Theorem ovoliunlem1 22447

Description: Lemma for ovoliun 22450. (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 5879	. . . . . . . . 9 |- ( j = ( J ` m ) -> ( 1st ` j ) = ( 1st ` ( J ` m ) ) )
2	1	fveq2d 5883	. . . . . . . 8 |- ( j = ( J ` m ) -> ( F ` ( 1st ` j ) ) = ( F ` ( 1st ` ( J ` m ) ) ) )
3		fveq2 5879	. . . . . . . 8 |- ( j = ( J ` m ) -> ( 2nd ` j ) = ( 2nd ` ( J ` m ) ) )
4	2, 3	fveq12d 5885	. . . . . . 7 |- ( j = ( J ` m ) -> ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) = ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) )
5	4	fveq2d 5883	. . . . . 6 |- ( j = ( J ` m ) -> ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) = ( 2nd ` ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) ) )
6	4	fveq2d 5883	. . . . . 6 |- ( j = ( J ` m ) -> ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) = ( 1st ` ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) ) )
7	5, 6	oveq12d 6321	. . . . 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 12187	. . . . 5 |- ( ph -> ( 1 ... K ) e. Fin )
9		ovoliun.j	. . . . . . 7 |- ( ph -> J : NN -1-1-onto-> ( NN X. NN ) )
10		f1of1 5828	. . . . . . 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 11830	. . . . . . 7 |- ( m e. ( 1 ... K ) -> m e. NN )
13	12	ssriv 3469	. . . . . 6 |- ( 1 ... K ) C_ NN
14		f1ores 5843	. . . . . 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 667	. . . . 5 |- ( ph -> ( J |` ( 1 ... K ) ) : ( 1 ... K ) -1-1-onto-> ( J " ( 1 ... K ) ) )
16		fvres 5893	. . . . . 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 3684	. . . . . . . . 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 5196	. . . . . . . . . . . . . . 15 |- ( J " ( 1 ... K ) ) C_ ran J
22		f1of 5829	. . . . . . . . . . . . . . . . 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 5750	. . . . . . . . . . . . . . . 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 3476	. . . . . . . . . . . . . 14 |- ( ph -> ( J " ( 1 ... K ) ) C_ ( NN X. NN ) )
27	26	sselda 3465	. . . . . . . . . . . . 13 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> j e. ( NN X. NN ) )
28		xp1st 6835	. . . . . . . . . . . . 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 6036	. . . . . . . . . . 11 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( F ` ( 1st ` j ) ) e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) )
31		reex 9632	. . . . . . . . . . . . . 14 |- RR e. _V
32	31, 31	xpex 6607	. . . . . . . . . . . . 13 |- ( RR X. RR ) e. _V
33	32	inex2 4564	. . . . . . . . . . . 12 |- ( <_ i^i ( RR X. RR ) ) e. _V
34		nnex 10617	. . . . . . . . . . . 12 |- NN e. _V
35	33, 34	elmap 7506	. . . . . . . . . . 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 6836	. . . . . . . . . . 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 6036	. . . . . . . . 9 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) e. ( <_ i^i ( RR X. RR ) ) )
40	18, 39	sseldi 3463	. . . . . . . 8 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) e. ( RR X. RR ) )
41		xp2nd 6836	. . . . . . . 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 6835	. . . . . . . 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 10049	. . . . . 6 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) - ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) ) e. RR )
46	45	recnd 9671	. . . . 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 13782	. . . 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 6035	. . . . . . . . . . . 12 |- ( ( ph /\ k e. NN ) -> ( J ` k ) e. ( NN X. NN ) )
