Theorem ovoliunlem1 22467

Description: Lemma for ovoliun 22470. (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 5870	. . . . . . . . 9 |- ( j = ( J ` m ) -> ( 1st ` j ) = ( 1st ` ( J ` m ) ) )
2	1	fveq2d 5874	. . . . . . . 8 |- ( j = ( J ` m ) -> ( F ` ( 1st ` j ) ) = ( F ` ( 1st ` ( J ` m ) ) ) )
3		fveq2 5870	. . . . . . . 8 |- ( j = ( J ` m ) -> ( 2nd ` j ) = ( 2nd ` ( J ` m ) ) )
4	2, 3	fveq12d 5876	. . . . . . 7 |- ( j = ( J ` m ) -> ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) = ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) )
5	4	fveq2d 5874	. . . . . 6 |- ( j = ( J ` m ) -> ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) = ( 2nd ` ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) ) )
6	4	fveq2d 5874	. . . . . 6 |- ( j = ( J ` m ) -> ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) = ( 1st ` ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) ) )
7	5, 6	oveq12d 6313	. . . . 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 12193	. . . . 5 |- ( ph -> ( 1 ... K ) e. Fin )
9		ovoliun.j	. . . . . . 7 |- ( ph -> J : NN -1-1-onto-> ( NN X. NN ) )
10		f1of1 5818	. . . . . . 7 |- ( J : NN -1-1-onto-> ( NN X. NN ) -> J : NN -1-1-> ( NN X. NN ) )
11	9, 10	syl 17	. . . . . 6 |- ( ph -> J : NN -1-1-> ( NN X. NN ) )
12		elfznn 11835	. . . . . . 7 |- ( m e. ( 1 ... K ) -> m e. NN )
13	12	ssriv 3438	. . . . . 6 |- ( 1 ... K ) C_ NN
14		f1ores 5833	. . . . . 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 669	. . . . 5 |- ( ph -> ( J |` ( 1 ... K ) ) : ( 1 ... K ) -1-1-onto-> ( J " ( 1 ... K ) ) )
16		fvres 5884	. . . . . 6 |- ( m e. ( 1 ... K ) -> ( ( J |` ( 1 ... K ) ) ` m ) = ( J ` m ) )
17	16	adantl 468	. . . . 5 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( ( J |` ( 1 ... K ) ) ` m ) = ( J ` m ) )
18		inss2 3655	. . . . . . . . 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 467	. . . . . . . . . . . 12 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> F : NN --> ( ( <_ i^i ( RR X. RR ) ) ^m NN ) )
21		imassrn 5182	. . . . . . . . . . . . . . 15 |- ( J " ( 1 ... K ) ) C_ ran J
22		f1of 5819	. . . . . . . . . . . . . . . . 17 |- ( J : NN -1-1-onto-> ( NN X. NN ) -> J : NN --> ( NN X. NN ) )
23	9, 22	syl 17	. . . . . . . . . . . . . . . 16 |- ( ph -> J : NN --> ( NN X. NN ) )
24		frn 5740	. . . . . . . . . . . . . . . 16 |- ( J : NN --> ( NN X. NN ) -> ran J C_ ( NN X. NN ) )
25	23, 24	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> ran J C_ ( NN X. NN ) )
26	21, 25	syl5ss 3445	. . . . . . . . . . . . . 14 |- ( ph -> ( J " ( 1 ... K ) ) C_ ( NN X. NN ) )
27	26	sselda 3434	. . . . . . . . . . . . 13 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> j e. ( NN X. NN ) )
28		xp1st 6828	. . . . . . . . . . . . 13 |- ( j e. ( NN X. NN ) -> ( 1st ` j ) e. NN )
29	27, 28	syl 17	. . . . . . . . . . . 12 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( 1st ` j ) e. NN )
30	20, 29	ffvelrnd 6028	. . . . . . . . . . 11 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( F ` ( 1st ` j ) ) e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) )
31		reex 9635	. . . . . . . . . . . . . 14 |- RR e. _V
32	31, 31	xpex 6600	. . . . . . . . . . . . 13 |- ( RR X. RR ) e. _V
33	32	inex2 4548	. . . . . . . . . . . 12 |- ( <_ i^i ( RR X. RR ) ) e. _V
34		nnex 10622	. . . . . . . . . . . 12 |- NN e. _V
35	33, 34	elmap 7505	. . . . . . . . . . 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 200	. . . . . . . . . 10 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( F ` ( 1st ` j ) ) : NN --> ( <_ i^i ( RR X. RR ) ) )
37		xp2nd 6829	. . . . . . . . . . 11 |- ( j e. ( NN X. NN ) -> ( 2nd ` j ) e. NN )
38	27, 37	syl 17	. . . . . . . . . 10 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( 2nd ` j ) e. NN )
39	36, 38	ffvelrnd 6028	. . . . . . . . 9 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) e. ( <_ i^i ( RR X. RR ) ) )
40	18, 39	sseldi 3432	. . . . . . . 8 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) e. ( RR X. RR ) )
41		xp2nd 6829	. . . . . . . 8 |- ( ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) e. ( RR X. RR ) -> ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) e. RR )
42	40, 41	syl 17	. . . . . . 7 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) e. RR )
43		xp1st 6828	. . . . . . . 8 |- ( ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) e. ( RR X. RR ) -> ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) e. RR )
44	40, 43	syl 17	. . . . . . 7 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) e. RR )
45	42, 44	resubcld 10054	. . . . . 6 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) - ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) ) e. RR )
