Theorem ovoliunlem1 21676

Description: Lemma for ovoliun 21679. (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 5866	. . . . . . . . 9 |- ( j = ( J ` m ) -> ( 1st ` j ) = ( 1st ` ( J ` m ) ) )
2	1	fveq2d 5870	. . . . . . . 8 |- ( j = ( J ` m ) -> ( F ` ( 1st ` j ) ) = ( F ` ( 1st ` ( J ` m ) ) ) )
3		fveq2 5866	. . . . . . . 8 |- ( j = ( J ` m ) -> ( 2nd ` j ) = ( 2nd ` ( J ` m ) ) )
4	2, 3	fveq12d 5872	. . . . . . 7 |- ( j = ( J ` m ) -> ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) = ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) )
5	4	fveq2d 5870	. . . . . 6 |- ( j = ( J ` m ) -> ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) = ( 2nd ` ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) ) )
6	4	fveq2d 5870	. . . . . 6 |- ( j = ( J ` m ) -> ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) = ( 1st ` ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) ) )
7	5, 6	oveq12d 6302	. . . . 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 12051	. . . . 5 |- ( ph -> ( 1 ... K ) e. Fin )
9		ovoliun.j	. . . . . . 7 |- ( ph -> J : NN -1-1-onto-> ( NN X. NN ) )
10		f1of1 5815	. . . . . . 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 11714	. . . . . . 7 |- ( m e. ( 1 ... K ) -> m e. NN )
13	12	ssriv 3508	. . . . . 6 |- ( 1 ... K ) C_ NN
14		f1ores 5830	. . . . . 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 662	. . . . 5 |- ( ph -> ( J |` ( 1 ... K ) ) : ( 1 ... K ) -1-1-onto-> ( J " ( 1 ... K ) ) )
16		fvres 5880	. . . . . 6 |- ( m e. ( 1 ... K ) -> ( ( J |` ( 1 ... K ) ) ` m ) = ( J ` m ) )
17	16	adantl 466	. . . . 5 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( ( J |` ( 1 ... K ) ) ` m ) = ( J ` m ) )
18		inss2 3719	. . . . . . . . 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 465	. . . . . . . . . . . 12 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> F : NN --> ( ( <_ i^i ( RR X. RR ) ) ^m NN ) )
21		imassrn 5348	. . . . . . . . . . . . . . 15 |- ( J " ( 1 ... K ) ) C_ ran J
22		f1of 5816	. . . . . . . . . . . . . . . . 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 5737	. . . . . . . . . . . . . . . 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 3515	. . . . . . . . . . . . . 14 |- ( ph -> ( J " ( 1 ... K ) ) C_ ( NN X. NN ) )
27	26	sselda 3504	. . . . . . . . . . . . 13 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> j e. ( NN X. NN ) )
28		xp1st 6814	. . . . . . . . . . . . 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 6022	. . . . . . . . . . 11 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( F ` ( 1st ` j ) ) e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) )
31		reex 9583	. . . . . . . . . . . . . 14 |- RR e. _V
32	31, 31	xpex 6588	. . . . . . . . . . . . 13 |- ( RR X. RR ) e. _V
33	32	inex2 4589	. . . . . . . . . . . 12 |- ( <_ i^i ( RR X. RR ) ) e. _V
34		nnex 10542	. . . . . . . . . . . 12 |- NN e. _V
35	33, 34	elmap 7447	. . . . . . . . . . 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 196	. . . . . . . . . 10 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( F ` ( 1st ` j ) ) : NN --> ( <_ i^i ( RR X. RR ) ) )
37		xp2nd 6815	. . . . . . . . . . 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 6022	. . . . . . . . 9 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) e. ( <_ i^i ( RR X. RR ) ) )
40	18, 39	sseldi 3502	. . . . . . . 8 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) e. ( RR X. RR ) )
41		xp2nd 6815	. . . . . . . 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 6814	. . . . . . . 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 9987	. . . . . 6 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) - ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) ) e. RR )
46	45	recnd 9622	. . . . 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 13508	. . . 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 465	. . . . . . . . . . 11 |- ( ( ph /\ k e. NN ) -> F : NN --> ( ( <_ i^i ( RR X. RR ) ) ^m NN ) )
49	23	ffvelrnda 6021	. . . . . . . . . . . 12 |- ( ( ph /\ k e. NN ) -> ( J ` k ) e. ( NN X. NN ) )
50		xp1st 6814	. . . . . . . . . . . 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 6022	. . . . . . . . . 10 |- ( ( ph /\ k e. NN ) -> ( F ` ( 1st ` ( J ` k ) ) ) e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) )