50		xp1st 6835	. . . . . . . . . . . 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 6036	. . . . . . . . . 10 |- ( ( ph /\ k e. NN ) -> ( F ` ( 1st ` ( J ` k ) ) ) e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) )
53	33, 34	elmap 7506	. . . . . . . . . 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 6836	. . . . . . . . . 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 6036	. . . . . . . 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 6059	. . . . . . 7 |- ( ph -> H : NN --> ( <_ i^i ( RR X. RR ) ) )
60		eqid 2423	. . . . . . . 8 |- ( ( abs o. - ) o. H ) = ( ( abs o. - ) o. H )
61	60	ovolfsval 22415	. . . . . . 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 5879	. . . . . . . . . . . . 13 |- ( k = m -> ( J ` k ) = ( J ` m ) )
65	64	fveq2d 5883	. . . . . . . . . . . 12 |- ( k = m -> ( 1st ` ( J ` k ) ) = ( 1st ` ( J ` m ) ) )
66	65	fveq2d 5883	. . . . . . . . . . 11 |- ( k = m -> ( F ` ( 1st ` ( J ` k ) ) ) = ( F ` ( 1st ` ( J ` m ) ) ) )
67	64	fveq2d 5883	. . . . . . . . . . 11 |- ( k = m -> ( 2nd ` ( J ` k ) ) = ( 2nd ` ( J ` m ) ) )
68	66, 67	fveq12d 5885	. . . . . . . . . 10 |- ( k = m -> ( ( F ` ( 1st ` ( J ` k ) ) ) ` ( 2nd ` ( J ` k ) ) ) = ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) )
69		fvex 5889	. . . . . . . . . 10 |- ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) e. _V
70	68, 58, 69	fvmpt 5962	. . . . . . . . 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 5883	. . . . . . 7 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( 2nd ` ( H ` m ) ) = ( 2nd ` ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) ) )
73	71	fveq2d 5883	. . . . . . 7 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( 1st ` ( H ` m ) ) = ( 1st ` ( ( F ` ( 1st ` ( J ` m ) ) ) ` ( 2nd ` ( J ` m ) ) ) ) )
74	72, 73	oveq12d 6321	. . . . . 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 2464	. . . . 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 11196	. . . . . 6 |- NN = ( ZZ>= ` 1 )
78	76, 77	syl6eleq 2521	. . . . 5 |- ( ph -> K e. ( ZZ>= ` 1 ) )
79		ffvelrn 6033	. . . . . . . . . . 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 3463	. . . . . . . . 9 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( H ` m ) e. ( RR X. RR ) )
82		xp2nd 6836	. . . . . . . . 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 6835	. . . . . . . . 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 10049	. . . . . . 7 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( ( 2nd ` ( H ` m ) ) - ( 1st ` ( H ` m ) ) ) e. RR )
87	86	recnd 9671	. . . . . 6 |- ( ( ph /\ m e. ( 1 ... K ) ) -> ( ( 2nd ` ( H ` m ) ) - ( 1st ` ( H ` m ) ) ) e. CC )
88	74, 87	eqeltrrd 2512	. . . . 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 13789	. . . 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 2464	. . 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 5880	. . 3 |- ( U ` K ) = ( seq 1 ( + , ( ( abs o. - ) o. H ) ) ` K )
93	90, 92	syl6eqr 2482	. 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 666	. . . . . 6 |- ( ph -> ( 1 ... K ) ~~ ( J " ( 1 ... K ) ) )
96	95	ensymd 7625	. . . . 5 |- ( ph -> ( J " ( 1 ... K ) ) ~~ ( 1 ... K ) )
97		enfii 7793	. . . . 5 |- ( ( ( 1 ... K ) e. Fin /\ ( J " ( 1 ... K ) ) ~~ ( 1 ... K ) ) -> ( J " ( 1 ... K ) ) e. Fin )
98	8, 96, 97	syl2anc 666	. . . 4 |- ( ph -> ( J " ( 1 ... K ) ) e. Fin )
