ) ) )
175	139, 174	eqtrd 2470	. . . . . . . . 9 |- ( ph -> U_ n e. ( 1 ... L ) ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) = ( J " ( 1 ... K ) ) )
176	175, 98	eqeltrd 2512	. . . . . . . 8 |- ( ph -> U_ n e. ( 1 ... L ) ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) e. Fin )
177		ssiun2 4136	. . . . . . . 8 |- ( n e. ( 1 ... L ) -> ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) C_ U_ n e. ( 1 ... L ) ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) )
178		ssfi 7332	. . . . . . . 8 |- ( ( U_ n e. ( 1 ... L ) ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) e. Fin /\ ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) C_ U_ n e. ( 1 ... L ) ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) ) -> ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) e. Fin )
179	176, 177, 178	syl2an 465	. . . . . . 7 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) e. Fin )
180		2ndconst 6439	. . . . . . . . . 10 |- ( n e. _V -> ( 2nd |` ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) ) : ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) -1-1-onto-> ( ( J " ( 1 ... K ) ) " { n } ) )
181	127, 180	ax-mp 5	. . . . . . . . 9 |- ( 2nd |` ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) ) : ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) -1-1-onto-> ( ( J " ( 1 ... K ) ) " { n } )
182		f1oeng 7129	. . . . . . . . 9 |- ( ( ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) e. Fin /\ ( 2nd |` ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) ) : ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) -1-1-onto-> ( ( J " ( 1 ... K ) ) " { n } ) ) -> ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) ~~ ( ( J " ( 1 ... K ) ) " { n } ) )
183	179, 181, 182	sylancl 645	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) ~~ ( ( J " ( 1 ... K ) ) " { n } ) )
184	183	ensymd 7161	. . . . . . 7 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( ( J " ( 1 ... K ) ) " { n } ) ~~ ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) )
185		enfii 7329	. . . . . . 7 |- ( ( ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) e. Fin /\ ( ( J " ( 1 ... K ) ) " { n } ) ~~ ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) ) -> ( ( J " ( 1 ... K ) ) " { n } ) e. Fin )
186	179, 184, 185	syl2anc 644	. . . . . 6 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( ( J " ( 1 ... K ) ) " { n } ) e. Fin )
187		ffvelrn 5871	. . . . . . . . . . . . . 14 |- ( ( F : NN --> ( ( <_ i^i ( RR X. RR ) ) ^m NN ) /\ n e. NN ) -> ( F ` n ) e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) )
188	19, 101, 187	syl2an 465	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( F ` n ) e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) )
189	33, 34	elmap 7045	. . . . . . . . . . . . 13 |- ( ( F ` n ) e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) <-> ( F ` n ) : NN --> ( <_ i^i ( RR X. RR ) ) )
190	188, 189	sylib 190	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( F ` n ) : NN --> ( <_ i^i ( RR X. RR ) ) )
191	190	adantrr 699	. . . . . . . . . . 11 |- ( ( ph /\ ( n e. ( 1 ... L ) /\ i e. ( ( J " ( 1 ... K ) ) " { n } ) ) ) -> ( F ` n ) : NN --> ( <_ i^i ( RR X. RR ) ) )