53	33, 34	elmap 7447	. . . . . . . . . 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 196	. . . . . . . . 9 |- ( ( ph /\ k e. NN ) -> ( F ` ( 1st ` ( J ` k ) ) ) : NN --> ( <_ i^i ( RR X. RR ) ) )
55		xp2nd 6815	. . . . . . . . . 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 6022	. . . . . . . 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 6045	. . . . . . 7 |- ( ph -> H : NN --> ( <_ i^i ( RR X. RR ) ) )
60		eqid 2467	. . . . . . . 8 |- ( ( abs o. - ) o. H ) = ( ( abs o. - ) o. H )
61	60	ovolfsval 21645	. . . . . . 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 477	. . . . . 6 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( ( ( abs o. - ) o. H ) ` m ) = ( ( 2nd ` ( H ` m ) ) - ( 1st ` ( H ` m ) ) ) )
63	12	adantl 466	. . . . . . . . 9 |- ( ( ph /\ m e. ( 1 ... K ) ) -> m e. NN )
64		fveq2 5866	. . . . . . . . . . . . 13 |- ( k = m -> ( J ` k ) = ( J ` m ) )
65	64	fveq2d 5870	. . . . . . . . . . . 12 |- ( k = m -> ( 1st ` ( J ` k ) ) = ( 1st ` ( J ` m ) ) )
66	65	fveq2d 5870	. . . . . . . . . . 11 |- ( k = m -> ( F ` ( 1st ` ( J ` k ) ) ) = ( F ` ( 1st ` ( J ` m ) ) ) )
67	64	fveq2d 5870	. . . . . . . . . . 11 |- ( k = m -> ( 2nd ` ( J ` k ) ) = ( 2nd ` ( J ` m ) ) )
68	66, 67	fveq12d 5872	. . . . . . . . . 10 |- ( k = m -> ( ( F ` ( 1st ` ( J ` k ) ) ) ` ( 2nd ` ( J ` k ) ) ) = ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) )
69		fvex 5876	. . . . . . . . . 10 |- ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) e. _V
70	68, 58, 69	fvmpt 5950	. . . . . . . . 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 5870	. . . . . . 7 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( 2nd ` ( H ` m ) ) = ( 2nd ` ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) ) )
73	71	fveq2d 5870	. . . . . . 7 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( 1st ` ( H ` m ) ) = ( 1st ` ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) ) )
74	72, 73	oveq12d 6302	. . . . . 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 2508	. . . . 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 11117	. . . . . 6 |- NN = ( ZZ>= ` 1 )
78	76, 77	syl6eleq 2565	. . . . 5 |- ( ph -> K e. ( ZZ>= ` 1 ) )
79		ffvelrn 6019	. . . . . . . . . . 11 |- ( ( H : NN --> ( <_ i^i ( RR X. RR ) ) /\ m e. NN ) -> ( H ` m ) e. ( <_ i^i ( RR X. RR ) ) )
80	59, 12, 79	syl2an 477	. . . . . . . . . 10 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( H ` m ) e. ( <_ i^i ( RR X. RR ) ) )
81	18, 80	sseldi 3502	. . . . . . . . 9 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( H ` m ) e. ( RR X. RR ) )
82		xp2nd 6815	. . . . . . . . 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 6814	. . . . . . . . 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 9987	. . . . . . 7 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( ( 2nd ` ( H ` m ) ) - ( 1st ` ( H ` m ) ) ) e. RR )
87	86	recnd 9622	. . . . . 6 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( ( 2nd ` ( H ` m ) ) - ( 1st ` ( H ` m ) ) ) e. CC )
88	74, 87	eqeltrrd 2556	. . . . 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 13515	. . . 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 2508	. . 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 5867	. . 3 |- ( U ` K ) = ( seq 1 ( + , ( ( abs o. - ) o. H ) ) ` K )
93	90, 92	syl6eqr 2526	. 2 |- ( ph -> sum_ j e. ( J " ( 1 ... K ) ) ( ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) - ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) ) = ( U ` K ) )
94		f1oeng 7534	. . . . . . 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 661	. . . . . 6 |- ( ph -> ( 1 ... K ) ~~ ( J " ( 1 ... K ) ) )
96	95	ensymd 7566	. . . . 5 |- ( ph -> ( J " ( 1 ... K ) ) ~~ ( 1 ... K ) )
97		enfii 7737	. . . . 5 |- ( ( ( 1 ... K ) e. Fin /\ ( J " ( 1 ... K ) ) ~~ ( 1 ... K ) ) -> ( J " ( 1 ... K ) ) e. Fin )
98	8, 96, 97	syl2anc 661	. . . 4 |- ( ph -> ( J " ( 1 ... K ) ) e. Fin )
99	98, 45	fsumrecl 13519	. . 3 |- ( ph -> sum_ j e. ( J " ( 1 ... K ) ) ( ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) - ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) ) e. RR )
100		fzfid 12051	. . . . 5 |- ( ph -> ( 1 ... L ) e. Fin )
101		elfznn 11714	. . . . . 6 |- ( n e. ( 1 ... L ) -> n e. NN )
102		ovoliun.v	. . . . . 6 |- ( ( ph /\ n e. NN ) -> ( vol* ` A ) e. RR )
103	101, 102	sylan2 474	. . . . 5 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( vol* ` A ) e. RR )
104	100, 103	fsumrecl 13519	. . . 4 |- ( ph -> sum_ n e. ( 1 ... L ) ( vol* ` A ) e. RR )