99	98, 45	fsumrecl 13793	. . 3 |- ( ph -> sum_ j e. ( J " ( 1 ... K ) ) ( ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) - ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) ) e. RR )
100		fzfid 12187	. . . . 5 |- ( ph -> ( 1 ... L ) e. Fin )
101		elfznn 11830	. . . . . 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 13793	. . . 4 |- ( ph -> sum_ n e. ( 1 ... L ) ( vol* ` A ) e. RR )
105		ovoliun.b	. . . . . . 7 |- ( ph -> B e. RR+ )
106	105	rpred 11343	. . . . . 6 |- ( ph -> B e. RR )
107		2nn 10769	. . . . . . . 8 |- 2 e. NN
108		nnnn0 10878	. . . . . . . 8 |- ( n e. NN -> n e. NN0 )
109		nnexpcl 12286	. . . . . . . 8 |- ( ( 2 e. NN /\ n e. NN0 ) -> ( 2 ^ n ) e. NN )
110	107, 108, 109	sylancr 668	. . . . . . 7 |- ( n e. NN -> ( 2 ^ n ) e. NN )
111	101, 110	syl 17	. . . . . 6 |- ( n e. ( 1 ... L ) -> ( 2 ^ n ) e. NN )
112		nndivre 10647	. . . . . 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 13793	. . . 4 |- ( ph -> sum_ n e. ( 1 ... L ) ( B / ( 2 ^ n ) ) e. RR )
115	104, 114	readdcld 9672	. . 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 9672	. . 3 |- ( ph -> ( sup ( ran T , RR* , < ) + B ) e. RR )
118		relxp 4959	. . . . . . . . . . . . . . 15 |- Rel ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) )
119		relres 5149	. . . . . . . . . . . . . . 15 |- Rel ( ( J " ( 1 ... K ) ) |` { n } )
120		opelxp 4881	. . . . . . . . . . . . . . . 16 |- ( <. x , y >. e. ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) <-> ( x e. { n } /\ y e. ( ( J " ( 1 ... K ) ) " { n } ) ) )
121		vex 3085	. . . . . . . . . . . . . . . . . 18 |- y e. _V
122	121	opelres 5127	. . . . . . . . . . . . . . . . 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 4022	. . . . . . . . . . . . . . . . . . . . 21 |- ( x e. { n } -> x = n )
125	124	opeq1d 4191	. . . . . . . . . . . . . . . . . . . 20 |- ( x e. { n } -> <. x , y >. = <. n , y >. )
126	125	eleq1d 2492	. . . . . . . . . . . . . . . . . . 19 |- ( x e. { n } -> ( <. x , y >. e. ( J " ( 1 ... K ) ) <-> <. n , y >. e. ( J " ( 1 ... K ) ) ) )
127		vex 3085	. . . . . . . . . . . . . . . . . . . 20 |- n e. _V
128	127, 121	elimasn 5210	. . . . . . . . . . . . . . . . . . 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 642	. . . . . . . . . . . . . . . . 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 4946	. . . . . . . . . . . . . 14 |- ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) = ( ( J " ( 1 ... K ) ) |` { n } )
134		df-res 4863	. . . . . . . . . . . . . 14 |- ( ( J " ( 1 ... K ) ) |` { n } ) = ( ( J " ( 1 ... K ) ) i^i ( { n } X. _V ) )
135	133, 134	eqtri 2452	. . . . . . . . . . . . 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 4319	. . . . . . . . . . 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 4361	. . . . . . . . . . 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 2480	. . . . . . . . . 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 4959	. . . . . . . . . . . . . 14 |- Rel ( NN X. NN )
141		relss 4939	. . . . . . . . . . . . . 14 |- ( ( J " ( 1 ... K ) ) C_ ( NN X. NN ) -> ( Rel ( NN X. NN ) -> Rel ( J " ( 1 ... K ) ) ) )
142	26, 140, 141	mpisyl 22	. . . . . . . . . . . . 13 |- ( ph -> Rel ( J " ( 1 ... K ) ) )
143		ovoliun.l2	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> A. w e. ( 1 ... K ) ( 1st ` ( J ` w ) ) <_ L )
144		ffn 5744	. . . . . . . . . . . . . . . . . . . . 21 |- ( J : NN --> ( NN X. NN ) -> J Fn NN )