192		imassrn 5219	. . . . . . . . . . . . . 14 |- ( ( J " ( 1 ... K ) ) " { n } ) C_ ran ( J " ( 1 ... K ) )
193	26	adantr 453	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( J " ( 1 ... K ) ) C_ ( NN X. NN ) )
194		rnss 5101	. . . . . . . . . . . . . . . 16 |- ( ( J " ( 1 ... K ) ) C_ ( NN X. NN ) -> ran ( J " ( 1 ... K ) ) C_ ran ( NN X. NN ) )
195	193, 194	syl 16	. . . . . . . . . . . . . . 15 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ran ( J " ( 1 ... K ) ) C_ ran ( NN X. NN ) )
196		rnxpid 5305	. . . . . . . . . . . . . . 15 |- ran ( NN X. NN ) = NN
197	195, 196	syl6sseq 3396	. . . . . . . . . . . . . 14 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ran ( J " ( 1 ... K ) ) C_ NN )
198	192, 197	syl5ss 3361	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( ( J " ( 1 ... K ) ) " { n } ) C_ NN )
199	198	sseld 3349	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( i e. ( ( J " ( 1 ... K ) ) " { n } ) -> i e. NN ) )
200	199	impr 604	. . . . . . . . . . 11 |- ( ( ph /\ ( n e. ( 1 ... L ) /\ i e. ( ( J " ( 1 ... K ) ) " { n } ) ) ) -> i e. NN )
201	191, 200	ffvelrnd 5874	. . . . . . . . . 10 |- ( ( ph /\ ( n e. ( 1 ... L ) /\ i e. ( ( J " ( 1 ... K ) ) " { n } ) ) ) -> ( ( F ` n ) ` i ) e. ( <_ i^i ( RR X. RR ) ) )
202	18, 201	sseldi 3348	. . . . . . . . 9 |- ( ( ph /\ ( n e. ( 1 ... L ) /\ i e. ( ( J " ( 1 ... K ) ) " { n } ) ) ) -> ( ( F ` n ) ` i ) e. ( RR X. RR ) )
203		xp2nd 6380	. . . . . . . . 9 |- ( ( ( F ` n ) ` i ) e. ( RR X. RR ) -> ( 2nd ` ( ( F ` n ) ` i ) ) e. RR )
204	202, 203	syl 16	. . . . . . . 8 |- ( ( ph /\ ( n e. ( 1 ... L ) /\ i e. ( ( J " ( 1 ... K ) ) " { n } ) ) ) -> ( 2nd ` ( ( F ` n ) ` i ) ) e. RR )
205		xp1st 6379	. . . . . . . . 9 |- ( ( ( F ` n ) ` i ) e. ( RR X. RR ) -> ( 1st ` ( ( F ` n ) ` i ) ) e. RR )
206	202, 205	syl 16	. . . . . . . 8 |- ( ( ph /\ ( n e. ( 1 ... L ) /\ i e. ( ( J " ( 1 ... K ) ) " { n } ) ) ) -> ( 1st ` ( ( F ` n ) ` i ) ) e. RR )
207	204, 206	resubcld 9470	. . . . . . 7 |- ( ( ph /\ ( n e. ( 1 ... L ) /\ i e. ( ( J " ( 1 ... K ) ) " { n } ) ) ) -> ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) e. RR )
208	207	anassrs 631	. . . . . 6 |- ( ( ( ph /\ n e. ( 1 ... L ) ) /\ i e. ( ( J " ( 1 ... K ) ) " { n } ) ) -> ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) e. RR )
209	186, 208	fsumrecl 12533	. . . . 5 |- ( ( ph /\ n e. ( 1 ... L ) ) -> sum_ i e. ( ( J " ( 1 ... K ) ) " { n } ) ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) e. RR )
210	106, 110, 112	syl2an 465	. . . . . . 7 |- ( ( ph /\ n e. NN ) -> ( B / ( 2 ^ n ) ) e. RR )
211	102, 210	readdcld 9120	. . . . . 6 |- ( ( ph /\ n e. NN ) -> ( ( vol * ` A ) + ( B / ( 2 ^ n ) ) ) e. RR )
212	101, 211	sylan2 462	. . . . 5 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( ( vol * ` A ) + ( B / ( 2 ^ n ) ) ) e. RR )
213		eqid 2438	. . . . . . . . . . . 12 |- ( ( abs o. - ) o. ( F ` n ) ) = ( ( abs o. - ) o. ( F ` n ) )
214		ovoliun.s	. . . . . . . . . . . 12 |- S = seq 1 ( + , ( ( abs o. - ) o. ( F ` n ) ) )
215	213, 214	ovolsf 19374	. . . . . . . . . . 11 |- ( ( F ` n ) : NN --> ( <_ i^i ( RR X. RR ) ) -> S : NN --> ( 0 [,) +oo ) )
216	190, 215	syl 16	. . . . . . . . . 10 |- ( ( ph /\ n e. ( 1 ... L ) ) -> S : NN --> ( 0 [,) +oo ) )
