Theorem vitalilem4 23380

Description: Lemma for vitali 23382. (Contributed by Mario Carneiro, 16-Jun-2014.)

Hypotheses

Ref	Expression
vitali.1	|- .~ = { <. x , y >. | ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) /\ ( x - y ) e. QQ ) }
vitali.2	|- S = ( ( 0 [,] 1 ) /. .~ )
vitali.3	|- ( ph -> F Fn S )
vitali.4	|- ( ph -> A. z e. S ( z =/= (/) -> ( F ` z ) e. z ) )
vitali.5	|- ( ph -> G : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) ) )
vitali.6	|- T = ( n e. NN |-> { s e. RR | ( s - ( G ` n ) ) e. ran F } )
vitali.7	|- ( ph -> -. ran F e. ( ~P RR \ dom vol ) )

Assertion

Ref	Expression
vitalilem4	|- ( ( ph /\ m e. NN ) -> ( vol* ` ( T ` m ) ) = 0 )

Distinct variable groups:   m, n, s, x, y, z, G   ph, m, n, x, z   z, S   T, m, x   m, F, n, s, x, y, z   .~ , m, n, s, x, y, z

Allowed substitution hints:   ph( y, s)   S( x, y, m, n, s)   T( y, z, n, s)

Proof of Theorem vitalilem4

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . . . . 9 |- ( n = m -> ( G ` n ) = ( G ` m ) )
2	1	oveq2d 6666	. . . . . . . 8 |- ( n = m -> ( s - ( G ` n ) ) = ( s - ( G ` m ) ) )
3	2	eleq1d 2686	. . . . . . 7 |- ( n = m -> ( ( s - ( G ` n ) ) e. ran F <-> ( s - ( G ` m ) ) e. ran F ) )
4	3	rabbidv 3189	. . . . . 6 |- ( n = m -> { s e. RR | ( s - ( G ` n ) ) e. ran F } = { s e. RR | ( s - ( G ` m ) ) e. ran F } )
5		vitali.6	. . . . . 6 |- T = ( n e. NN |-> { s e. RR | ( s - ( G ` n ) ) e. ran F } )
6		reex 10027	. . . . . . 7 |- RR e. _V
7	6	rabex 4813	. . . . . 6 |- { s e. RR | ( s - ( G ` m ) ) e. ran F } e. _V
8	4, 5, 7	fvmpt 6282	. . . . 5 |- ( m e. NN -> ( T ` m ) = { s e. RR | ( s - ( G ` m ) ) e. ran F } )
9	8	adantl 482	. . . 4 |- ( ( ph /\ m e. NN ) -> ( T ` m ) = { s e. RR | ( s - ( G ` m ) ) e. ran F } )
10	9	fveq2d 6195	. . 3 |- ( ( ph /\ m e. NN ) -> ( vol* ` ( T ` m ) ) = ( vol* ` { s e. RR | ( s - ( G ` m ) ) e. ran F } ) )
11		vitali.1	. . . . . . . 8 |- .~ = { <. x , y >. | ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) /\ ( x - y ) e. QQ ) }
12		vitali.2	. . . . . . . 8 |- S = ( ( 0 [,] 1 ) /. .~ )
13		vitali.3	. . . . . . . 8 |- ( ph -> F Fn S )
14		vitali.4	. . . . . . . 8 |- ( ph -> A. z e. S ( z =/= (/) -> ( F ` z ) e. z ) )
15		vitali.5	. . . . . . . 8 |- ( ph -> G : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) ) )
16		vitali.7	. . . . . . . 8 |- ( ph -> -. ran F e. ( ~P RR \ dom vol ) )
17	11, 12, 13, 14, 15, 5, 16	vitalilem2 23378	. . . . . . 7 |- ( ph -> ( ran F C_ ( 0 [,] 1 ) /\ ( 0 [,] 1 ) C_ U_ m e. NN ( T ` m ) /\ U_ m e. NN ( T ` m ) C_ ( -u 1 [,] 2 ) ) )
18	17	simp1d 1073	. . . . . 6 |- ( ph -> ran F C_ ( 0 [,] 1 ) )
19		unitssre 12319	. . . . . 6 |- ( 0 [,] 1 ) C_ RR