145	23, 144	syl 17	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> J Fn NN )
146		fveq2 5879	. . . . . . . . . . . . . . . . . . . . . 22 |- ( j = ( J ` w ) -> ( 1st ` j ) = ( 1st ` ( J ` w ) ) )
147	146	breq1d 4431	. . . . . . . . . . . . . . . . . . . . 21 |- ( j = ( J ` w ) -> ( ( 1st ` j ) <_ L <-> ( 1st ` ( J ` w ) ) <_ L ) )
148	147	ralima 6158	. . . . . . . . . . . . . . . . . . . 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 667	. . . . . . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ j e. ( J " ( 1 ... K ) ) ) -> ( 1st ` j ) <_ L )
152	29, 77	syl6eleq 2521	. . . . . . . . . . . . . . . . . 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 11794	. . . . . . . . . . . . . . . . . 18 |- ( ( ( 1st ` j ) e. ( ZZ>= ` 1 ) /\ L e. ZZ ) -> ( ( 1st ` j ) e. ( 1 ... L ) <-> ( 1st ` j ) <_ L ) )
156	152, 154, 155	syl2anc 666	. . . . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . . 15 |- ( ph -> A. j e. ( J " ( 1 ... K ) ) ( 1st ` j ) e. ( 1 ... L ) )
159		vex 3085	. . . . . . . . . . . . . . . . . 18 |- x e. _V
160	159, 121	op1std 6815	. . . . . . . . . . . . . . . . 17 |- ( j = <. x , y >. -> ( 1st ` j ) = x )
161	160	eleq1d 2492	. . . . . . . . . . . . . . . 16 |- ( j = <. x , y >. -> ( ( 1st ` j ) e. ( 1 ... L ) <-> x e. ( 1 ... L ) ) )
162	161	rspccv 3180	. . . . . . . . . . . . . . 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 4881	. . . . . . . . . . . . . . 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 4352	. . . . . . . . . . . . . . . 16 |- U_ n e. ( 1 ... L ) { n } = ( 1 ... L )
167	166	eleq2i 2501	. . . . . . . . . . . . . . 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 4944	. . . . . . . . . . . 12 |- ( ph -> ( J " ( 1 ... K ) ) C_ ( U_ n e. ( 1 ... L ) { n } X. _V ) )
171		xpiundir 4907	. . . . . . . . . . . 12 |- ( U_ n e. ( 1 ... L ) { n } X. _V ) = U_ n e. ( 1 ... L ) ( { n } X. _V )
172	170, 171	syl6sseq 3511	. . . . . . . . . . 11 |- ( ph -> ( J " ( 1 ... K ) ) C_ U_ n e. ( 1 ... L ) ( { n } X. _V ) )
173		df-ss 3451	. . . . . . . . . . 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 2464	. . . . . . . . 9 |- ( ph -> U_ n e. ( 1 ... L ) ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) = ( J " ( 1 ... K ) ) )
176	175, 98	eqeltrd 2511	. . . . . . . 8 |- ( ph -> U_ n e. ( 1 ... L ) ( { n } X. ( ( J " ( 1 ... K ) ) " { n } ) ) e. Fin )
177		ssiun2 4340	. . . . . . . 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 7796	. . . . . . . 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 6894	. . . . . . . . . 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 667	. . . . . . . 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 7793	. . . . . . 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 666	. . . . . 6 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( ( J " ( 1 ... K ) ) " { n } ) e. Fin )
187		ffvelrn 6033	. . . . . . . . . . . . . 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 7506	. . . . . . . . . . . . 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 722	. . . . . . . . . . 11 |- ( ( ph /\ ( n e. ( 1 ... L ) /\ i e. ( ( J " ( 1 ... K ) ) " { n } ) ) ) -> ( F ` n ) : NN --> ( <_ i^i ( RR X. RR ) ) )
192		imassrn 5196	. . . . . . . . . . . . . 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 5080	. . . . . . . . . . . . . . . 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 5287	. . . . . . . . . . . . . . 15 |- ran ( NN X. NN ) = NN