105		ovoliun.b	. . . . . . 7 |- ( ph -> B e. RR+ )
106	105	rpred 11256	. . . . . 6 |- ( ph -> B e. RR )
107		2nn 10693	. . . . . . . 8 |- 2 e. NN
108		nnnn0 10802	. . . . . . . 8 |- ( n e. NN -> n e. NN0 )
109		nnexpcl 12147	. . . . . . . 8 |- ( ( 2 e. NN /\ n e. NN0 ) -> ( 2 ^ n ) e. NN )
110	107, 108, 109	sylancr 663	. . . . . . 7 |- ( n e. NN -> ( 2 ^ n ) e. NN )
111	101, 110	syl 16	. . . . . 6 |- ( n e. ( 1 ... L ) -> ( 2 ^ n ) e. NN )
112		nndivre 10571	. . . . . 6 |- ( ( B e. RR /\ ( 2 ^ n ) e. NN ) -> ( B / ( 2 ^ n ) ) e. RR )
113	106, 111, 112	syl2an 477	. . . . 5 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( B / ( 2 ^ n ) ) e. RR )
114	100, 113	fsumrecl 13519	. . . 4 |- ( ph -> sum_ n e. ( 1 ... L ) ( B / ( 2 ^ n ) ) e. RR )
115	104, 114	readdcld 9623	. . 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 9623	. . 3 |- ( ph -> ( sup ( ran T , RR* , < ) + B ) e. RR )
118		relxp 5110	. . . . . . . . . . . . . . 15 |- Rel ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) )
119		relres 5301	. . . . . . . . . . . . . . 15 |- Rel ( ( J " ( 1 ... K ) ) |` { n } )
120		opelxp 5029	. . . . . . . . . . . . . . . 16 |- ( <. x , y >. e. ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) <-> ( x e. { n } /\ y e. ( ( J " ( 1 ... K ) ) " { n } ) ) )
121		vex 3116	. . . . . . . . . . . . . . . . . 18 |- y e. _V
122	121	opelres 5279	. . . . . . . . . . . . . . . . 17 |- ( <. x , y >. e. ( ( J " ( 1 ... K ) ) |` { n } ) <-> ( <. x , y >. e. ( J " ( 1 ... K ) ) /\ x e. { n } ) )
123		ancom 450	. . . . . . . . . . . . . . . . 17 |- ( ( x e. { n } /\ <. x , y >. e. ( J " ( 1 ... K ) ) ) <-> ( <. x , y >. e. ( J " ( 1 ... K ) ) /\ x e. { n } ) )
124		elsni 4052	. . . . . . . . . . . . . . . . . . . . 21 |- ( x e. { n } -> x = n )
125	124	opeq1d 4219	. . . . . . . . . . . . . . . . . . . 20 |- ( x e. { n } -> <. x , y >. = <. n , y >. )
126	125	eleq1d 2536	. . . . . . . . . . . . . . . . . . 19 |- ( x e. { n } -> ( <. x , y >. e. ( J " ( 1 ... K ) ) <-> <. n , y >. e. ( J " ( 1 ... K ) ) ) )
127		vex 3116	. . . . . . . . . . . . . . . . . . . 20 |- n e. _V
128	127, 121	elimasn 5362	. . . . . . . . . . . . . . . . . . 19 |- ( y e. ( ( J " ( 1 ... K ) ) " { n } ) <-> <. n , y >. e. ( J " ( 1 ... K ) ) )
129	126, 128	syl6bbr 263	. . . . . . . . . . . . . . . . . 18 |- ( x e. { n } -> ( <. x , y >. e. ( J " ( 1 ... K ) ) <-> y e. ( ( J " ( 1 ... K ) ) " { n } ) ) )
130	129	pm5.32i 637	. . . . . . . . . . . . . . . . 17 |- ( ( x e. { n } /\ <. x , y >. e. ( J " ( 1 ... K ) ) ) <-> ( x e. { n } /\ y e. ( ( J " ( 1 ... K ) ) " { n } ) ) )
131	122, 123, 130	3bitr2ri 274	. . . . . . . . . . . . . . . 16 |- ( ( x e. { n } /\ y e. ( ( J " ( 1 ... K ) ) " { n } ) ) <-> <. x , y >. e. ( ( J " ( 1 ... K ) ) |` { n } ) )
132	120, 131	bitri 249	. . . . . . . . . . . . . . 15 |- ( <. x , y >. e. ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) <-> <. x , y >. e. ( ( J " ( 1 ... K ) ) |` { n } ) )
133	118, 119, 132	eqrelriiv 5097	. . . . . . . . . . . . . 14 |- ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) = ( ( J " ( 1 ... K ) ) |` { n } )
134		df-res 5011	. . . . . . . . . . . . . 14 |- ( ( J " ( 1 ... K ) ) |` { n } ) = ( ( J " ( 1 ... K ) ) i^i ( { n } X. _V ) )
135	133, 134	eqtri 2496	. . . . . . . . . . . . 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 4347	. . . . . . . . . . 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 4389	. . . . . . . . . . 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 2524	. . . . . . . . . 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 5110	. . . . . . . . . . . . . 14 |- Rel ( NN X. NN )
141		relss 5090	. . . . . . . . . . . . . 14 |- ( ( J " ( 1 ... K ) ) C_ ( NN X. NN ) -> ( Rel ( NN X. NN ) -> Rel ( J " ( 1 ... K ) ) ) )
142	26, 140, 141	mpisyl 18	. . . . . . . . . . . . 13 |- ( ph -> Rel ( J " ( 1 ... K ) ) )
143		ovoliun.l2	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> A. w e. ( 1 ... K ) ( 1st ` ( J ` w ) ) <_ L )
144		ffn 5731	. . . . . . . . . . . . . . . . . . . . 21 |- ( J : NN --> ( NN X. NN ) -> J Fn NN )
145	23, 144	syl 16	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> J Fn NN )
146		fveq2 5866	. . . . . . . . . . . . . . . . . . . . . 22 |- ( j = ( J ` w ) -> ( 1st ` j ) = ( 1st ` ( J ` w ) ) )
147	146	breq1d 4457	. . . . . . . . . . . . . . . . . . . . 21 |- ( j = ( J ` w ) -> ( ( 1st ` j ) <_ L <-> ( 1st ` ( J ` w ) ) <_ L ) )