217		frn 5600	. . . . . . . . . 10 |- ( S : NN --> ( 0 [,) +oo ) -> ran S C_ ( 0 [,) +oo ) )
218	216, 217	syl 16	. . . . . . . . 9 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ran S C_ ( 0 [,) +oo ) )
219		icossxr 11000	. . . . . . . . 9 |- ( 0 [,) +oo ) C_ RR*
220	218, 219	syl6ss 3362	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ran S C_ RR* )
221		supxrcl 10898	. . . . . . . 8 |- ( ran S C_ RR* -> sup ( ran S , RR* , < ) e. RR* )
222	220, 221	syl 16	. . . . . . 7 |- ( ( ph /\ n e. ( 1 ... L ) ) -> sup ( ran S , RR* , < ) e. RR* )
223		mnfxr 10719	. . . . . . . . 9 |- -oo e. RR*
224	223	a1i 11	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> -oo e. RR* )
225	103	rexrd 9139	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( vol * ` A ) e. RR* )
226		mnflt 10727	. . . . . . . . 9 |- ( ( vol * ` A ) e. RR -> -oo < ( vol * ` A ) )
227	103, 226	syl 16	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> -oo < ( vol * ` A ) )
228		ovoliun.x1	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> A C_ U. ran ( (,) o. ( F ` n ) ) )
229	101, 228	sylan2 462	. . . . . . . . 9 |- ( ( ph /\ n e. ( 1 ... L ) ) -> A C_ U. ran ( (,) o. ( F ` n ) ) )
230	214	ovollb 19380	. . . . . . . . 9 |- ( ( ( F ` n ) : NN --> ( <_ i^i ( RR X. RR ) ) /\ A C_ U. ran ( (,) o. ( F ` n ) ) ) -> ( vol * ` A ) <_ sup ( ran S , RR* , < ) )
231	190, 229, 230	syl2anc 644	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( vol * ` A ) <_ sup ( ran S , RR* , < ) )
232	224, 225, 222, 227, 231	xrltletrd 10756	. . . . . . 7 |- ( ( ph /\ n e. ( 1 ... L ) ) -> -oo < sup ( ran S , RR* , < ) )
233		ovoliun.x2	. . . . . . . 8 |- ( ( ph /\ n e. NN ) -> sup ( ran S , RR* , < ) <_ ( ( vol * ` A ) + ( B / ( 2 ^ n ) ) ) )
234	101, 233	sylan2 462	. . . . . . 7 |- ( ( ph /\ n e. ( 1 ... L ) ) -> sup ( ran S , RR* , < ) <_ ( ( vol * ` A ) + ( B / ( 2 ^ n ) ) ) )
235		xrre 10762	. . . . . . 7 |- ( ( ( sup ( ran S , RR* , < ) e. RR* /\ ( ( vol * ` A ) + ( B / ( 2 ^ n ) ) ) e. RR ) /\ ( -oo < sup ( ran S , RR* , < ) /\ sup ( ran S , RR* , < ) <_ ( ( vol * ` A ) + ( B / ( 2 ^ n ) ) ) ) ) -> sup ( ran S , RR* , < ) e. RR )
236	222, 212, 232, 234, 235	syl22anc 1186	. . . . . 6 |- ( ( ph /\ n e. ( 1 ... L ) ) -> sup ( ran S , RR* , < ) e. RR )
237		1z 10316	. . . . . . . . 9 |- 1 e. ZZ
238	237	a1i 11	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> 1 e. ZZ )
239	213	ovolfsval 19372	. . . . . . . . 9 |- ( ( ( F ` n ) : NN --> ( <_ i^i ( RR X. RR ) ) /\ i e. NN ) -> ( ( ( abs o. - ) o. ( F ` n ) ) ` i ) = ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) )
240	190, 239	sylan 459	. . . . . . . 8 |- ( ( ( ph /\ n e. ( 1 ... L ) ) /\ i e. NN ) -> ( ( ( abs o. - ) o. ( F ` n ) ) ` i ) = ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) )
241	213	ovolfsf 19373	. . . . . . . . . . . . 13 |- ( ( F ` n ) : NN --> ( <_ i^i ( RR X. RR ) ) -> ( ( abs o. - ) o. ( F ` n ) ) : NN --> ( 0 [,) +oo ) )
242	190, 241	syl 16	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( ( abs o. - ) o. ( F ` n ) ) : NN --> ( 0 [,) +oo ) )
243	242	ffvelrnda 5873	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. ( 1 ... L ) ) /\ i e. NN ) -> ( ( ( abs o. - ) o. ( F ` n ) ) ` i ) e. ( 0 [,) +oo ) )