46	45	recnd 9674	. . . . 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 13801	. . . 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 467	. . . . . . . . . . 11 |- ( ( ph /\ k e. NN ) -> F : NN --> ( ( <_ i^i ( RR X. RR ) ) ^m NN ) )
49	23	ffvelrnda 6027	. . . . . . . . . . . 12 |- ( ( ph /\ k e. NN ) -> ( J ` k ) e. ( NN X. NN ) )
50		xp1st 6828	. . . . . . . . . . . 12 |- ( ( J ` k ) e. ( NN X. NN ) -> ( 1st ` ( J ` k ) ) e. NN )
51	49, 50	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ k e. NN ) -> ( 1st ` ( J ` k ) ) e. NN )
52	48, 51	ffvelrnd 6028	. . . . . . . . . 10 |- ( ( ph /\ k e. NN ) -> ( F ` ( 1st ` ( J ` k ) ) ) e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) )
53	33, 34	elmap 7505	. . . . . . . . . 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 200	. . . . . . . . 9 |- ( ( ph /\ k e. NN ) -> ( F ` ( 1st ` ( J ` k ) ) ) : NN --> ( <_ i^i ( RR X. RR ) ) )
55		xp2nd 6829	. . . . . . . . . 10 |- ( ( J ` k ) e. ( NN X. NN ) -> ( 2nd ` ( J ` k ) ) e. NN )
56	49, 55	syl 17	. . . . . . . . 9 |- ( ( ph /\ k e. NN ) -> ( 2nd ` ( J ` k ) ) e. NN )
57	54, 56	ffvelrnd 6028	. . . . . . . 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 6051	. . . . . . 7 |- ( ph -> H : NN --> ( <_ i^i ( RR X. RR ) ) )
60		eqid 2453	. . . . . . . 8 |- ( ( abs o. - ) o. H ) = ( ( abs o. - ) o. H )
61	60	ovolfsval 22435	. . . . . . 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 480	. . . . . 6 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( ( ( abs o. - ) o. H ) ` m ) = ( ( 2nd ` ( H ` m ) ) - ( 1st ` ( H ` m ) ) ) )
63	12	adantl 468	. . . . . . . . 9 |- ( ( ph /\ m e. ( 1 ... K ) ) -> m e. NN )
64		fveq2 5870	. . . . . . . . . . . . 13 |- ( k = m -> ( J ` k ) = ( J ` m ) )
65	64	fveq2d 5874	. . . . . . . . . . . 12 |- ( k = m -> ( 1st ` ( J ` k ) ) = ( 1st ` ( J ` m ) ) )
66	65	fveq2d 5874	. . . . . . . . . . 11 |- ( k = m -> ( F ` ( 1st ` ( J ` k ) ) ) = ( F ` ( 1st ` ( J ` m ) ) ) )
67	64	fveq2d 5874	. . . . . . . . . . 11 |- ( k = m -> ( 2nd ` ( J ` k ) ) = ( 2nd ` ( J ` m ) ) )
68	66, 67	fveq12d 5876	. . . . . . . . . 10 |- ( k = m -> ( ( F ` ( 1st ` ( J ` k ) ) ) ` ( 2nd ` ( J ` k ) ) ) = ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) )
69		fvex 5880	. . . . . . . . . 10 |- ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) e. _V
70	68, 58, 69	fvmpt 5953	. . . . . . . . 9 |- ( m e. NN -> ( H ` m ) = ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) )
71	63, 70	syl 17	. . . . . . . 8 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( H ` m ) = ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) )
72	71	fveq2d 5874	. . . . . . 7 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( 2nd ` ( H ` m ) ) = ( 2nd ` ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) ) )
73	71	fveq2d 5874	. . . . . . 7 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( 1st ` ( H ` m ) ) = ( 1st ` ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) ) )
74	72, 73	oveq12d 6313	. . . . . 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 2487	. . . . 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 11201	. . . . . 6 |- NN = ( ZZ>= ` 1 )
78	76, 77	syl6eleq 2541	. . . . 5 |- ( ph -> K e. ( ZZ>= ` 1 ) )
79		ffvelrn 6025	. . . . . . . . . . 11 |- ( ( H : NN --> ( <_ i^i ( RR X. RR ) ) /\ m e. NN ) -> ( H ` m ) e. ( <_ i^i ( RR X. RR ) ) )
80	59, 12, 79	syl2an 480	. . . . . . . . . 10 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( H ` m ) e. ( <_ i^i ( RR X. RR ) ) )
81	18, 80	sseldi 3432	. . . . . . . . 9 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( H ` m ) e. ( RR X. RR ) )
82		xp2nd 6829	. . . . . . . . 9 |- ( ( H ` m ) e. ( RR X. RR ) -> ( 2nd ` ( H ` m ) ) e. RR )
83	81, 82	syl 17	. . . . . . . 8 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( 2nd ` ( H ` m ) ) e. RR )
84		xp1st 6828	. . . . . . . . 9 |- ( ( H ` m ) e. ( RR X. RR ) -> ( 1st ` ( H ` m ) ) e. RR )
85	81, 84	syl 17	. . . . . . . 8 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( 1st ` ( H ` m ) ) e. RR )
86	83, 85	resubcld 10054	. . . . . . 7 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( ( 2nd ` ( H ` m ) ) - ( 1st ` ( H ` m ) ) ) e. RR )
87	86	recnd 9674	. . . . . 6 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( ( 2nd ` ( H ` m ) ) - ( 1st ` ( H ` m ) ) ) e. CC )
88	74, 87	eqeltrrd 2532	. . . . 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 13808	. . . 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 2487	. . 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 5871	. . 3 |- ( U ` K ) = ( seq 1 ( + , ( ( abs o. - ) o. H ) ) ` K )
93	90, 92	syl6eqr 2505	. 2 |- ( ph -> sum_ j e. ( J " ( 1 ... K ) ) ( ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) - ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) ) = ( U ` K ) )
94		f1oeng 7593	. . . . . . 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 667	. . . . . 6 |- ( ph -> ( 1 ... K ) ~~ ( J " ( 1 ... K ) ) )