148	147	ralima 6140	. . . . . . . . . . . . . . . . . . . 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 662	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( A. j e. ( J " ( 1 ... K ) ) ( 1st ` j ) <_ L <-> A. w e. ( 1 ... K ) ( 1st ` ( J ` w ) ) <_ L ) )
150	143, 149	mpbird 232	. . . . . . . . . . . . . . . . . 18 |- ( ph -> A. j e. ( J " ( 1 ... K ) ) ( 1st ` j ) <_ L )
151	150	r19.21bi 2833	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( 1st ` j ) <_ L )
152	29, 77	syl6eleq 2565	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( 1st ` j ) e. ( ZZ>= ` 1 ) )
153		ovoliun.l1	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> L e. ZZ )
154	153	adantr 465	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> L e. ZZ )
155		elfz5 11680	. . . . . . . . . . . . . . . . . 18 |- ( ( ( 1st ` j ) e. ( ZZ>= ` 1 ) /\ L e. ZZ ) -> ( ( 1st ` j ) e. ( 1 ... L ) <-> ( 1st ` j ) <_ L ) )
156	152, 154, 155	syl2anc 661	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( ( 1st ` j ) e. ( 1 ... L ) <-> ( 1st ` j ) <_ L ) )
157	151, 156	mpbird 232	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( 1st ` j ) e. ( 1 ... L ) )
158	157	ralrimiva 2878	. . . . . . . . . . . . . . 15 |- ( ph -> A. j e. ( J " ( 1 ... K ) ) ( 1st ` j ) e. ( 1 ... L ) )
159		vex 3116	. . . . . . . . . . . . . . . . . 18 |- x e. _V
160	159, 121	op1std 6794	. . . . . . . . . . . . . . . . 17 |- ( j = <. x , y >. -> ( 1st ` j ) = x )
161	160	eleq1d 2536	. . . . . . . . . . . . . . . 16 |- ( j = <. x , y >. -> ( ( 1st ` j ) e. ( 1 ... L ) <-> x e. ( 1 ... L ) ) )
162	161	rspccv 3211	. . . . . . . . . . . . . . 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 5029	. . . . . . . . . . . . . . 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 505	. . . . . . . . . . . . . . 15 |- ( x e. U_ n e. ( 1 ... L ) { n } <-> ( x e. U_ n e. ( 1 ... L ) { n } /\ y e. _V ) )
166		iunid 4380	. . . . . . . . . . . . . . . 16 |- U_ n e. ( 1 ... L ) { n } = ( 1 ... L )
167	166	eleq2i 2545	. . . . . . . . . . . . . . 15 |- ( x e. U_ n e. ( 1 ... L ) { n } <-> x e. ( 1 ... L ) )
168	164, 165, 167	3bitr2i 273	. . . . . . . . . . . . . 14 |- ( <. x , y >. e. ( U_ n e. ( 1 ... L ) { n } X. _V ) <-> x e. ( 1 ... L ) )
169	163, 168	syl6ibr 227	. . . . . . . . . . . . 13 |- ( ph -> ( <. x , y >. e. ( J " ( 1 ... K ) ) -> <. x , y >. e. ( U_ n e. ( 1 ... L ) { n } X. _V ) ) )
170	142, 169	relssdv 5095	. . . . . . . . . . . 12 |- ( ph -> ( J " ( 1 ... K ) ) C_ ( U_ n e. ( 1 ... L ) { n } X. _V ) )
171		xpiundir 5055	. . . . . . . . . . . 12 |- ( U_ n e. ( 1 ... L ) { n } X. _V ) = U_ n e. ( 1 ... L ) ( { n } X. _V )
172	170, 171	syl6sseq 3550	. . . . . . . . . . 11 |- ( ph -> ( J " ( 1 ... K ) ) C_ U_ n e. ( 1 ... L ) ( { n } X. _V ) )
173		df-ss 3490	. . . . . . . . . . 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 196	. . . . . . . . . 10 |- ( ph -> ( ( J " ( 1 ... K ) ) i^i U_ n e. ( 1 ... L ) ( { n } X. _V ) ) = ( J " ( 1 ... K ) ) )
175	139, 174	eqtrd 2508	. . . . . . . . 9 |- ( ph -> U_ n e. ( 1 ... L ) ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) = ( J " ( 1 ... K ) ) )
176	175, 98	eqeltrd 2555	. . . . . . . 8 |- ( ph -> U_ n e. ( 1 ... L ) ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) e. Fin )
177		ssiun2 4368	. . . . . . . 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 7740	. . . . . . . 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 477	. . . . . . 7 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) e. Fin )
180		2ndconst 6872	. . . . . . . . . 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 7534	. . . . . . . . 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 662	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) ~~ ( ( J " ( 1 ... K ) ) " { n } ) )
184	183	ensymd 7566	. . . . . . 7 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( ( J " ( 1 ... K ) ) " { n } ) ~~ ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) )
185		enfii 7737	. . . . . . 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 661	. . . . . 6 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( ( J " ( 1 ... K ) ) " { n } ) e. Fin )
187		ffvelrn 6019	. . . . . . . . . . . . . 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 477	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( F ` n ) e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) )