197	195, 196	syl6sseq 3511	. . . . . . . . . . . . . 14 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ran ( J " ( 1 ... K ) ) C_ NN )
198	192, 197	syl5ss 3476	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( ( J " ( 1 ... K ) ) " { n } ) C_ NN )
199	198	sseld 3464	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( i e. ( ( J " ( 1 ... K ) ) " { n } ) -> i e. NN ) )
200	199	impr 624	. . . . . . . . . . 11 |- ( ( ph /\ ( n e. ( 1 ... L ) /\ i e. ( ( J " ( 1 ... K ) ) " { n } ) ) ) -> i e. NN )
201	191, 200	ffvelrnd 6036	. . . . . . . . . 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 3463	. . . . . . . . 9 |- ( ( ph /\ ( n e. ( 1 ... L ) /\ i e. ( ( J " ( 1 ... K ) ) " { n } ) ) ) -> ( ( F ` n ) ` i ) e. ( RR X. RR ) )
203		xp2nd 6836	. . . . . . . . 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 6835	. . . . . . . . 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 10049	. . . . . . 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 653	. . . . . 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 13793	. . . . 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 9672	. . . . . 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 2423	. . . . . . . . . . . 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 22417	. . . . . . . . . . 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 5750	. . . . . . . . . 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 11721	. . . . . . . . 9 |- ( 0 [,) +oo ) C_ RR*
220	218, 219	syl6ss 3477	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ran S C_ RR* )
221		supxrcl 11602	. . . . . . . 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 11416	. . . . . . . . 9 |- -oo e. RR*
224	223	a1i 11	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> -oo e. RR* )
225	103	rexrd 9692	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( vol* ` A ) e. RR* )
226		mnflt 11427	. . . . . . . . 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 22424	. . . . . . . . 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 666	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( vol* ` A ) <_ sup ( ran S , RR* , < ) )
232	224, 225, 222, 227, 231	xrltletrd 11460	. . . . . . 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 11466	. . . . . . 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 1266	. . . . . 6 |- ( ( ph /\ n e. ( 1 ... L ) ) -> sup ( ran S , RR* , < ) e. RR )
237		1zzd 10970	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> 1 e. ZZ )
238	213	ovolfsval 22415	. . . . . . . . 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 22416	. . . . . . . . . . . . 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 6035	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. ( 1 ... L ) ) /\ i e. NN ) -> ( ( ( abs o. - ) o. ( F ` n ) ) ` i ) e. ( 0 [,) +oo ) )
243	239, 242	eqeltrrd 2512	. . . . . . . . . 10 |- ( ( ( ph /\ n e. ( 1 ... L ) ) /\ i e. NN ) -> ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) e. ( 0 [,) +oo ) )
244		elrege0 11740	. . . . . . . . . 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 11612	. . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. ( 1 ... L ) ) -> A. z e. ran S z <_ sup ( ran S , RR* , < ) )
251		breq2 4425	. . . . . . . . . . . . . . 15 |- ( x = sup ( ran S , RR* , < ) -> ( z <_ x <-> z <_ sup ( ran S , RR* , < ) ) )
252	251	ralbidv 2865	. . . . . . . . . . . . . 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 3183	. . . . . . . . . . . . 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 666	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( 1 ... L ) ) -> E. x e. RR A. z e. ran S z <_ x )