244	240, 243	eqeltrrd 2513	. . . . . . . . . 10 |- ( ( ( ph /\ n e. ( 1 ... L ) ) /\ i e. NN ) -> ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) e. ( 0 [,) +oo ) )
245		elrege0 11012	. . . . . . . . . 10 |- ( ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) e. ( 0 [,) +oo ) <-> ( ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) e. RR /\ 0 <_ ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) ) )
246	244, 245	sylib 190	. . . . . . . . 9 |- ( ( ( ph /\ n e. ( 1 ... L ) ) /\ i e. NN ) -> ( ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) e. RR /\ 0 <_ ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) ) )
247	246	simpld 447	. . . . . . . 8 |- ( ( ( ph /\ n e. ( 1 ... L ) ) /\ i e. NN ) -> ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) e. RR )
248	246	simprd 451	. . . . . . . 8 |- ( ( ( ph /\ n e. ( 1 ... L ) ) /\ i e. NN ) -> 0 <_ ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) )
249		supxrub 10908	. . . . . . . . . . . . . . 15 |- ( ( ran S C_ RR* /\ z e. ran S ) -> z <_ sup ( ran S , RR* , < ) )
250	220, 249	sylan 459	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ n e. ( 1 ... L ) ) /\ z e. ran S ) -> z <_ sup ( ran S , RR* , < ) )
251	250	ralrimiva 2791	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. ( 1 ... L ) ) -> A. z e. ran S z <_ sup ( ran S , RR* , < ) )
252		breq2 4219	. . . . . . . . . . . . . . 15 |- ( x = sup ( ran S , RR* , < ) -> ( z <_ x <-> z <_ sup ( ran S , RR* , < ) ) )
253	252	ralbidv 2727	. . . . . . . . . . . . . 14 |- ( x = sup ( ran S , RR* , < ) -> ( A. z e. ran S z <_ x <-> A. z e. ran S z <_ sup ( ran S , RR* , < ) ) )
254	253	rspcev 3054	. . . . . . . . . . . . 13 |- ( ( sup ( ran S , RR* , < ) e. RR /\ A. z e. ran S z <_ sup ( ran S , RR* , < ) ) -> E. x e. RR A. z e. ran S z <_ x )
255	236, 251, 254	syl2anc 644	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( 1 ... L ) ) -> E. x e. RR A. z e. ran S z <_ x )
256		ffn 5594	. . . . . . . . . . . . . . 15 |- ( S : NN --> ( 0 [,) +oo ) -> S Fn NN )
257	216, 256	syl 16	. . . . . . . . . . . . . 14 |- ( ( ph /\ n e. ( 1 ... L ) ) -> S Fn NN )
258		breq1 4218	. . . . . . . . . . . . . . 15 |- ( z = ( S ` k ) -> ( z <_ x <-> ( S ` k ) <_ x ) )
259	258	ralrn 5876	. . . . . . . . . . . . . 14 |- ( S Fn NN -> ( A. z e. ran S z <_ x <-> A. k e. NN ( S ` k ) <_ x ) )
260	257, 259	syl 16	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( A. z e. ran S z <_ x <-> A. k e. NN ( S ` k ) <_ x ) )
261	260	rexbidv 2728	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( E. x e. RR A. z e. ran S z <_ x <-> E. x e. RR A. k e. NN ( S ` k ) <_ x ) )
262	255, 261	mpbid 203	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( 1 ... L ) ) -> E. x e. RR A. k e. NN ( S ` k ) <_ x )
263	77, 214, 238, 240, 247, 248, 262	isumsup2 12631	. . . . . . . . . 10 |- ( ( ph /\ n e. ( 1 ... L ) ) -> S ~~> sup ( ran S , RR , < ) )
264	214, 263	syl5eqbrr 4249	. . . . . . . . 9 |- ( ( ph /\ n e. ( 1 ... L ) ) -> seq 1 ( + , ( ( abs o. - ) o. ( F ` n ) ) ) ~~> sup ( ran S , RR , < ) )
265		climrel 12291	. . . . . . . . . 10 |- Rel ~~>