96	95	ensymd 7625	. . . . 5 |- ( ph -> ( J " ( 1 ... K ) ) ~~ ( 1 ... K ) )
97		enfii 7794	. . . . 5 |- ( ( ( 1 ... K ) e. Fin /\ ( J " ( 1 ... K ) ) ~~ ( 1 ... K ) ) -> ( J " ( 1 ... K ) ) e. Fin )
98	8, 96, 97	syl2anc 667	. . . 4 |- ( ph -> ( J " ( 1 ... K ) ) e. Fin )
99	98, 45	fsumrecl 13812	. . 3 |- ( ph -> sum_ j e. ( J " ( 1 ... K ) ) ( ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) - ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) ) e. RR )
100		fzfid 12193	. . . . 5 |- ( ph -> ( 1 ... L ) e. Fin )
101		elfznn 11835	. . . . . 6 |- ( n e. ( 1 ... L ) -> n e. NN )
102		ovoliun.v	. . . . . 6 |- ( ( ph /\ n e. NN ) -> ( vol* ` A ) e. RR )
103	101, 102	sylan2 477	. . . . 5 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( vol* ` A ) e. RR )
104	100, 103	fsumrecl 13812	. . . 4 |- ( ph -> sum_ n e. ( 1 ... L ) ( vol* ` A ) e. RR )
105		ovoliun.b	. . . . . . 7 |- ( ph -> B e. RR+ )
106	105	rpred 11348	. . . . . 6 |- ( ph -> B e. RR )
107		2nn 10774	. . . . . . . 8 |- 2 e. NN
108		nnnn0 10883	. . . . . . . 8 |- ( n e. NN -> n e. NN0 )
109		nnexpcl 12292	. . . . . . . 8 |- ( ( 2 e. NN /\ n e. NN0 ) -> ( 2 ^ n ) e. NN )
110	107, 108, 109	sylancr 670	. . . . . . 7 |- ( n e. NN -> ( 2 ^ n ) e. NN )
111	101, 110	syl 17	. . . . . 6 |- ( n e. ( 1 ... L ) -> ( 2 ^ n ) e. NN )
112		nndivre 10652	. . . . . 6 |- ( ( B e. RR /\ ( 2 ^ n ) e. NN ) -> ( B / ( 2 ^ n ) ) e. RR )
113	106, 111, 112	syl2an 480	. . . . 5 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( B / ( 2 ^ n ) ) e. RR )
114	100, 113	fsumrecl 13812	. . . 4 |- ( ph -> sum_ n e. ( 1 ... L ) ( B / ( 2 ^ n ) ) e. RR )
115	104, 114	readdcld 9675	. . 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 9675	. . 3 |- ( ph -> ( sup ( ran T , RR* , < ) + B ) e. RR )
118		relxp 4945	. . . . . . . . . . . . . . 15 |- Rel ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) )
119		relres 5135	. . . . . . . . . . . . . . 15 |- Rel ( ( J " ( 1 ... K ) ) |` { n } )
120		opelxp 4867	. . . . . . . . . . . . . . . 16 |- ( <. x , y >. e. ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) <-> ( x e. { n } /\ y e. ( ( J " ( 1 ... K ) ) " { n } ) ) )
121		vex 3050	. . . . . . . . . . . . . . . . . 18 |- y e. _V
122	121	opelres 5113	. . . . . . . . . . . . . . . . 17 |- ( <. x , y >. e. ( ( J " ( 1 ... K ) ) |` { n } ) <-> ( <. x , y >. e. ( J " ( 1 ... K ) ) /\ x e. { n } ) )
123		ancom 452	. . . . . . . . . . . . . . . . 17 |- ( ( x e. { n } /\ <. x , y >. e. ( J " ( 1 ... K ) ) ) <-> ( <. x , y >. e. ( J " ( 1 ... K ) ) /\ x e. { n } ) )
124		elsni 3995	. . . . . . . . . . . . . . . . . . . . 21 |- ( x e. { n } -> x = n )
125	124	opeq1d 4175	. . . . . . . . . . . . . . . . . . . 20 |- ( x e. { n } -> <. x , y >. = <. n , y >. )
126	125	eleq1d 2515	. . . . . . . . . . . . . . . . . . 19 |- ( x e. { n } -> ( <. x , y >. e. ( J " ( 1 ... K ) ) <-> <. n , y >. e. ( J " ( 1 ... K ) ) ) )
127		vex 3050	. . . . . . . . . . . . . . . . . . . 20 |- n e. _V
128	127, 121	elimasn 5196	. . . . . . . . . . . . . . . . . . 19 |- ( y e. ( ( J " ( 1 ... K ) ) " { n } ) <-> <. n , y >. e. ( J " ( 1 ... K ) ) )
129	126, 128	syl6bbr 267	. . . . . . . . . . . . . . . . . 18 |- ( x e. { n } -> ( <. x , y >. e. ( J " ( 1 ... K ) ) <-> y e. ( ( J " ( 1 ... K ) ) " { n } ) ) )
130	129	pm5.32i 643	. . . . . . . . . . . . . . . . 17 |- ( ( x e. { n } /\ <. x , y >. e. ( J " ( 1 ... K ) ) ) <-> ( x e. { n } /\ y e. ( ( J " ( 1 ... K ) ) " { n } ) ) )
131	122, 123, 130	3bitr2ri 278	. . . . . . . . . . . . . . . 16 |- ( ( x e. { n } /\ y e. ( ( J " ( 1 ... K ) ) " { n } ) ) <-> <. x , y >. e. ( ( J " ( 1 ... K ) ) |` { n } ) )
132	120, 131	bitri 253	. . . . . . . . . . . . . . 15 |- ( <. x , y >. e. ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) <-> <. x , y >. e. ( ( J " ( 1 ... K ) ) |` { n } ) )
133	118, 119, 132	eqrelriiv 4932	. . . . . . . . . . . . . 14 |- ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) = ( ( J " ( 1 ... K ) ) |` { n } )
134		df-res 4849	. . . . . . . . . . . . . 14 |- ( ( J " ( 1 ... K ) ) |` { n } ) = ( ( J " ( 1 ... K ) ) i^i ( { n } X. _V ) )
135	133, 134	eqtri 2475	. . . . . . . . . . . . 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 4303	. . . . . . . . . . 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 4345	. . . . . . . . . . 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 2503	. . . . . . . . . 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 4945	. . . . . . . . . . . . . 14 |- Rel ( NN X. NN )
141		relss 4925	. . . . . . . . . . . . . 14 |- ( ( J " ( 1 ... K ) ) C_ ( NN X. NN ) -> ( Rel ( NN X. NN ) -> Rel ( J " ( 1 ... K ) ) ) )
142	26, 140, 141	mpisyl 21	. . . . . . . . . . . . 13 |- ( ph -> Rel ( J " ( 1 ... K ) ) )