20	18, 19	syl6ss 3615	. . . . 5 |- ( ph -> ran F C_ RR )
21	20	adantr 481	. . . 4 |- ( ( ph /\ m e. NN ) -> ran F C_ RR )
22		neg1rr 11125	. . . . . 6 |- -u 1 e. RR
23		1re 10039	. . . . . 6 |- 1 e. RR
24		iccssre 12255	. . . . . 6 |- ( ( -u 1 e. RR /\ 1 e. RR ) -> ( -u 1 [,] 1 ) C_ RR )
25	22, 23, 24	mp2an 708	. . . . 5 |- ( -u 1 [,] 1 ) C_ RR
26		inss2 3834	. . . . . 6 |- ( QQ i^i ( -u 1 [,] 1 ) ) C_ ( -u 1 [,] 1 )
27		f1of 6137	. . . . . . . 8 |- ( G : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) ) -> G : NN --> ( QQ i^i ( -u 1 [,] 1 ) ) )
28	15, 27	syl 17	. . . . . . 7 |- ( ph -> G : NN --> ( QQ i^i ( -u 1 [,] 1 ) ) )
29	28	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ m e. NN ) -> ( G ` m ) e. ( QQ i^i ( -u 1 [,] 1 ) ) )
30	26, 29	sseldi 3601	. . . . 5 |- ( ( ph /\ m e. NN ) -> ( G ` m ) e. ( -u 1 [,] 1 ) )
31	25, 30	sseldi 3601	. . . 4 |- ( ( ph /\ m e. NN ) -> ( G ` m ) e. RR )
32		eqidd 2623	. . . 4 |- ( ( ph /\ m e. NN ) -> { s e. RR | ( s - ( G ` m ) ) e. ran F } = { s e. RR | ( s - ( G ` m ) ) e. ran F } )
33	21, 31, 32	ovolshft 23279	. . 3 |- ( ( ph /\ m e. NN ) -> ( vol* ` ran F ) = ( vol* ` { s e. RR | ( s - ( G ` m ) ) e. ran F } ) )
34	10, 33	eqtr4d 2659	. 2 |- ( ( ph /\ m e. NN ) -> ( vol* ` ( T ` m ) ) = ( vol* ` ran F ) )
35		3re 11094	. . . . . . . 8 |- 3 e. RR
36	35	rexri 10097	. . . . . . 7 |- 3 e. RR*
37	36	a1i 11	. . . . . 6 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> 3 e. RR* )
38		3nn 11186	. . . . . . . . . . . . . 14 |- 3 e. NN
39		nnrp 11842	. . . . . . . . . . . . . 14 |- ( 3 e. NN -> 3 e. RR+ )
40	38, 39	ax-mp 5	. . . . . . . . . . . . 13 |- 3 e. RR+
41		0re 10040	. . . . . . . . . . . . . . . . . . . 20 |- 0 e. RR
42		0le1 10551	. . . . . . . . . . . . . . . . . . . 20 |- 0 <_ 1
43		ovolicc 23291	. . . . . . . . . . . . . . . . . . . 20 |- ( ( 0 e. RR /\ 1 e. RR /\ 0 <_ 1 ) -> ( vol* ` ( 0 [,] 1 ) ) = ( 1 - 0 ) )
44	41, 23, 42, 43	mp3an 1424	. . . . . . . . . . . . . . . . . . 19 |- ( vol* ` ( 0 [,] 1 ) ) = ( 1 - 0 )
45		1m0e1 11131	. . . . . . . . . . . . . . . . . . 19 |- ( 1 - 0 ) = 1
46	44, 45	eqtri 2644	. . . . . . . . . . . . . . . . . 18 |- ( vol* ` ( 0 [,] 1 ) ) = 1
47	46, 23	eqeltri 2697	. . . . . . . . . . . . . . . . 17 |- ( vol* ` ( 0 [,] 1 ) ) e. RR
48		ovolsscl 23254	. . . . . . . . . . . . . . . . 17 |- ( ( ran F C_ ( 0 [,] 1 ) /\ ( 0 [,] 1 ) C_ RR /\ ( vol* ` ( 0 [,] 1 ) ) e. RR ) -> ( vol* ` ran F ) e. RR )
49	19, 47, 48	mp3an23 1416	. . . . . . . . . . . . . . . 16 |- ( ran F C_ ( 0 [,] 1 ) -> ( vol* ` ran F ) e. RR )
50	18, 49	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> ( vol* ` ran F ) e. RR )