266	265	releldmi 5109	. . . . . . . . 9 |- ( seq 1 ( + , ( ( abs o. - ) o. ( F ` n ) ) ) ~~> sup ( ran S , RR , < ) -> seq 1 ( + , ( ( abs o. - ) o. ( F ` n ) ) ) e. dom ~~> )
267	264, 266	syl 16	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> seq 1 ( + , ( ( abs o. - ) o. ( F ` n ) ) ) e. dom ~~> )
268	77, 238, 186, 198, 240, 247, 248, 267	isumless 12630	. . . . . . 7 |- ( ( ph /\ n e. ( 1 ... L ) ) -> sum_ i e. ( ( J " ( 1 ... K ) ) " { n } ) ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) <_ sum_ i e. NN ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) )
269	77, 214, 238, 240, 247, 248, 262	isumsup 12632	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> sum_ i e. NN ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) = sup ( ran S , RR , < ) )
270		0re 9096	. . . . . . . . . . 11 |- 0 e. RR
271		pnfxr 10718	. . . . . . . . . . 11 |- +oo e. RR*
272		icossre 10996	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ +oo e. RR* ) -> ( 0 [,) +oo ) C_ RR )
273	270, 271, 272	mp2an 655	. . . . . . . . . 10 |- ( 0 [,) +oo ) C_ RR
274	218, 273	syl6ss 3362	. . . . . . . . 9 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ran S C_ RR )
275		1nn 10016	. . . . . . . . . . . 12 |- 1 e. NN
276		fdm 5598	. . . . . . . . . . . . 13 |- ( S : NN --> ( 0 [,) +oo ) -> dom S = NN )
277	216, 276	syl 16	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( 1 ... L ) ) -> dom S = NN )
278	275, 277	syl5eleqr 2525	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( 1 ... L ) ) -> 1 e. dom S )
279		ne0i 3636	. . . . . . . . . . 11 |- ( 1 e. dom S -> dom S =/= (/) )
280	278, 279	syl 16	. . . . . . . . . 10 |- ( ( ph /\ n e. ( 1 ... L ) ) -> dom S =/= (/) )
281		dm0rn0 5089	. . . . . . . . . . 11 |- ( dom S = (/) <-> ran S = (/) )
282	281	necon3bii 2635	. . . . . . . . . 10 |- ( dom S =/= (/) <-> ran S =/= (/) )
283	280, 282	sylib 190	. . . . . . . . 9 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ran S =/= (/) )
284		supxrre 10911	. . . . . . . . 9 |- ( ( ran S C_ RR /\ ran S =/= (/) /\ E. x e. RR A. z e. ran S z <_ x ) -> sup ( ran S , RR* , < ) = sup ( ran S , RR , < ) )
285	274, 283, 255, 284	syl3anc 1185	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> sup ( ran S , RR* , < ) = sup ( ran S , RR , < ) )
286	269, 285	eqtr4d 2473	. . . . . . 7 |- ( ( ph /\ n e. ( 1 ... L ) ) -> sum_ i e. NN ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) = sup ( ran S , RR* , < ) )
287	268, 286	breqtrd 4239	. . . . . 6 |- ( ( ph /\ n e. ( 1 ... L ) ) -> sum_ i e. ( ( J " ( 1 ... K ) ) " { n } ) ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) <_ sup ( ran S , RR* , < ) )
288	209, 236, 212, 287, 234	letrd 9232	. . . . 5 |- ( ( ph /\ n e. ( 1 ... L ) ) -> sum_ i e. ( ( J " ( 1 ... K ) ) " { n } ) ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) <_ ( ( vol * ` A ) + ( B / ( 2 ^ n ) ) ) )
289	100, 209, 212, 288	fsumle 12583	. . . 4 |- ( ph -> sum_ n e. ( 1 ... L ) sum_ i e. ( ( J " ( 1 ... K ) ) " { n } ) ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) <_ sum_ n e. ( 1 ... L ) ( ( vol * ` A ) + ( B / ( 2 ^ n ) ) ) )
290		vex 2961	. . . . . . . . . . 11 |- i e. _V
291	127, 290	op1std 6360	. . . . . . . . . 10 |- ( j = <. n , i >. -> ( 1st ` j ) = n )