143		ovoliun.l2	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> A. w e. ( 1 ... K ) ( 1st ` ( J ` w ) ) <_ L )
144		ffn 5733	. . . . . . . . . . . . . . . . . . . . 21 |- ( J : NN --> ( NN X. NN ) -> J Fn NN )
145	23, 144	syl 17	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> J Fn NN )
146		fveq2 5870	. . . . . . . . . . . . . . . . . . . . . 22 |- ( j = ( J ` w ) -> ( 1st ` j ) = ( 1st ` ( J ` w ) ) )
147	146	breq1d 4415	. . . . . . . . . . . . . . . . . . . . 21 |- ( j = ( J ` w ) -> ( ( 1st ` j ) <_ L <-> ( 1st ` ( J ` w ) ) <_ L ) )
148	147	ralima 6150	. . . . . . . . . . . . . . . . . . . 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 669	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( A. j e. ( J " ( 1 ... K ) ) ( 1st ` j ) <_ L <-> A. w e. ( 1 ... K ) ( 1st ` ( J ` w ) ) <_ L ) )
150	143, 149	mpbird 236	. . . . . . . . . . . . . . . . . 18 |- ( ph -> A. j e. ( J " ( 1 ... K ) ) ( 1st ` j ) <_ L )
151	150	r19.21bi 2759	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( 1st ` j ) <_ L )
152	29, 77	syl6eleq 2541	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( 1st ` j ) e. ( ZZ>= ` 1 ) )
153		ovoliun.l1	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> L e. ZZ )
154	153	adantr 467	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> L e. ZZ )
155		elfz5 11799	. . . . . . . . . . . . . . . . . 18 |- ( ( ( 1st ` j ) e. ( ZZ>= ` 1 ) /\ L e. ZZ ) -> ( ( 1st ` j ) e. ( 1 ... L ) <-> ( 1st ` j ) <_ L ) )
156	152, 154, 155	syl2anc 667	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( ( 1st ` j ) e. ( 1 ... L ) <-> ( 1st ` j ) <_ L ) )
157	151, 156	mpbird 236	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( 1st ` j ) e. ( 1 ... L ) )
158	157	ralrimiva 2804	. . . . . . . . . . . . . . 15 |- ( ph -> A. j e. ( J " ( 1 ... K ) ) ( 1st ` j ) e. ( 1 ... L ) )
159		vex 3050	. . . . . . . . . . . . . . . . . 18 |- x e. _V
160	159, 121	op1std 6808	. . . . . . . . . . . . . . . . 17 |- ( j = <. x , y >. -> ( 1st ` j ) = x )
161	160	eleq1d 2515	. . . . . . . . . . . . . . . 16 |- ( j = <. x , y >. -> ( ( 1st ` j ) e. ( 1 ... L ) <-> x e. ( 1 ... L ) ) )
162	161	rspccv 3149	. . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . 14 |- ( ph -> ( <. x , y >. e. ( J " ( 1 ... K ) ) -> x e. ( 1 ... L ) ) )
164		opelxp 4867	. . . . . . . . . . . . . . 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 508	. . . . . . . . . . . . . . 15 |- ( x e. U_ n e. ( 1 ... L ) { n } <-> ( x e. U_ n e. ( 1 ... L ) { n } /\ y e. _V ) )
166		iunid 4336	. . . . . . . . . . . . . . . 16 |- U_ n e. ( 1 ... L ) { n } = ( 1 ... L )
167	166	eleq2i 2523	. . . . . . . . . . . . . . 15 |- ( x e. U_ n e. ( 1 ... L ) { n } <-> x e. ( 1 ... L ) )
168	164, 165, 167	3bitr2i 277	. . . . . . . . . . . . . 14 |- ( <. x , y >. e. ( U_ n e. ( 1 ... L ) { n } X. _V ) <-> x e. ( 1 ... L ) )
169	163, 168	syl6ibr 231	. . . . . . . . . . . . 13 |- ( ph -> ( <. x , y >. e. ( J " ( 1 ... K ) ) -> <. x , y >. e. ( U_ n e. ( 1 ... L ) { n } X. _V ) ) )
170	142, 169	relssdv 4930	. . . . . . . . . . . 12 |- ( ph -> ( J " ( 1 ... K ) ) C_ ( U_ n e. ( 1 ... L ) { n } X. _V ) )
171		xpiundir 4893	. . . . . . . . . . . 12 |- ( U_ n e. ( 1 ... L ) { n } X. _V ) = U_ n e. ( 1 ... L ) ( { n } X. _V )
172	170, 171	syl6sseq 3480	. . . . . . . . . . 11 |- ( ph -> ( J " ( 1 ... K ) ) C_ U_ n e. ( 1 ... L ) ( { n } X. _V ) )
173		df-ss 3420	. . . . . . . . . . 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 200	. . . . . . . . . 10 |- ( ph -> ( ( J " ( 1 ... K ) ) i^i U_ n e. ( 1 ... L ) ( { n } X. _V ) ) = ( J " ( 1 ... K ) ) )
175	139, 174	eqtrd 2487	. . . . . . . . 9 |- ( ph -> U_ n e. ( 1 ... L ) ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) = ( J " ( 1 ... K ) ) )
176	175, 98	eqeltrd 2531	. . . . . . . 8 |- ( ph -> U_ n e. ( 1 ... L ) ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) e. Fin )
177		ssiun2 4324	. . . . . . . 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 7797	. . . . . . . 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 480	. . . . . . 7 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) e. Fin )
180		2ndconst 6890	. . . . . . . . . 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 7593	. . . . . . . . 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 669	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) ~~ ( ( J " ( 1 ... K ) ) " { n } ) )
184	183	ensymd 7625	. . . . . . 7 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( ( J " ( 1 ... K ) ) " { n } ) ~~ ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) )
185		enfii 7794	. . . . . . 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 667	. . . . . 6 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( ( J " ( 1 ... K ) ) " { n } ) e. Fin )
187		ffvelrn 6025	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( F ` n ) e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) )