255		ffn 5744	. . . . . . . . . . . . . . 15 |- ( S : NN --> ( 0 [,) +oo ) -> S Fn NN )
256	216, 255	syl 17	. . . . . . . . . . . . . 14 |- ( ( ph /\ n e. ( 1 ... L ) ) -> S Fn NN )
257		breq1 4424	. . . . . . . . . . . . . . 15 |- ( z = ( S ` k ) -> ( z <_ x <-> ( S ` k ) <_ x ) )
258	257	ralrn 6038	. . . . . . . . . . . . . 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 2940	. . . . . . . . . . . 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 13897	. . . . . . . . . 10 |- ( ( ph /\ n e. ( 1 ... L ) ) -> S ~~> sup ( ran S , RR , < ) )
263	214, 262	syl5eqbrr 4456	. . . . . . . . 9 |- ( ( ph /\ n e. ( 1 ... L ) ) -> seq 1 ( + , ( ( abs o. - ) o. ( F ` n ) ) ) ~~> sup ( ran S , RR , < ) )
264		climrel 13549	. . . . . . . . . 10 |- Rel ~~>
265	264	releldmi 5088	. . . . . . . . 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 13896	. . . . . . 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 13898	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> sum_ i e. NN ( ( 2nd ` ( ( F ` n ) ` i ) ) - ( 1st ` ( ( F ` n ) ` i ) ) ) = sup ( ran S , RR , < ) )
269		rge0ssre 11742	. . . . . . . . . 10 |- ( 0 [,) +oo ) C_ RR
270	218, 269	syl6ss 3477	. . . . . . . . 9 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ran S C_ RR )
271		1nn 10622	. . . . . . . . . . . 12 |- 1 e. NN
272		fdm 5748	. . . . . . . . . . . . 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 2518	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( 1 ... L ) ) -> 1 e. dom S )
275		ne0i 3768	. . . . . . . . . . 11 |- ( 1 e. dom S -> dom S =/= (/) )
276	274, 275	syl 17	. . . . . . . . . 10 |- ( ( ph /\ n e. ( 1 ... L ) ) -> dom S =/= (/) )
277		dm0rn0 5068	. . . . . . . . . . 11 |- ( dom S = (/) <-> ran S = (/) )
278	277	necon3bii 2693	. . . . . . . . . 10 |- ( dom S =/= (/) <-> ran S =/= (/) )
279	276, 278	sylib 200	. . . . . . . . 9 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ran S =/= (/) )
280		supxrre 11615	. . . . . . . . 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 1265	. . . . . . . 8 |- ( ( ph /\ n e. ( 1 ... L ) ) -> sup ( ran S , RR* , < ) = sup ( ran S , RR , < ) )
282	268, 281	eqtr4d 2467	. . . . . . 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 4446	. . . . . 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 9794	. . . . 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 13852	. . . 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 3085	. . . . . . . . . . 11 |- i e. _V
287	127, 286	op1std 6815	. . . . . . . . . 10 |- ( j = <. n , i >. -> ( 1st ` j ) = n )
288	287	fveq2d 5883	. . . . . . . . 9 |- ( j = <. n , i >. -> ( F ` ( 1st ` j ) ) = ( F ` n ) )
289	127, 286	op2ndd 6816	. . . . . . . . 9 |- ( j = <. n , i >. -> ( 2nd ` j ) = i )
290	288, 289	fveq12d 5885	. . . . . . . 8 |- ( j = <. n , i >. -> ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) = ( ( F ` n ) ` i ) )
291	290	fveq2d 5883	. . . . . . 7 |- ( j = <. n , i >. -> ( 2nd ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) = ( 2nd ` ( ( F ` n ) ` i ) ) )
292	290	fveq2d 5883	. . . . . . 7 |- ( j = <. n , i >. -> ( 1st ` ( ( F ` ( 1st ` j ) ) ` ( 2nd ` j ) ) ) = ( 1st ` ( ( F ` n ) ` i ) ) )