292	291	fveq2d 5735	. . . . . . . . 9 |- ( j = <. n , i >. -> ( F ` ( 1st ` j ) ) = ( F ` n ) )
293	127, 290	op2ndd 6361	. . . . . . . . 9 |- ( j = <. n , i >. -> ( 2nd ` j ) = i )
294	292, 293	fveq12d 5737	. . . . . . . 8 |- ( j = <. n , i >. -> ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) = ( ( F ` n ) ` i ) )
295	294	fveq2d 5735	. . . . . . 7 |- ( j = <. n , i >. -> ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) = ( 2nd ` ( ( F ` n ) ` i ) ) )
296	294	fveq2d 5735	. . . . . . 7 |- ( j = <. n , i >. -> ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) = ( 1st ` ( ( F ` n ) ` i ) ) )
297	295, 296	oveq12d 6102	. . . . . 6 |- ( j = <. n , i >. -> ( ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) - ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) ) = ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) )
298	207	recnd 9119	. . . . . 6 |- ( ( ph /\ ( n e. ( 1 ... L ) /\ i e. ( ( J " ( 1 ... K ) ) " { n } ) ) ) -> ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) e. CC )
299	297, 100, 186, 298	fsum2d 12560	. . . . 5 |- ( ph -> sum_ n e. ( 1 ... L ) sum_ i e. ( ( J " ( 1 ... K ) ) " { n } ) ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) = sum_ j e. U_ n e. ( 1 ... L ) ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) ( ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) - ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) ) )
300	175	sumeq1d 12500	. . . . 5 |- ( ph -> sum_ j e. U_ n e. ( 1 ... L ) ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) ( ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) - ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) ) = sum_ j e. ( J " ( 1 ... K ) ) ( ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) - ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) ) )
301	299, 300	eqtrd 2470	. . . 4 |- ( ph -> sum_ n e. ( 1 ... L ) sum_ i e. ( ( J " ( 1 ... K ) ) " { n } ) ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) = sum_ j e. ( J " ( 1 ... K ) ) ( ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) - ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) ) )
302	103	recnd 9119	. . . . 5 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( vol * ` A ) e. CC )
303	113	recnd 9119	. . . . 5 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( B / ( 2 ^ n ) ) e. CC )
304	100, 302, 303	fsumadd 12537	. . . 4 |- ( ph -> sum_ n e. ( 1 ... L ) ( ( vol * ` A ) + ( B / ( 2 ^ n ) ) ) = ( sum_ n e. ( 1 ... L ) ( vol * ` A ) + sum_ n e. ( 1 ... L ) ( B / ( 2 ^ n ) ) ) )
305	289, 301, 304	3brtr3d 4244	. . 3 |- ( ph -> sum_ j e. ( J " ( 1 ... K ) ) ( ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) - ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) ) <_ ( sum_ n e. ( 1 ... L ) ( vol * ` A ) + sum_ n e. ( 1 ... L ) ( B / ( 2 ^ n ) ) ) )
306	237	a1i 11	. . . . . . . . 9 |- ( ph -> 1 e. ZZ )
307		simpr 449	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> n e. NN )
308		ovoliun.g	. . . . . . . . . . . 12 |- G = ( n e. NN |-> ( vol * ` A ) )
309	308	fvmpt2 5815	. . . . . . . . . . 11 |- ( ( n e. NN /\ ( vol * ` A ) e. RR ) -> ( G ` n ) = ( vol * ` A ) )
310	307, 102, 309	syl2anc 644	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( G ` n ) = ( vol * ` A ) )