189	33, 34	elmap 7505	. . . . . . . . . . . . 13 |- ( ( F ` n ) e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) <-> ( F ` n ) : NN --> ( <_ i^i ( RR X. RR ) ) )
190	188, 189	sylib 200	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( F ` n ) : NN --> ( <_ i^i ( RR X. RR ) ) )
191	190	adantrr 724	. . . . . . . . . . 11 |- ( ( ph /\ ( n e. ( 1 ... L ) /\ i e. ( ( J " ( 1 ... K ) ) " { n } ) ) ) -> ( F ` n ) : NN --> ( <_ i^i ( RR X. RR ) ) )
192		imassrn 5182	. . . . . . . . . . . . . 14 |- ( ( J " ( 1 ... K ) ) " { n } ) C_ ran ( J " ( 1 ... K ) )
193	26	adantr 467	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( J " ( 1 ... K ) ) C_ ( NN X. NN ) )
194		rnss 5066	. . . . . . . . . . . . . . . 16 |- ( ( J " ( 1 ... K ) ) C_ ( NN X. NN ) -> ran ( J " ( 1 ... K ) ) C_ ran ( NN X. NN ) )
195	193, 194	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ran ( J " ( 1 ... K ) ) C_ ran ( NN X. NN ) )
196		rnxpid 5273	. . . . . . . . . . . . . . 15 |- ran ( NN X. NN ) = NN
197	195, 196	syl6sseq 3480	. . . . . . . . . . . . . 14 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ran ( J " ( 1 ... K ) ) C_ NN )
198	192, 197	syl5ss 3445	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( ( J " ( 1 ... K ) ) " { n } ) C_ NN )
199	198	sseld 3433	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( i e. ( ( J " ( 1 ... K ) ) " { n } ) -> i e. NN ) )
200	199	impr 625	. . . . . . . . . . 11 |- ( ( ph /\ ( n e. ( 1 ... L ) /\ i e. ( ( J " ( 1 ... K ) ) " { n } ) ) ) -> i e. NN )
201	191, 200	ffvelrnd 6028	. . . . . . . . . 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 3432	. . . . . . . . 9 |- ( ( ph /\ ( n e. ( 1 ... L ) /\ i e. ( ( J " ( 1 ... K ) ) " { n } ) ) ) -> ( ( F ` n ) ` i ) e. ( RR X. RR ) )
203		xp2nd 6829	. . . . . . . . 9 |- ( ( ( F ` n ) ` i ) e. ( RR X. RR ) -> ( 2nd ` ( ( F ` n ) ` i ) ) e. RR )
204	202, 203	syl 17	. . . . . . . 8 |- ( ( ph /\ ( n e. ( 1 ... L ) /\ i e. ( ( J " ( 1 ... K ) ) " { n } ) ) ) -> ( 2nd ` ( ( F ` n ) ` i ) ) e. RR )
205		xp1st 6828	. . . . . . . . 9 |- ( ( ( F ` n ) ` i ) e. ( RR X. RR ) -> ( 1st ` ( ( F ` n ) ` i ) ) e. RR )
206	202, 205	syl 17	. . . . . . . 8 |- ( ( ph /\ ( n e. ( 1 ... L ) /\ i e. ( ( J " ( 1 ... K ) ) " { n } ) ) ) -> ( 1st ` ( ( F ` n ) ` i ) ) e. RR )
207	204, 206	resubcld 10054	. . . . . . 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 654	. . . . . 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 13812	. . . . 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 480	. . . . . . 7 |- ( ( ph /\ n e. NN ) -> ( B / ( 2 ^ n ) ) e. RR )
211	102, 210	readdcld 9675	. . . . . 6 |- ( ( ph /\ n e. NN ) -> ( ( vol* ` A ) + ( B / ( 2 ^ n ) ) ) e. RR )
212	101, 211	sylan2 477	. . . . 5 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( ( vol* ` A ) + ( B / ( 2 ^ n ) ) ) e. RR )
213		eqid 2453	. . . . . . . . . . . 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 22437	. . . . . . . . . . 11 |- ( ( F ` n ) : NN --> ( <_ i^i ( RR X. RR ) ) -> S : NN --> ( 0 [,) +oo ) )
216	190, 215	syl 17	. . . . . . . . . 10 |- ( ( ph /\ n e. ( 1 ... L ) ) -> S : NN --> ( 0 [,) +oo ) )
217		frn 5740	. . . . . . . . . 10 |- ( S : NN --> ( 0 [,) +oo ) -> ran S C_ ( 0 [,) +oo ) )
218	216, 217	syl 17	. . . . . . . . 9 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ran S C_ ( 0 [,) +oo ) )
219		icossxr 11726	. . . . . . . . 9 |- ( 0 [,) +oo ) C_ RR*
220	218, 219	syl6ss 3446	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ran S C_ RR* )
221		supxrcl 11607	. . . . . . . 8 |- ( ran S C_ RR* -> sup ( ran S , RR* , < ) e. RR* )
222	220, 221	syl 17	. . . . . . 7 |- ( ( ph /\ n e. ( 1 ... L ) ) -> sup ( ran S , RR* , < ) e. RR* )
223		mnfxr 11421	. . . . . . . . 9 |- -oo e. RR*
224	223	a1i 11	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> -oo e. RR* )
225	103	rexrd 9695	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( vol* ` A ) e. RR* )
226		mnflt 11432	. . . . . . . . 9 |- ( ( vol* ` A ) e. RR -> -oo < ( vol* ` A ) )
227	103, 226	syl 17	. . . . . . . 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 477	. . . . . . . . 9 |- ( ( ph /\ n e. ( 1 ... L ) ) -> A C_ U. ran ( (,) o. ( F ` n ) ) )
230	214	ovollb 22444	. . . . . . . . 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 667	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( vol* ` A ) <_ sup ( ran S , RR* , < ) )
232	224, 225, 222, 227, 231	xrltletrd 11465	. . . . . . 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 477	. . . . . . 7 |- ( ( ph /\ n e. ( 1 ... L ) ) -> sup ( ran S , RR* , < ) <_ ( ( vol* ` A ) + ( B / ( 2 ^ n ) ) ) )
235		xrre 11471	. . . . . . 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 1270	. . . . . 6 |- ( ( ph /\ n e. ( 1 ... L ) ) -> sup ( ran S , RR* , < ) e. RR )