51	50	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ( vol* ` ran F ) e. RR )
52		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> 0 < ( vol* ` ran F ) )
53	51, 52	elrpd 11869	. . . . . . . . . . . . 13 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ( vol* ` ran F ) e. RR+ )
54		rpdivcl 11856	. . . . . . . . . . . . 13 |- ( ( 3 e. RR+ /\ ( vol* ` ran F ) e. RR+ ) -> ( 3 / ( vol* ` ran F ) ) e. RR+ )
55	40, 53, 54	sylancr 695	. . . . . . . . . . . 12 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ( 3 / ( vol* ` ran F ) ) e. RR+ )
56	55	rpred 11872	. . . . . . . . . . 11 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ( 3 / ( vol* ` ran F ) ) e. RR )
57	55	rpge0d 11876	. . . . . . . . . . 11 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> 0 <_ ( 3 / ( vol* ` ran F ) ) )
58		flge0nn0 12621	. . . . . . . . . . 11 |- ( ( ( 3 / ( vol* ` ran F ) ) e. RR /\ 0 <_ ( 3 / ( vol* ` ran F ) ) ) -> ( |_ ` ( 3 / ( vol* ` ran F ) ) ) e. NN0 )
59	56, 57, 58	syl2anc 693	. . . . . . . . . 10 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ( |_ ` ( 3 / ( vol* ` ran F ) ) ) e. NN0 )
60		nn0p1nn 11332	. . . . . . . . . 10 |- ( ( |_ ` ( 3 / ( vol* ` ran F ) ) ) e. NN0 -> ( ( |_ ` ( 3 / ( vol* ` ran F ) ) ) + 1 ) e. NN )
61	59, 60	syl 17	. . . . . . . . 9 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ( ( |_ ` ( 3 / ( vol* ` ran F ) ) ) + 1 ) e. NN )
62	61	nnred 11035	. . . . . . . 8 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ( ( |_ ` ( 3 / ( vol* ` ran F ) ) ) + 1 ) e. RR )
63	62, 51	remulcld 10070	. . . . . . 7 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ( ( ( |_ ` ( 3 / ( vol* ` ran F ) ) ) + 1 ) x. ( vol* ` ran F ) ) e. RR )
64	63	rexrd 10089	. . . . . 6 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ( ( ( |_ ` ( 3 / ( vol* ` ran F ) ) ) + 1 ) x. ( vol* ` ran F ) ) e. RR* )
65	6	elpw2 4828	. . . . . . . . . . . . . . . . . . 19 |- ( ran F e. ~P RR <-> ran F C_ RR )
66	20, 65	sylibr 224	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ran F e. ~P RR )
67	66	anim1i 592	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ -. ran F e. dom vol ) -> ( ran F e. ~P RR /\ -. ran F e. dom vol ) )
68		eldif 3584	. . . . . . . . . . . . . . . . 17 |- ( ran F e. ( ~P RR \ dom vol ) <-> ( ran F e. ~P RR /\ -. ran F e. dom vol ) )
69	67, 68	sylibr 224	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ -. ran F e. dom vol ) -> ran F e. ( ~P RR \ dom vol ) )
70	69	ex 450	. . . . . . . . . . . . . . 15 |- ( ph -> ( -. ran F e. dom vol -> ran F e. ( ~P RR \ dom vol ) ) )
71	16, 70	mt3d 140	. . . . . . . . . . . . . 14 |- ( ph -> ran F e. dom vol )