311	310, 102	eqeltrd 2512	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( G ` n ) e. RR )
312	77, 306, 311	serfre 11357	. . . . . . . 8 |- ( ph -> seq 1 ( + , G ) : NN --> RR )
313		ovoliun.t	. . . . . . . . 9 |- T = seq 1 ( + , G )
314	313	feq1i 5588	. . . . . . . 8 |- ( T : NN --> RR <-> seq 1 ( + , G ) : NN --> RR )
315	312, 314	sylibr 205	. . . . . . 7 |- ( ph -> T : NN --> RR )
316		frn 5600	. . . . . . 7 |- ( T : NN --> RR -> ran T C_ RR )
317	315, 316	syl 16	. . . . . 6 |- ( ph -> ran T C_ RR )
318		ressxr 9134	. . . . . 6 |- RR C_ RR*
319	317, 318	syl6ss 3362	. . . . 5 |- ( ph -> ran T C_ RR* )
320	101, 310	sylan2 462	. . . . . . 7 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( G ` n ) = ( vol * ` A ) )
321		1re 9095	. . . . . . . . . . 11 |- 1 e. RR
322	321	a1i 11	. . . . . . . . . 10 |- ( ph -> 1 e. RR )
323		ffvelrn 5871	. . . . . . . . . . . . 13 |- ( ( J : NN --> ( NN X. NN ) /\ 1 e. NN ) -> ( J ` 1 ) e. ( NN X. NN ) )
324	23, 275, 323	sylancl 645	. . . . . . . . . . . 12 |- ( ph -> ( J ` 1 ) e. ( NN X. NN ) )
325		xp1st 6379	. . . . . . . . . . . 12 |- ( ( J ` 1 ) e. ( NN X. NN ) -> ( 1st ` ( J ` 1 ) ) e. NN )
326	324, 325	syl 16	. . . . . . . . . . 11 |- ( ph -> ( 1st ` ( J ` 1 ) ) e. NN )
327	326	nnred 10020	. . . . . . . . . 10 |- ( ph -> ( 1st ` ( J ` 1 ) ) e. RR )
328	153	zred 10380	. . . . . . . . . 10 |- ( ph -> L e. RR )
329	326	nnge1d 10047	. . . . . . . . . 10 |- ( ph -> 1 <_ ( 1st ` ( J ` 1 ) ) )
330		eluzfz1 11069	. . . . . . . . . . . 12 |- ( K e. ( ZZ>= ` 1 ) -> 1 e. ( 1 ... K ) )
331	78, 330	syl 16	. . . . . . . . . . 11 |- ( ph -> 1 e. ( 1 ... K ) )
332		fveq2 5731	. . . . . . . . . . . . . 14 |- ( w = 1 -> ( J ` w ) = ( J ` 1 ) )
333	332	fveq2d 5735	. . . . . . . . . . . . 13 |- ( w = 1 -> ( 1st ` ( J ` w ) ) = ( 1st ` ( J ` 1 ) ) )
334	333	breq1d 4225	. . . . . . . . . . . 12 |- ( w = 1 -> ( ( 1st ` ( J ` w ) ) <_ L <-> ( 1st ` ( J ` 1 ) ) <_ L ) )
335	334	rspcv 3050	. . . . . . . . . . 11 |- ( 1 e. ( 1 ... K ) -> ( A. w e. ( 1 ... K ) ( 1st ` ( J ` w ) ) <_ L -> ( 1st ` ( J ` 1 ) ) <_ L ) )
336	331, 143, 335	sylc 59	. . . . . . . . . 10 |- ( ph -> ( 1st ` ( J ` 1 ) ) <_ L )
337	322, 327, 328, 329, 336	letrd 9232	. . . . . . . . 9 |- ( ph -> 1 <_ L )
338		elnnz1 10312	. . . . . . . . 9 |- ( L e. NN <-> ( L e. ZZ /\ 1 <_ L ) )
339	153, 337, 338	sylanbrc 647	. . . . . . . 8 |- ( ph -> L e. NN )
340	339, 77	syl6eleq 2528	. . . . . . 7 |- ( ph -> L e. ( ZZ>= ` 1 ) )
341	320, 340, 302	fsumser 12529	. . . . . 6 |- ( ph -> sum_ n e. ( 1 ... L ) ( vol * ` A ) = ( seq 1 ( + , G ) ` L ) )
342		seqfn 11340	. . . . . . . . 9 |- ( 1 e. ZZ -> seq 1 ( + , G ) Fn ( ZZ>= ` 1 ) )
343	306, 342	syl 16	. . . . . . . 8 |- ( ph -> seq 1 ( + , G ) Fn ( ZZ>= ` 1 ) )
344		fnfvelrn 5870	. . . . . . . 8 |- ( ( seq 1 ( + , G ) Fn ( ZZ>= ` 1 ) /\ L e. ( ZZ>= ` 1 ) ) -> ( seq 1 ( + , G ) ` L ) e. ran seq 1 ( + , G ) )
345	343, 340, 344	syl2anc 644	. . . . . . 7 |- ( ph -> ( seq 1 ( + , G ) ` L ) e. ran seq 1 ( + , G ) )
346	313	rneqi 5099	. . . . . . 7 |- ran T = ran seq 1 ( + , G )
347	345, 346	syl6eleqr 2529	. . . . . 6 |- ( ph -> ( seq 1 ( + , G ) ` L ) e. ran T )