237		1zzd 10975	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> 1 e. ZZ )
238	213	ovolfsval 22435	. . . . . . . . 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 ) ) ) )
239	190, 238	sylan 474	. . . . . . . 8 |- ( ( ( ph /\ n e. ( 1 ... L ) ) /\ i e. NN ) -> ( ( ( abs o. - ) o. ( F ` n ) ) ` i ) = ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) )
240	213	ovolfsf 22436	. . . . . . . . . . . . 13 |- ( ( F ` n ) : NN --> ( <_ i^i ( RR X. RR ) ) -> ( ( abs o. - ) o. ( F ` n ) ) : NN --> ( 0 [,) +oo ) )
241	190, 240	syl 17	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( ( abs o. - ) o. ( F ` n ) ) : NN --> ( 0 [,) +oo ) )
242	241	ffvelrnda 6027	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. ( 1 ... L ) ) /\ i e. NN ) -> ( ( ( abs o. - ) o. ( F ` n ) ) ` i ) e. ( 0 [,) +oo ) )
243	239, 242	eqeltrrd 2532	. . . . . . . . . 10 |- ( ( ( ph /\ n e. ( 1 ... L ) ) /\ i e. NN ) -> ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) e. ( 0 [,) +oo ) )
244		elrege0 11745	. . . . . . . . . 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 ) ) ) ) )
245	243, 244	sylib 200	. . . . . . . . 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 ) ) ) ) )
246	245	simpld 461	. . . . . . . 8 |- ( ( ( ph /\ n e. ( 1 ... L ) ) /\ i e. NN ) -> ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) e. RR )
247	245	simprd 465	. . . . . . . 8 |- ( ( ( ph /\ n e. ( 1 ... L ) ) /\ i e. NN ) -> 0 <_ ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) )
248		supxrub 11617	. . . . . . . . . . . . . . 15 |- ( ( ran S C_ RR* /\ z e. ran S ) -> z <_ sup ( ran S , RR* , < ) )
249	220, 248	sylan 474	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ n e. ( 1 ... L ) ) /\ z e. ran S ) -> z <_ sup ( ran S , RR* , < ) )
250	249	ralrimiva 2804	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. ( 1 ... L ) ) -> A. z e. ran S z <_ sup ( ran S , RR* , < ) )
251		breq2 4409	. . . . . . . . . . . . . . 15 |- ( x = sup ( ran S , RR* , < ) -> ( z <_ x <-> z <_ sup ( ran S , RR* , < ) ) )
252	251	ralbidv 2829	. . . . . . . . . . . . . 14 |- ( x = sup ( ran S , RR* , < ) -> ( A. z e. ran S z <_ x <-> A. z e. ran S z <_ sup ( ran S , RR* , < ) ) )
253	252	rspcev 3152	. . . . . . . . . . . . 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 )
254	236, 250, 253	syl2anc 667	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( 1 ... L ) ) -> E. x e. RR A. z e. ran S z <_ x )
255		ffn 5733	. . . . . . . . . . . . . . 15 |- ( S : NN --> ( 0 [,) +oo ) -> S Fn NN )
256	216, 255	syl 17	. . . . . . . . . . . . . 14 |- ( ( ph /\ n e. ( 1 ... L ) ) -> S Fn NN )
257		breq1 4408	. . . . . . . . . . . . . . 15 |- ( z = ( S ` k ) -> ( z <_ x <-> ( S ` k ) <_ x ) )
258	257	ralrn 6030	. . . . . . . . . . . . . 14 |- ( S Fn NN -> ( A. z e. ran S z <_ x <-> A. k e. NN ( S ` k ) <_ x ) )
259	256, 258	syl 17	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( A. z e. ran S z <_ x <-> A. k e. NN ( S ` k ) <_ x ) )
260	259	rexbidv 2903	. . . . . . . . . . . 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 ) )
261	254, 260	mpbid 214	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( 1 ... L ) ) -> E. x e. RR A. k e. NN ( S ` k ) <_ x )
262	77, 214, 237, 239, 246, 247, 261	isumsup2 13916	. . . . . . . . . 10 |- ( ( ph /\ n e. ( 1 ... L ) ) -> S ~~> sup ( ran S , RR , < ) )
263	214, 262	syl5eqbrr 4440	. . . . . . . . 9 |- ( ( ph /\ n e. ( 1 ... L ) ) -> seq 1 ( + , ( ( abs o. - ) o. ( F ` n ) ) ) ~~> sup ( ran S , RR , < ) )
264		climrel 13568	. . . . . . . . . 10 |- Rel ~~>
265	264	releldmi 5074	. . . . . . . . 9 |- ( seq 1 ( + , ( ( abs o. - ) o. ( F ` n ) ) ) ~~> sup ( ran S , RR , < ) -> seq 1 ( + , ( ( abs o. - ) o. ( F ` n ) ) ) e. dom ~~> )
266	263, 265	syl 17	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> seq 1 ( + , ( ( abs o. - ) o. ( F ` n ) ) ) e. dom ~~> )
267	77, 237, 186, 198, 239, 246, 247, 266	isumless 13915	. . . . . . 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 ) ) ) )
268	77, 214, 237, 239, 246, 247, 261	isumsup 13917	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> sum_ i e. NN ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) = sup ( ran S , RR , < ) )
269		rge0ssre 11747	. . . . . . . . . 10 |- ( 0 [,) +oo ) C_ RR
270	218, 269	syl6ss 3446	. . . . . . . . 9 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ran S C_ RR )
271		1nn 10627	. . . . . . . . . . . 12 |- 1 e. NN
272		fdm 5738	. . . . . . . . . . . . 13 |- ( S : NN --> ( 0 [,) +oo ) -> dom S = NN )
273	216, 272	syl 17	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( 1 ... L ) ) -> dom S = NN )
274	271, 273	syl5eleqr 2538	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( 1 ... L ) ) -> 1 e. dom S )
275		ne0i 3739	. . . . . . . . . . 11 |- ( 1 e. dom S -> dom S =/= (/) )