72	71	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN ) -> ran F e. dom vol )
73		inss1 3833	. . . . . . . . . . . . . . . 16 |- ( QQ i^i ( -u 1 [,] 1 ) ) C_ QQ
74		qssre 11798	. . . . . . . . . . . . . . . 16 |- QQ C_ RR
75	73, 74	sstri 3612	. . . . . . . . . . . . . . 15 |- ( QQ i^i ( -u 1 [,] 1 ) ) C_ RR
76		fss 6056	. . . . . . . . . . . . . . 15 |- ( ( G : NN --> ( QQ i^i ( -u 1 [,] 1 ) ) /\ ( QQ i^i ( -u 1 [,] 1 ) ) C_ RR ) -> G : NN --> RR )
77	28, 75, 76	sylancl 694	. . . . . . . . . . . . . 14 |- ( ph -> G : NN --> RR )
78	77	ffvelrnda 6359	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. NN ) -> ( G ` n ) e. RR )
79		shftmbl 23306	. . . . . . . . . . . . 13 |- ( ( ran F e. dom vol /\ ( G ` n ) e. RR ) -> { s e. RR | ( s - ( G ` n ) ) e. ran F } e. dom vol )
80	72, 78, 79	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ph /\ n e. NN ) -> { s e. RR | ( s - ( G ` n ) ) e. ran F } e. dom vol )
81	80, 5	fmptd 6385	. . . . . . . . . . 11 |- ( ph -> T : NN --> dom vol )
82	81	ffvelrnda 6359	. . . . . . . . . 10 |- ( ( ph /\ m e. NN ) -> ( T ` m ) e. dom vol )
83	82	ralrimiva 2966	. . . . . . . . 9 |- ( ph -> A. m e. NN ( T ` m ) e. dom vol )
84		iunmbl 23321	. . . . . . . . 9 |- ( A. m e. NN ( T ` m ) e. dom vol -> U_ m e. NN ( T ` m ) e. dom vol )
85	83, 84	syl 17	. . . . . . . 8 |- ( ph -> U_ m e. NN ( T ` m ) e. dom vol )
86		mblss 23299	. . . . . . . 8 |- ( U_ m e. NN ( T ` m ) e. dom vol -> U_ m e. NN ( T ` m ) C_ RR )
87		ovolcl 23246	. . . . . . . 8 |- ( U_ m e. NN ( T ` m ) C_ RR -> ( vol* ` U_ m e. NN ( T ` m ) ) e. RR* )
88	85, 86, 87	3syl 18	. . . . . . 7 |- ( ph -> ( vol* ` U_ m e. NN ( T ` m ) ) e. RR* )
89	88	adantr 481	. . . . . 6 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ( vol* ` U_ m e. NN ( T ` m ) ) e. RR* )
90		flltp1 12601	. . . . . . . 8 |- ( ( 3 / ( vol* ` ran F ) ) e. RR -> ( 3 / ( vol* ` ran F ) ) < ( ( |_ ` ( 3 / ( vol* ` ran F ) ) ) + 1 ) )
91	56, 90	syl 17	. . . . . . 7 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ( 3 / ( vol* ` ran F ) ) < ( ( |_ ` ( 3 / ( vol* ` ran F ) ) ) + 1 ) )
92	35	a1i 11	. . . . . . . 8 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> 3 e. RR )
93	92, 62, 53	ltdivmul2d 11924	. . . . . . 7 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ( ( 3 / ( vol* ` ran F ) ) < ( ( |_ ` ( 3 / ( vol* ` ran F ) ) ) + 1 ) <-> 3 < ( ( ( |_ ` ( 3 / ( vol* ` ran F ) ) ) + 1 ) x. ( vol* ` ran F ) ) ) )
94	91, 93	mpbid 222	. . . . . 6 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> 3 < ( ( ( |_ ` ( 3 / ( vol* ` ran F ) ) ) + 1 ) x. ( vol* ` ran F ) ) )
95		nnuz 11723	. . . . . . . . . . 11 |- NN = ( ZZ>= ` 1 )