348	341, 347	eqeltrd 2512	. . . . 5 |- ( ph -> sum_ n e. ( 1 ... L ) ( vol * ` A ) e. ran T )
349		supxrub 10908	. . . . 5 |- ( ( ran T C_ RR* /\ sum_ n e. ( 1 ... L ) ( vol * ` A ) e. ran T ) -> sum_ n e. ( 1 ... L ) ( vol * ` A ) <_ sup ( ran T , RR* , < ) )
350	319, 348, 349	syl2anc 644	. . . 4 |- ( ph -> sum_ n e. ( 1 ... L ) ( vol * ` A ) <_ sup ( ran T , RR* , < ) )
351	106	recnd 9119	. . . . . 6 |- ( ph -> B e. CC )
352		geo2sum 12655	. . . . . 6 |- ( ( L e. NN /\ B e. CC ) -> sum_ n e. ( 1 ... L ) ( B / ( 2 ^ n ) ) = ( B - ( B / ( 2 ^ L ) ) ) )
353	339, 351, 352	syl2anc 644	. . . . 5 |- ( ph -> sum_ n e. ( 1 ... L ) ( B / ( 2 ^ n ) ) = ( B - ( B / ( 2 ^ L ) ) ) )
354	339	nnnn0d 10279	. . . . . . . . . 10 |- ( ph -> L e. NN0 )
355		nnexpcl 11399	. . . . . . . . . 10 |- ( ( 2 e. NN /\ L e. NN0 ) -> ( 2 ^ L ) e. NN )
356	107, 354, 355	sylancr 646	. . . . . . . . 9 |- ( ph -> ( 2 ^ L ) e. NN )
357	356	nnrpd 10652	. . . . . . . 8 |- ( ph -> ( 2 ^ L ) e. RR+ )
358	105, 357	rpdivcld 10670	. . . . . . 7 |- ( ph -> ( B / ( 2 ^ L ) ) e. RR+ )
359	358	rpge0d 10657	. . . . . 6 |- ( ph -> 0 <_ ( B / ( 2 ^ L ) ) )
360	106, 356	nndivred 10053	. . . . . . 7 |- ( ph -> ( B / ( 2 ^ L ) ) e. RR )
361	106, 360	subge02d 9623	. . . . . 6 |- ( ph -> ( 0 <_ ( B / ( 2 ^ L ) ) <-> ( B - ( B / ( 2 ^ L ) ) ) <_ B ) )
362	359, 361	mpbid 203	. . . . 5 |- ( ph -> ( B - ( B / ( 2 ^ L ) ) ) <_ B )
363	353, 362	eqbrtrd 4235	. . . 4 |- ( ph -> sum_ n e. ( 1 ... L ) ( B / ( 2 ^ n ) ) <_ B )
364	104, 114, 116, 106, 350, 363	le2addd 9649	. . 3 |- ( ph -> ( sum_ n e. ( 1 ... L ) ( vol * ` A ) + sum_ n e. ( 1 ... L ) ( B / ( 2 ^ n ) ) ) <_ ( sup ( ran T , RR* , < ) + B ) )
365	99, 115, 117, 305, 364	letrd 9232	. 2 |- ( ph -> sum_ j e. ( J " ( 1 ... K ) ) ( ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) - ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) ) <_ ( sup ( ran T , RR* , < ) + B ) )
366	93, 365	eqbrtrrd 4237	1 |- ( ph -> ( U ` K ) <_ ( sup ( ran T , RR* , < ) + B ) )

Syntax hints:   -> wi 4   <-> wb 178   /\ wa 360   = wceq 1653   e. wcel 1726   =/= wne 2601   A.wral 2707   E.wrex 2708   _Vcvv 2958   i^i cin 3321   C_ wss 3322   (/)c0 3630   {csn 3816   <.cop 3819   U.cuni 4017   U_ciun 4095   class class class wbr 4215   e. cmpt 4269   X. cxp 4879   dom cdm 4881   ran crn 4882   |` cres 4883   "cima 4884   o. ccom 4885   Rel wrel 4886   Fn wfn 5452   -->wf 5453   -1-1->wf1 5454   -1-1-onto->wf1o 5456   ` cfv 5457  (class class class)co 6084   1stc1st 6350   2ndc2nd 6351   ^m cmap 7021   ~~ cen 7109   Fincfn 7112   supcsup 7448   CCcc 8993   RRcr 8994   0cc0 8995   1c1 8996   + caddc 8998   +oocpnf 9122   -oocmnf 9123   RR*cxr 9124   < clt 9125   <_ cle 9126   - cmin 9296   / cdiv 9682   NNcn 10005   2c2 10054   NN0cn0 10226   ZZcz 10287   ZZ>=cuz 10493   RR+crp 10617   (,)cioo 10921   [,)cico 10923   ...cfz 11048   seq cseq 11328   ^cexp 11387   abscabs 12044   ~~> cli 12283   sum_csu 12484   vol *covol 19364

This theorem is referenced by:  ovoliunlem2  19404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-ioo 10925  df-ico 10927  df-fz 11049  df-fzo 11141  df-fl 11207  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-rlim 12288  df-sum 12485  df-ovol 19366