276	274, 275	syl 17	. . . . . . . . . 10 |- ( ( ph /\ n e. ( 1 ... L ) ) -> dom S =/= (/) )
277		dm0rn0 5054	. . . . . . . . . . 11 |- ( dom S = (/) <-> ran S = (/) )
278	277	necon3bii 2678	. . . . . . . . . 10 |- ( dom S =/= (/) <-> ran S =/= (/) )
279	276, 278	sylib 200	. . . . . . . . 9 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ran S =/= (/) )
280		supxrre 11620	. . . . . . . . 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 , < ) )
281	270, 279, 254, 280	syl3anc 1269	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> sup ( ran S , RR* , < ) = sup ( ran S , RR , < ) )
282	268, 281	eqtr4d 2490	. . . . . . 7 |- ( ( ph /\ n e. ( 1 ... L ) ) -> sum_ i e. NN ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) = sup ( ran S , RR* , < ) )
283	267, 282	breqtrd 4430	. . . . . 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* , < ) )
284	209, 236, 212, 283, 234	letrd 9797	. . . . 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 ) ) ) )
285	100, 209, 212, 284	fsumle 13871	. . . 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 ) ) ) )
286		vex 3050	. . . . . . . . . . 11 |- i e. _V
287	127, 286	op1std 6808	. . . . . . . . . 10 |- ( j = <. n , i >. -> ( 1st ` j ) = n )
288	287	fveq2d 5874	. . . . . . . . 9 |- ( j = <. n , i >. -> ( F ` ( 1st ` j ) ) = ( F ` n ) )
289	127, 286	op2ndd 6809	. . . . . . . . 9 |- ( j = <. n , i >. -> ( 2nd ` j ) = i )
290	288, 289	fveq12d 5876	. . . . . . . 8 |- ( j = <. n , i >. -> ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) = ( ( F ` n ) ` i ) )
291	290	fveq2d 5874	. . . . . . 7 |- ( j = <. n , i >. -> ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) = ( 2nd ` ( ( F ` n ) ` i ) ) )
292	290	fveq2d 5874	. . . . . . 7 |- ( j = <. n , i >. -> ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) = ( 1st ` ( ( F ` n ) ` i ) ) )
293	291, 292	oveq12d 6313	. . . . . 6 |- ( j = <. n , i >. -> ( ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) - ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) ) = ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) )
294	207	recnd 9674	. . . . . 6 |- ( ( ph /\ ( n e. ( 1 ... L ) /\ i e. ( ( J " ( 1 ... K ) ) " { n } ) ) ) -> ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) e. CC )
295	293, 100, 186, 294	fsum2d 13844	. . . . 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 ) ) ) ) )
296	175	sumeq1d 13779	. . . . 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 ) ) ) ) )
297	295, 296	eqtrd 2487	. . . 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 ) ) ) ) )
298	103	recnd 9674	. . . . 5 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( vol* ` A ) e. CC )
299	113	recnd 9674	. . . . 5 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( B / ( 2 ^ n ) ) e. CC )
300	100, 298, 299	fsumadd 13817	. . . 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 ) ) ) )
301	285, 297, 300	3brtr3d 4435	. . 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 ) ) ) )
302		1zzd 10975	. . . . . . . . 9 |- ( ph -> 1 e. ZZ )
303		simpr 463	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> n e. NN )
304		ovoliun.g	. . . . . . . . . . . 12 |- G = ( n e. NN |-> ( vol* ` A ) )
305	304	fvmpt2 5962	. . . . . . . . . . 11 |- ( ( n e. NN /\ ( vol* ` A ) e. RR ) -> ( G ` n ) = ( vol* ` A ) )
306	303, 102, 305	syl2anc 667	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( G ` n ) = ( vol* ` A ) )
307	306, 102	eqeltrd 2531	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( G ` n ) e. RR )
308	77, 302, 307	serfre 12249	. . . . . . . 8 |- ( ph -> seq 1 ( + , G ) : NN --> RR )
309		ovoliun.t	. . . . . . . . 9 |- T = seq 1 ( + , G )
310	309	feq1i 5725	. . . . . . . 8 |- ( T : NN --> RR <-> seq 1 ( + , G ) : NN --> RR )
311	308, 310	sylibr 216	. . . . . . 7 |- ( ph -> T : NN --> RR )
312		frn 5740	. . . . . . 7 |- ( T : NN --> RR -> ran T C_ RR )
313	311, 312	syl 17	. . . . . 6 |- ( ph -> ran T C_ RR )
314		ressxr 9689	. . . . . 6 |- RR C_ RR*
315	313, 314	syl6ss 3446	. . . . 5 |- ( ph -> ran T C_ RR* )
316	101, 306	sylan2 477	. . . . . . 7 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( G ` n ) = ( vol* ` A ) )
317		1red 9663	. . . . . . . . . 10 |- ( ph -> 1 e. RR )
318		ffvelrn 6025	. . . . . . . . . . . . 13 |- ( ( J : NN --> ( NN X. NN ) /\ 1 e. NN ) -> ( J ` 1 ) e. ( NN X. NN ) )
319	23, 271, 318	sylancl 669	. . . . . . . . . . . 12 |- ( ph -> ( J ` 1 ) e. ( NN X. NN ) )
320		xp1st 6828	. . . . . . . . . . . 12 |- ( ( J ` 1 ) e. ( NN X. NN ) -> ( 1st ` ( J ` 1 ) ) e. NN )
321	319, 320	syl 17	. . . . . . . . . . 11 |- ( ph -> ( 1st ` ( J ` 1 ) ) e. NN )
322	321	nnred 10631	. . . . . . . . . 10 |- ( ph -> ( 1st ` ( J ` 1 ) ) e. RR )
323	153	zred 11047	. . . . . . . . . 10 |- ( ph -> L e. RR )
324	321	nnge1d 10659	. . . . . . . . . 10 |- ( ph -> 1 <_ ( 1st ` ( J ` 1 ) ) )
325		eluzfz1 11813	. . . . . . . . . . . 12 |- ( K e. ( ZZ>= ` 1 ) -> 1 e. ( 1 ... K ) )