293	291, 292	oveq12d 6321	. . . . . 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 9671	. . . . . 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 13825	. . . . 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 13760	. . . . 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 2464	. . . 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 9671	. . . . 5 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( vol* ` A ) e. CC )
299	113	recnd 9671	. . . . 5 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( B / ( 2 ^ n ) ) e. CC )
300	100, 298, 299	fsumadd 13798	. . . 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 4451	. . 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 10970	. . . . . . . . 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 5971	. . . . . . . . . . 11 |- ( ( n e. NN /\ ( vol* ` A ) e. RR ) -> ( G ` n ) = ( vol* ` A ) )
306	303, 102, 305	syl2anc 666	. . . . . . . . . 10 |- ( ( ph /\ n e. NN ) -> ( G ` n ) = ( vol* ` A ) )
307	306, 102	eqeltrd 2511	. . . . . . . . 9 |- ( ( ph /\ n e. NN ) -> ( G ` n ) e. RR )
308	77, 302, 307	serfre 12243	. . . . . . . 8 |- ( ph -> seq 1 ( + , G ) : NN --> RR )
309		ovoliun.t	. . . . . . . . 9 |- T = seq 1 ( + , G )
310	309	feq1i 5736	. . . . . . . 8 |- ( T : NN --> RR <-> seq 1 ( + , G ) : NN --> RR )
311	308, 310	sylibr 216	. . . . . . 7 |- ( ph -> T : NN --> RR )
312		frn 5750	. . . . . . 7 |- ( T : NN --> RR -> ran T C_ RR )
313	311, 312	syl 17	. . . . . 6 |- ( ph -> ran T C_ RR )
314		ressxr 9686	. . . . . 6 |- RR C_ RR*
315	313, 314	syl6ss 3477	. . . . 5 |- ( ph -> ran T C_ RR* )
316	101, 306	sylan2 477	. . . . . . 7 |- ( ( ph /\ n e. ( 1 ... L ) ) -> ( G ` n ) = ( vol* ` A ) )
317		1red 9660	. . . . . . . . . 10 |- ( ph -> 1 e. RR )
318		ffvelrn 6033	. . . . . . . . . . . . 13 |- ( ( J : NN --> ( NN X. NN ) /\ 1 e. NN ) -> ( J ` 1 ) e. ( NN X. NN ) )
319	23, 271, 318	sylancl 667	. . . . . . . . . . . 12 |- ( ph -> ( J ` 1 ) e. ( NN X. NN ) )
320		xp1st 6835	. . . . . . . . . . . 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 10626	. . . . . . . . . 10 |- ( ph -> ( 1st ` ( J ` 1 ) ) e. RR )
323	153	zred 11042	. . . . . . . . . 10 |- ( ph -> L e. RR )
324	321	nnge1d 10654	. . . . . . . . . 10 |- ( ph -> 1 <_ ( 1st ` ( J ` 1 ) ) )
325		eluzfz1 11808	. . . . . . . . . . . 12 |- ( K e. ( ZZ>= ` 1 ) -> 1 e. ( 1 ... K ) )
326	78, 325	syl 17	. . . . . . . . . . 11 |- ( ph -> 1 e. ( 1 ... K ) )
327		fveq2 5879	. . . . . . . . . . . . . 14 |- ( w = 1 -> ( J ` w ) = ( J ` 1 ) )
328	327	fveq2d 5883	. . . . . . . . . . . . 13 |- ( w = 1 -> ( 1st ` ( J ` w ) ) = ( 1st ` ( J ` 1 ) ) )
329	328	breq1d 4431	. . . . . . . . . . . 12 |- ( w = 1 -> ( ( 1st ` ( J ` w ) ) <_ L <-> ( 1st ` ( J ` 1 ) ) <_ L ) )
330	329	rspcv 3179	. . . . . . . . . . 11 |- ( 1 e. ( 1 ... K ) -> ( A. w e. ( 1 ... K ) ( 1st ` ( J ` w ) ) <_ L -> ( 1st ` ( J ` 1 ) ) <_ L ) )
331	326, 143, 330	sylc 63	. . . . . . . . . 10 |- ( ph -> ( 1st ` ( J ` 1 ) ) <_ L )
332	317, 322, 323, 324, 331	letrd 9794	. . . . . . . . 9 |- ( ph -> 1 <_ L )
333		elnnz1 10965	. . . . . . . . 9 |- ( L e. NN <-> ( L e. ZZ /\ 1 <_ L ) )
334	153, 332, 333	sylanbrc 669	. . . . . . . 8 |- ( ph -> L e. NN )
335	334, 77	syl6eleq 2521	. . . . . . 7 |- ( ph -> L e. ( ZZ>= ` 1 ) )
336	316, 335, 298	fsumser 13789	. . . . . 6 |- ( ph -> sum_ n e. ( 1 ... L ) ( vol* ` A ) = ( seq 1 ( + , G ) ` L ) )
337		seqfn 12226	. . . . . . . . 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 6032	. . . . . . . 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 666	. . . . . . 7 |- ( ph -> ( seq 1 ( + , G ) ` L ) e. ran seq 1 ( + , G ) )