189	33, 34	elmap 7447	. . . . . . . . . . . . 13 |- ( ( F ` n ) e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) <-> ( F ` n ) : NN --> ( <_ i^i ( RR X. RR ) ) )
190	188, 189	sylib 196	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( F ` n ) : NN --> ( <_ i^i ( RR X. RR ) ) )
191	190	adantrr 716	. . . . . . . . . . 11 |- ( ( ph /\ ( n e. ( 1 ... L ) /\ i e. ( ( J " ( 1 ... K ) ) " { n } ) ) ) -> ( F ` n ) : NN --> ( <_ i^i ( RR X. RR ) ) )
192		imassrn 5348	. . . . . . . . . . . . . 14 |- ( ( J " ( 1 ... K ) ) " { n } ) C_ ran ( J " ( 1 ... K ) )
193	26	adantr 465	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( J " ( 1 ... K ) ) C_ ( NN X. NN ) )
194		rnss 5231	. . . . . . . . . . . . . . . 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 5440	. . . . . . . . . . . . . . 15 |- ran ( NN X. NN ) = NN
197	195, 196	syl6sseq 3550	. . . . . . . . . . . . . 14 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ran ( J " ( 1 ... K ) ) C_ NN )
198	192, 197	syl5ss 3515	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( ( J " ( 1 ... K ) ) " { n } ) C_ NN )
199	198	sseld 3503	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( i e. ( ( J " ( 1 ... K ) ) " { n } ) -> i e. NN ) )
200	199	impr 619	. . . . . . . . . . 11 |- ( ( ph /\ ( n e. ( 1 ... L ) /\ i e. ( ( J " ( 1 ... K ) ) " { n } ) ) ) -> i e. NN )
201	191, 200	ffvelrnd 6022	. . . . . . . . . 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 3502	. . . . . . . . 9 |- ( ( ph /\ ( n e. ( 1 ... L ) /\ i e. ( ( J " ( 1 ... K ) ) " { n } ) ) ) -> ( ( F ` n ) ` i ) e. ( RR X. RR ) )
203		xp2nd 6815	. . . . . . . . 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 6814	. . . . . . . . 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 9987	. . . . . . 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 648	. . . . . 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 13519	. . . . 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 477	. . . . . . 7 |- ( ( ph /\ n e. NN ) -> ( B / ( 2 ^ n ) ) e. RR )
211	102, 210	readdcld 9623	. . . . . 6 |- ( ( ph /\ n e. NN ) -> ( ( vol* ` A ) + ( B / ( 2 ^ n ) ) ) e. RR )
212	101, 211	sylan2 474	. . . . 5 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( ( vol* ` A ) + ( B / ( 2 ^ n ) ) ) e. RR )
213		eqid 2467	. . . . . . . . . . . 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 21647	. . . . . . . . . . 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 5737	. . . . . . . . . 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 11609	. . . . . . . . 9 |- ( 0 [,) +oo ) C_ RR*
220	218, 219	syl6ss 3516	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ran S C_ RR* )
221		supxrcl 11506	. . . . . . . 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 11323	. . . . . . . . 9 |- -oo e. RR*
224	223	a1i 11	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> -oo e. RR* )
225	103	rexrd 9643	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( vol* ` A ) e. RR* )
226		mnflt 11333	. . . . . . . . 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 474	. . . . . . . . 9 |- ( ( ph /\ n e. ( 1 ... L ) ) -> A C_ U. ran ( (,) o. ( F ` n ) ) )
230	214	ovollb 21653	. . . . . . . . 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 661	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( vol* ` A ) <_ sup ( ran S , RR* , < ) )
232	224, 225, 222, 227, 231	xrltletrd 11364	. . . . . . 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 474	. . . . . . 7 |- ( ( ph /\ n e. ( 1 ... L ) ) -> sup ( ran S , RR* , < ) <_ ( ( vol* ` A ) + ( B / ( 2 ^ n ) ) ) )
235		xrre 11370	. . . . . . 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 1229	. . . . . 6 |- ( ( ph /\ n e. ( 1 ... L ) ) -> sup ( ran S , RR* , < ) e. RR )
237		1zzd 10895	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> 1 e. ZZ )
238	213	ovolfsval 21645	. . . . . . . . 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 471	. . . . . . . 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 21646	. . . . . . . . . . . . 13 |- ( ( F ` n ) : NN --> ( <_ i^i ( RR X. RR ) ) -> ( ( abs o. - ) o. ( F ` n ) ) : NN --> ( 0 [,) +oo ) )
241	190, 240	syl 16	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( ( abs o. - ) o. ( F ` n ) ) : NN --> ( 0 [,) +oo ) )
242	241	ffvelrnda 6021	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. ( 1 ... L ) ) /\ i e. NN ) -> ( ( ( abs o. - ) o. ( F ` n ) ) ` i ) e. ( 0 [,) +oo ) )