326	78, 325	syl 17	. . . . . . . . . . 11 |- ( ph -> 1 e. ( 1 ... K ) )
327		fveq2 5870	. . . . . . . . . . . . . 14 |- ( w = 1 -> ( J ` w ) = ( J ` 1 ) )
328	327	fveq2d 5874	. . . . . . . . . . . . 13 |- ( w = 1 -> ( 1st ` ( J ` w ) ) = ( 1st ` ( J ` 1 ) ) )
329	328	breq1d 4415	. . . . . . . . . . . 12 |- ( w = 1 -> ( ( 1st ` ( J ` w ) ) <_ L <-> ( 1st ` ( J ` 1 ) ) <_ L ) )
330	329	rspcv 3148	. . . . . . . . . . 11 |- ( 1 e. ( 1 ... K ) -> ( A. w e. ( 1 ... K ) ( 1st ` ( J ` w ) ) <_ L -> ( 1st ` ( J ` 1 ) ) <_ L ) )
331	326, 143, 330	sylc 62	. . . . . . . . . 10 |- ( ph -> ( 1st ` ( J ` 1 ) ) <_ L )
332	317, 322, 323, 324, 331	letrd 9797	. . . . . . . . 9 |- ( ph -> 1 <_ L )
333		elnnz1 10970	. . . . . . . . 9 |- ( L e. NN <-> ( L e. ZZ /\ 1 <_ L ) )
334	153, 332, 333	sylanbrc 671	. . . . . . . 8 |- ( ph -> L e. NN )
335	334, 77	syl6eleq 2541	. . . . . . 7 |- ( ph -> L e. ( ZZ>= ` 1 ) )
336	316, 335, 298	fsumser 13808	. . . . . 6 |- ( ph -> sum_ n e. ( 1 ... L ) ( vol* ` A ) = ( seq 1 ( + , G ) ` L ) )
337		seqfn 12232	. . . . . . . . 9 |- ( 1 e. ZZ -> seq 1 ( + , G ) Fn ( ZZ>= ` 1 ) )
338	302, 337	syl 17	. . . . . . . 8 |- ( ph -> seq 1 ( + , G ) Fn ( ZZ>= ` 1 ) )
339		fnfvelrn 6024	. . . . . . . 8 |- ( ( seq 1 ( + , G ) Fn ( ZZ>= ` 1 ) /\ L e. ( ZZ>= ` 1 ) ) -> ( seq 1 ( + , G ) ` L ) e. ran seq 1 ( + , G ) )
340	338, 335, 339	syl2anc 667	. . . . . . 7 |- ( ph -> ( seq 1 ( + , G ) ` L ) e. ran seq 1 ( + , G ) )
341	309	rneqi 5064	. . . . . . 7 |- ran T = ran seq 1 ( + , G )
342	340, 341	syl6eleqr 2542	. . . . . 6 |- ( ph -> ( seq 1 ( + , G ) ` L ) e. ran T )
343	336, 342	eqeltrd 2531	. . . . 5 |- ( ph -> sum_ n e. ( 1 ... L ) ( vol* ` A ) e. ran T )
344		supxrub 11617	. . . . 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* , < ) )
345	315, 343, 344	syl2anc 667	. . . 4 |- ( ph -> sum_ n e. ( 1 ... L ) ( vol* ` A ) <_ sup ( ran T , RR* , < ) )
346	106	recnd 9674	. . . . . 6 |- ( ph -> B e. CC )
347		geo2sum 13941	. . . . . 6 |- ( ( L e. NN /\ B e. CC ) -> sum_ n e. ( 1 ... L ) ( B / ( 2 ^ n ) ) = ( B - ( B / ( 2 ^ L ) ) ) )
348	334, 346, 347	syl2anc 667	. . . . 5 |- ( ph -> sum_ n e. ( 1 ... L ) ( B / ( 2 ^ n ) ) = ( B - ( B / ( 2 ^ L ) ) ) )
349	334	nnnn0d 10932	. . . . . . . . . 10 |- ( ph -> L e. NN0 )
350		nnexpcl 12292	. . . . . . . . . 10 |- ( ( 2 e. NN /\ L e. NN0 ) -> ( 2 ^ L ) e. NN )
351	107, 349, 350	sylancr 670	. . . . . . . . 9 |- ( ph -> ( 2 ^ L ) e. NN )
352	351	nnrpd 11346	. . . . . . . 8 |- ( ph -> ( 2 ^ L ) e. RR+ )
353	105, 352	rpdivcld 11365	. . . . . . 7 |- ( ph -> ( B / ( 2 ^ L ) ) e. RR+ )
354	353	rpge0d 11352	. . . . . 6 |- ( ph -> 0 <_ ( B / ( 2 ^ L ) ) )
355	106, 351	nndivred 10665	. . . . . . 7 |- ( ph -> ( B / ( 2 ^ L ) ) e. RR )
356	106, 355	subge02d 10212	. . . . . 6 |- ( ph -> ( 0 <_ ( B / ( 2 ^ L ) ) <-> ( B - ( B / ( 2 ^ L ) ) ) <_ B ) )
357	354, 356	mpbid 214	. . . . 5 |- ( ph -> ( B - ( B / ( 2 ^ L ) ) ) <_ B )
358	348, 357	eqbrtrd 4426	. . . 4 |- ( ph -> sum_ n e. ( 1 ... L ) ( B / ( 2 ^ n ) ) <_ B )
359	104, 114, 116, 106, 345, 358	le2addd 10239	. . 3 |- ( ph -> ( sum_ n e. ( 1 ... L ) ( vol* ` A ) + sum_ n e. ( 1 ... L ) ( B / ( 2 ^ n ) ) ) <_ ( sup ( ran T , RR* , < ) + B ) )
360	99, 115, 117, 301, 359	letrd 9797	. 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 ) )
361	93, 360	eqbrtrrd 4428	1 |- ( ph -> ( U ` K ) <_ ( sup ( ran T , RR* , < ) + B ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1446   e. wcel 1889   =/= wne 2624   A.wral 2739   E.wrex 2740   _Vcvv 3047   i^i cin 3405   C_ wss 3406   (/)c0 3733   {csn 3970   <.cop 3976   U.cuni 4201   U_ciun 4281   class class class wbr 4405   |-> cmpt 4464   X. cxp 4835   dom cdm 4837   ran crn 4838   |` cres 4839   "cima 4840   o. ccom 4841   Rel wrel 4842   Fn wfn 5580   -->wf 5581   -1-1->wf1 5582   -1-1-onto->wf1o 5584   ` cfv 5585  (class class class)co 6295   1stc1st 6796   2ndc2nd 6797   ^m cmap 7477   ~~ cen 7571   Fincfn 7574   supcsup 7959   CCcc 9542   RRcr 9543   0cc0 9544   1c1 9545   + caddc 9547   +oocpnf 9677   -oocmnf 9678   RR*cxr 9679   < clt 9680   <_ cle 9681   - cmin 9865   / cdiv 10276   NNcn 10616   2c2 10666   NN0cn0 10876   ZZcz 10944   ZZ>=cuz 11166   RR+crp 11309   (,)cioo 11642   [,)cico 11644   ...cfz 11791   seqcseq 12220   ^cexp 12279   abscabs 13309   ~~> cli 13560   sum_csu 13764   vol*covol 22425

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-sup 7961  df-inf 7962  df-oi 8030  df-card 8378  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-rp 11310  df-ioo 11646  df-ico 11648  df-fz 11792  df-fzo 11923  df-fl 12035  df-seq 12221  df-exp 12280  df-hash 12523  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-clim 13564  df-rlim 13565  df-sum 13765  df-ovol 22428

This theorem is referenced by:  ovoliunlem2  22468