341	309	rneqi 5078	. . . . . . 7 |- ran T = ran seq 1 ( + , G )
342	340, 341	syl6eleqr 2522	. . . . . 6 |- ( ph -> ( seq 1 ( + , G ) ` L ) e. ran T )
343	336, 342	eqeltrd 2511	. . . . 5 |- ( ph -> sum_ n e. ( 1 ... L ) ( vol* ` A ) e. ran T )
344		supxrub 11612	. . . . 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 666	. . . 4 |- ( ph -> sum_ n e. ( 1 ... L ) ( vol* ` A ) <_ sup ( ran T , RR* , < ) )
346	106	recnd 9671	. . . . . 6 |- ( ph -> B e. CC )
347		geo2sum 13922	. . . . . 6 |- ( ( L e. NN /\ B e. CC ) -> sum_ n e. ( 1 ... L ) ( B / ( 2 ^ n ) ) = ( B - ( B / ( 2 ^ L ) ) ) )
348	334, 346, 347	syl2anc 666	. . . . 5 |- ( ph -> sum_ n e. ( 1 ... L ) ( B / ( 2 ^ n ) ) = ( B - ( B / ( 2 ^ L ) ) ) )
349	334	nnnn0d 10927	. . . . . . . . . 10 |- ( ph -> L e. NN0 )
350		nnexpcl 12286	. . . . . . . . . 10 |- ( ( 2 e. NN /\ L e. NN0 ) -> ( 2 ^ L ) e. NN )
351	107, 349, 350	sylancr 668	. . . . . . . . 9 |- ( ph -> ( 2 ^ L ) e. NN )
352	351	nnrpd 11341	. . . . . . . 8 |- ( ph -> ( 2 ^ L ) e. RR+ )
353	105, 352	rpdivcld 11360	. . . . . . 7 |- ( ph -> ( B / ( 2 ^ L ) ) e. RR+ )
354	353	rpge0d 11347	. . . . . 6 |- ( ph -> 0 <_ ( B / ( 2 ^ L ) ) )
355	106, 351	nndivred 10660	. . . . . . 7 |- ( ph -> ( B / ( 2 ^ L ) ) e. RR )
356	106, 355	subge02d 10207	. . . . . 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 4442	. . . 4 |- ( ph -> sum_ n e. ( 1 ... L ) ( B / ( 2 ^ n ) ) <_ B )
359	104, 114, 116, 106, 345, 358	le2addd 10234	. . 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 9794	. 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 4444	1 |- ( ph -> ( U ` K ) <_ ( sup ( ran T , RR* , < ) + B ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1438   e. wcel 1869   =/= wne 2619   A.wral 2776   E.wrex 2777   _Vcvv 3082   i^i cin 3436   C_ wss 3437   (/)c0 3762   {csn 3997   <.cop 4003   U.cuni 4217   U_ciun 4297   class class class wbr 4421   |-> cmpt 4480   X. cxp 4849   dom cdm 4851   ran crn 4852   |` cres 4853   "cima 4854   o. ccom 4855   Rel wrel 4856   Fn wfn 5594   -->wf 5595   -1-1->wf1 5596   -1-1-onto->wf1o 5598   ` cfv 5599  (class class class)co 6303   1stc1st 6803   2ndc2nd 6804   ^m cmap 7478   ~~ cen 7572   Fincfn 7575   supcsup 7958   CCcc 9539   RRcr 9540   0cc0 9541   1c1 9542   + caddc 9544   +oocpnf 9674   -oocmnf 9675   RR*cxr 9676   < clt 9677   <_ cle 9678   - cmin 9862   / cdiv 10271   NNcn 10611   2c2 10661   NN0cn0 10871   ZZcz 10939   ZZ>=cuz 11161   RR+crp 11304   (,)cioo 11637   [,)cico 11639   ...cfz 11786   seqcseq 12214   ^cexp 12273   abscabs 13291   ~~> cli 13541   sum_csu 13745   vol*covol 22405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-inf2 8150  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-pre-sup 9619

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-se 4811  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-oadd 7192  df-er 7369  df-map 7480  df-pm 7481  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-sup 7960  df-inf 7961  df-oi 8029  df-card 8376  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-2 10670  df-3 10671  df-n0 10872  df-z 10940  df-uz 11162  df-rp 11305  df-ioo 11641  df-ico 11643  df-fz 11787  df-fzo 11918  df-fl 12029  df-seq 12215  df-exp 12274  df-hash 12517  df-cj 13156  df-re 13157  df-im 13158  df-sqrt 13292  df-abs 13293  df-clim 13545  df-rlim 13546  df-sum 13746  df-ovol 22408

This theorem is referenced by:  ovoliunlem2  22448