243	239, 242	eqeltrrd 2556	. . . . . . . . . 10 |- ( ( ( ph /\ n e. ( 1 ... L ) ) /\ i e. NN ) -> ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) e. ( 0 [,) +oo ) )
244		elrege0 11627	. . . . . . . . . 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 196	. . . . . . . . 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 459	. . . . . . . 8 |- ( ( ( ph /\ n e. ( 1 ... L ) ) /\ i e. NN ) -> ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) e. RR )
247	245	simprd 463	. . . . . . . 8 |- ( ( ( ph /\ n e. ( 1 ... L ) ) /\ i e. NN ) -> 0 <_ ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) )
248		supxrub 11516	. . . . . . . . . . . . . . 15 |- ( ( ran S C_ RR* /\ z e. ran S ) -> z <_ sup ( ran S , RR* , < ) )
249	220, 248	sylan 471	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ n e. ( 1 ... L ) ) /\ z e. ran S ) -> z <_ sup ( ran S , RR* , < ) )
250	249	ralrimiva 2878	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. ( 1 ... L ) ) -> A. z e. ran S z <_ sup ( ran S , RR* , < ) )
251		breq2 4451	. . . . . . . . . . . . . . 15 |- ( x = sup ( ran S , RR* , < ) -> ( z <_ x <-> z <_ sup ( ran S , RR* , < ) ) )
252	251	ralbidv 2903	. . . . . . . . . . . . . 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 3214	. . . . . . . . . . . . 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 661	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( 1 ... L ) ) -> E. x e. RR A. z e. ran S z <_ x )
255		ffn 5731	. . . . . . . . . . . . . . 15 |- ( S : NN --> ( 0 [,) +oo ) -> S Fn NN )
256	216, 255	syl 16	. . . . . . . . . . . . . 14 |- ( ( ph /\ n e. ( 1 ... L ) ) -> S Fn NN )
257		breq1 4450	. . . . . . . . . . . . . . 15 |- ( z = ( S ` k ) -> ( z <_ x <-> ( S ` k ) <_ x ) )
258	257	ralrn 6024	. . . . . . . . . . . . . 14 |- ( S Fn NN -> ( A. z e. ran S z <_ x <-> A. k e. NN ( S ` k ) <_ x ) )
259	256, 258	syl 16	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( A. z e. ran S z <_ x <-> A. k e. NN ( S ` k ) <_ x ) )
260	259	rexbidv 2973	. . . . . . . . . . . 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 210	. . . . . . . . . . 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 13621	. . . . . . . . . 10 |- ( ( ph /\ n e. ( 1 ... L ) ) -> S ~~> sup ( ran S , RR , < ) )
263	214, 262	syl5eqbrr 4481	. . . . . . . . 9 |- ( ( ph /\ n e. ( 1 ... L ) ) -> seq 1 ( + , ( ( abs o. - ) o. ( F ` n ) ) ) ~~> sup ( ran S , RR , < ) )
264		climrel 13278	. . . . . . . . . 10 |- Rel ~~>
265	264	releldmi 5239	. . . . . . . . 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 16	. . . . . . . 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 13620	. . . . . . 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 13622	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> sum_ i e. NN ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) = sup ( ran S , RR , < ) )
269		0re 9596	. . . . . . . . . . 11 |- 0 e. RR
270		pnfxr 11321	. . . . . . . . . . 11 |- +oo e. RR*
271		icossre 11605	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ +oo e. RR* ) -> ( 0 [,) +oo ) C_ RR )
272	269, 270, 271	mp2an 672	. . . . . . . . . 10 |- ( 0 [,) +oo ) C_ RR
273	218, 272	syl6ss 3516	. . . . . . . . 9 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ran S C_ RR )
274		1nn 10547	. . . . . . . . . . . 12 |- 1 e. NN
275		fdm 5735	. . . . . . . . . . . . 13 |- ( S : NN --> ( 0 [,) +oo ) -> dom S = NN )
276	216, 275	syl 16	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( 1 ... L ) ) -> dom S = NN )
277	274, 276	syl5eleqr 2562	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( 1 ... L ) ) -> 1 e. dom S )
278		ne0i 3791	. . . . . . . . . . 11 |- ( 1 e. dom S -> dom S =/= (/) )
279	277, 278	syl 16	. . . . . . . . . 10 |- ( ( ph /\ n e. ( 1 ... L ) ) -> dom S =/= (/) )
280		dm0rn0 5219	. . . . . . . . . . 11 |- ( dom S = (/) <-> ran S = (/) )
281	280	necon3bii 2735	. . . . . . . . . 10 |- ( dom S =/= (/) <-> ran S =/= (/) )
282	279, 281	sylib 196	. . . . . . . . 9 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ran S =/= (/) )
283		supxrre 11519	. . . . . . . . 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 , < ) )
284	273, 282, 254, 283	syl3anc 1228	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> sup ( ran S , RR* , < ) = sup ( ran S , RR , < ) )
285	268, 284	eqtr4d 2511	. . . . . . 7 |- ( ( ph /\ n e. ( 1 ... L ) ) -> sum_ i e. NN ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) = sup ( ran S , RR* , < ) )
286	267, 285	breqtrd 4471	. . . . . 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* , < ) )
287	209, 236, 212, 286, 234	letrd 9738	. . . . 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 ) ) ) )
288	100, 209, 212, 287	fsumle 13576	. . . 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 ) ) ) )
289		vex 3116	. . . . . . . . . . 11 |- i e. _V
290	127, 289	op1std 6794	. . . . . . . . . 10 |- ( j = <. n , i >. -> ( 1st ` j ) = n )
291	290	fveq2d 5870	. . . . . . . . 9 |- ( j = <. n , i >. -> ( F ` ( 1st ` j ) ) = ( F ` n ) )
292	127, 289	op2ndd 6795	. . . . . . . . 9 |- ( j = <. n , i >. -> ( 2nd ` j ) = i )
293	291, 292	fveq12d 5872	. . . . . . . 8 |- ( j = <. n , i >. -> ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) = ( ( F ` n ) ` i ) )
294	293	fveq2d 5870	. . . . . . 7 |- ( j = <. n , i >. -> ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) = ( 2nd ` ( ( F ` n ) ` i ) ) )
295	293	fveq2d 5870	. . . . . . 7 |- ( j = <. n , i >. -> ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) = ( 1st ` ( ( F ` n ) ` i ) ) )
296	294, 295	oveq12d 6302	. . . . . 6 |- ( j = <. n , i >. -> ( ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) - ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) ) = ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) )
297	207	recnd 9622	. . . . . 6 |- ( ( ph /\ ( n e. ( 1 ... L ) /\ i e. ( ( J " ( 1 ... K ) ) " { n } ) ) ) -> ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) e. CC )
298	296, 100, 186, 297	fsum2d 13549	. . . . 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 ) ) ) ) )
299	175	sumeq1d 13486	. . . . 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 ) ) ) ) )
300	298, 299	eqtrd 2508	. . . 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 ) ) ) ) )
301	103	recnd 9622	. . . . 5 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( vol* ` A ) e. CC )
302	113	recnd 9622	. . . . 5 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( B / ( 2 ^ n ) ) e. CC )
303	100, 301, 302	fsumadd 13524	. . . 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 ) ) ) )
304	288, 300, 303	3brtr3d 4476	. . 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 ) ) ) )
305		1zzd 10895	. . . . . . . . 9 |- ( ph -> 1 e. ZZ )
306		simpr 461	. . . . . . . . . . 11 |- ( ( ph /\ n e. NN ) -> n e. NN )
307		ovoliun.g	. . . . . . . . . . . 12 |- G = ( n e. NN |-> ( vol* ` A ) )
308	307	fvmpt2 5957	. . . . . . . . . . 11 |- ( ( n e. NN /\ ( vol* ` A ) e. RR ) -> ( G ` n ) = ( vol* ` A ) )
309	306, 102, 308	syl2anc 661	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( G ` n ) = ( vol* ` A ) )
310	309, 102	eqeltrd 2555	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( G ` n ) e. RR )
311	77, 305, 310	serfre 12104	. . . . . . . 8 |- ( ph -> seq 1 ( + , G ) : NN --> RR )
312		ovoliun.t	. . . . . . . . 9 |- T = seq 1 ( + , G )
313	312	feq1i 5723	. . . . . . . 8 |- ( T : NN --> RR <-> seq 1 ( + , G ) : NN --> RR )
314	311, 313	sylibr 212	. . . . . . 7 |- ( ph -> T : NN --> RR )
315		frn 5737	. . . . . . 7 |- ( T : NN --> RR -> ran T C_ RR )
316	314, 315	syl 16	. . . . . 6 |- ( ph -> ran T C_ RR )
317		ressxr 9637	. . . . . 6 |- RR C_ RR*
318	316, 317	syl6ss 3516	. . . . 5 |- ( ph -> ran T C_ RR* )
319	101, 309	sylan2 474	. . . . . . 7 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( G ` n ) = ( vol* ` A ) )
320		1red 9611	. . . . . . . . . 10 |- ( ph -> 1 e. RR )
321		ffvelrn 6019	. . . . . . . . . . . . 13 |- ( ( J : NN --> ( NN X. NN ) /\ 1 e. NN ) -> ( J ` 1 ) e. ( NN X. NN ) )
322	23, 274, 321	sylancl 662	. . . . . . . . . . . 12 |- ( ph -> ( J ` 1 ) e. ( NN X. NN ) )
323		xp1st 6814	. . . . . . . . . . . 12 |- ( ( J ` 1 ) e. ( NN X. NN ) -> ( 1st ` ( J ` 1 ) ) e. NN )
324	322, 323	syl 16	. . . . . . . . . . 11 |- ( ph -> ( 1st ` ( J ` 1 ) ) e. NN )
325	324	nnred 10551	. . . . . . . . . 10 |- ( ph -> ( 1st ` ( J ` 1 ) ) e. RR )
326	153	zred 10966	. . . . . . . . . 10 |- ( ph -> L e. RR )
327	324	nnge1d 10578	. . . . . . . . . 10 |- ( ph -> 1 <_ ( 1st ` ( J ` 1 ) ) )
328		eluzfz1 11693	. . . . . . . . . . . 12 |- ( K e. ( ZZ>= ` 1 ) -> 1 e. ( 1 ... K ) )
329	78, 328	syl 16	. . . . . . . . . . 11 |- ( ph -> 1 e. ( 1 ... K ) )
330		fveq2 5866	. . . . . . . . . . . . . 14 |- ( w = 1 -> ( J ` w ) = ( J ` 1 ) )
331	330	fveq2d 5870	. . . . . . . . . . . . 13 |- ( w = 1 -> ( 1st ` ( J ` w ) ) = ( 1st ` ( J ` 1 ) ) )
332	331	breq1d 4457	. . . . . . . . . . . 12 |- ( w = 1 -> ( ( 1st ` ( J ` w ) ) <_ L <-> ( 1st ` ( J ` 1 ) ) <_ L ) )
333	332	rspcv 3210	. . . . . . . . . . 11 |- ( 1 e. ( 1 ... K ) -> ( A. w e. ( 1 ... K ) ( 1st ` ( J ` w ) ) <_ L -> ( 1st ` ( J ` 1 ) ) <_ L ) )
334	329, 143, 333	sylc 60	. . . . . . . . . 10 |- ( ph -> ( 1st ` ( J ` 1 ) ) <_ L )
335	320, 325, 326, 327, 334	letrd 9738	. . . . . . . . 9 |- ( ph -> 1 <_ L )
336		elnnz1 10890	. . . . . . . . 9 |- ( L e. NN <-> ( L e. ZZ /\ 1 <_ L ) )
337	153, 335, 336	sylanbrc 664	. . . . . . . 8 |- ( ph -> L e. NN )
338	337, 77	syl6eleq 2565	. . . . . . 7 |- ( ph -> L e. ( ZZ>= ` 1 ) )
339	319, 338, 301	fsumser 13515	. . . . . 6 |- ( ph -> sum_ n e. ( 1 ... L ) ( vol* ` A ) = ( seq 1 ( + , G ) ` L ) )
340		seqfn 12087	. . . . . . . . 9 |- ( 1 e. ZZ -> seq 1 ( + , G ) Fn ( ZZ>= ` 1 ) )
341	305, 340	syl 16	. . . . . . . 8 |- ( ph -> seq 1 ( + , G ) Fn ( ZZ>= ` 1 ) )
342		fnfvelrn 6018	. . . . . . . 8 |- ( ( seq 1 ( + , G ) Fn ( ZZ>= ` 1 ) /\ L e. ( ZZ>= ` 1 ) ) -> ( seq 1 ( + , G ) ` L ) e. ran seq 1 ( + , G ) )
343	341, 338, 342	syl2anc 661	. . . . . . 7 |- ( ph -> ( seq 1 ( + , G ) ` L ) e. ran seq 1 ( + , G ) )
344	312	rneqi 5229	. . . . . . 7 |- ran T = ran seq 1 ( + , G )
345	343, 344	syl6eleqr 2566	. . . . . 6 |- ( ph -> ( seq 1 ( + , G ) ` L ) e. ran T )
346	339, 345	eqeltrd 2555	. . . . 5 |- ( ph -> sum_ n e. ( 1 ... L ) ( vol* ` A ) e. ran T )
347		supxrub 11516	. . . . 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* , < ) )
348	318, 346, 347	syl2anc 661	. . . 4 |- ( ph -> sum_ n e. ( 1 ... L ) ( vol* ` A ) <_ sup ( ran T , RR* , < ) )
349	106	recnd 9622	. . . . . 6 |- ( ph -> B e. CC )
350		geo2sum 13645	. . . . . 6 |- ( ( L e. NN /\ B e. CC ) -> sum_ n e. ( 1 ... L ) ( B / ( 2 ^ n ) ) = ( B - ( B / ( 2 ^ L ) ) ) )
351	337, 349, 350	syl2anc 661	. . . . 5 |- ( ph -> sum_ n e. ( 1 ... L ) ( B / ( 2 ^ n ) ) = ( B - ( B / ( 2 ^ L ) ) ) )
352	337	nnnn0d 10852	. . . . . . . . . 10 |- ( ph -> L e. NN0 )
353		nnexpcl 12147	. . . . . . . . . 10 |- ( ( 2 e. NN /\ L e. NN0 ) -> ( 2 ^ L ) e. NN )
354	107, 352, 353	sylancr 663	. . . . . . . . 9 |- ( ph -> ( 2 ^ L ) e. NN )
355	354	nnrpd 11255	. . . . . . . 8 |- ( ph -> ( 2 ^ L ) e. RR+ )
356	105, 355	rpdivcld 11273	. . . . . . 7 |- ( ph -> ( B / ( 2 ^ L ) ) e. RR+ )
357	356	rpge0d 11260	. . . . . 6 |- ( ph -> 0 <_ ( B / ( 2 ^ L ) ) )
358	106, 354	nndivred 10584	. . . . . . 7 |- ( ph -> ( B / ( 2 ^ L ) ) e. RR )
359	106, 358	subge02d 10144	. . . . . 6 |- ( ph -> ( 0 <_ ( B / ( 2 ^ L ) ) <-> ( B - ( B / ( 2 ^ L ) ) ) <_ B ) )
360	357, 359	mpbid 210	. . . . 5 |- ( ph -> ( B - ( B / ( 2 ^ L ) ) ) <_ B )
361	351, 360	eqbrtrd 4467	. . . 4 |- ( ph -> sum_ n e. ( 1 ... L ) ( B / ( 2 ^ n ) ) <_ B )
362	104, 114, 116, 106, 348, 361	le2addd 10170	. . 3 |- ( ph -> ( sum_ n e. ( 1 ... L ) ( vol* ` A ) + sum_ n e. ( 1 ... L ) ( B / ( 2 ^ n ) ) ) <_ ( sup ( ran T , RR* , < ) + B ) )
363	99, 115, 117, 304, 362	letrd 9738	. 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 ) )
364	93, 363	eqbrtrrd 4469	1 |- ( ph -> ( U ` K ) <_ ( sup ( ran T , RR* , < ) + B ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   =/= wne 2662   A.wral 2814   E.wrex 2815   _Vcvv 3113   i^i cin 3475   C_ wss 3476   (/)c0 3785   {csn 4027   <.cop 4033   U.cuni 4245   U_ciun 4325   class class class wbr 4447   |-> cmpt 4505   X. cxp 4997   dom cdm 4999   ran crn 5000   |` cres 5001   "cima 5002   o. ccom 5003   Rel wrel 5004   Fn wfn 5583   -->wf 5584   -1-1->wf1 5585   -1-1-onto->wf1o 5587   ` cfv 5588  (class class class)co 6284   1stc1st 6782   2ndc2nd 6783   ^m cmap 7420   ~~ cen 7513   Fincfn 7516   supcsup 7900   CCcc 9490   RRcr 9491   0cc0 9492   1c1 9493   + caddc 9495   +oocpnf 9625   -oocmnf 9626   RR*cxr 9627   < clt 9628   <_ cle 9629   - cmin 9805   / cdiv 10206   NNcn 10536   2c2 10585   NN0cn0 10795   ZZcz 10864   ZZ>=cuz 11082   RR+crp 11220   (,)cioo 11529   [,)cico 11531   ...cfz 11672   seqcseq 12075   ^cexp 12134   abscabs 13030   ~~> cli 13270   sum_csu 13471   vol*covol 21637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-oi 7935  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-rp 11221  df-ioo 11533  df-ico 11535  df-fz 11673  df-fzo 11793  df-fl 11897  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-rlim 13275  df-sum 13472  df-ovol 21639

This theorem is referenced by:  ovoliunlem2  21677