96		1zzd 11408	. . . . . . . . . . 11 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> 1 e. ZZ )
97		mblvol 23298	. . . . . . . . . . . . . . . . 17 |- ( ( T ` m ) e. dom vol -> ( vol ` ( T ` m ) ) = ( vol* ` ( T ` m ) ) )
98	82, 97	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ m e. NN ) -> ( vol ` ( T ` m ) ) = ( vol* ` ( T ` m ) ) )
99	98, 34	eqtrd 2656	. . . . . . . . . . . . . . 15 |- ( ( ph /\ m e. NN ) -> ( vol ` ( T ` m ) ) = ( vol* ` ran F ) )
100	50	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ m e. NN ) -> ( vol* ` ran F ) e. RR )
101	99, 100	eqeltrd 2701	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN ) -> ( vol ` ( T ` m ) ) e. RR )
102	101	adantlr 751	. . . . . . . . . . . . 13 |- ( ( ( ph /\ 0 < ( vol* ` ran F ) ) /\ m e. NN ) -> ( vol ` ( T ` m ) ) e. RR )
103		eqid 2622	. . . . . . . . . . . . 13 |- ( m e. NN |-> ( vol ` ( T ` m ) ) ) = ( m e. NN |-> ( vol ` ( T ` m ) ) )
104	102, 103	fmptd 6385	. . . . . . . . . . . 12 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ( m e. NN |-> ( vol ` ( T ` m ) ) ) : NN --> RR )
105	104	ffvelrnda 6359	. . . . . . . . . . 11 |- ( ( ( ph /\ 0 < ( vol* ` ran F ) ) /\ k e. NN ) -> ( ( m e. NN |-> ( vol ` ( T ` m ) ) ) ` k ) e. RR )
106	95, 96, 105	serfre 12830	. . . . . . . . . 10 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> seq 1 ( + , ( m e. NN |-> ( vol ` ( T ` m ) ) ) ) : NN --> RR )
107		frn 6053	. . . . . . . . . 10 |- ( seq 1 ( + , ( m e. NN |-> ( vol ` ( T ` m ) ) ) ) : NN --> RR -> ran seq 1 ( + , ( m e. NN |-> ( vol ` ( T ` m ) ) ) ) C_ RR )
108	106, 107	syl 17	. . . . . . . . 9 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ran seq 1 ( + , ( m e. NN |-> ( vol ` ( T ` m ) ) ) ) C_ RR )
109		ressxr 10083	. . . . . . . . 9 |- RR C_ RR*
110	108, 109	syl6ss 3615	. . . . . . . 8 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ran seq 1 ( + , ( m e. NN |-> ( vol ` ( T ` m ) ) ) ) C_ RR* )
111	99	adantlr 751	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ 0 < ( vol* ` ran F ) ) /\ m e. NN ) -> ( vol ` ( T ` m ) ) = ( vol* ` ran F ) )
112	111	mpteq2dva 4744	. . . . . . . . . . . . 13 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ( m e. NN |-> ( vol ` ( T ` m ) ) ) = ( m e. NN |-> ( vol* ` ran F ) ) )
113		fconstmpt 5163	. . . . . . . . . . . . 13 |- ( NN X. { ( vol* ` ran F ) } ) = ( m e. NN |-> ( vol* ` ran F ) )
114	112, 113	syl6eqr 2674	. . . . . . . . . . . 12 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ( m e. NN |-> ( vol ` ( T ` m ) ) ) = ( NN X. { ( vol* ` ran F ) } ) )
115	114	seqeq3d 12809	. . . . . . . . . . 11 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> seq 1 ( + , ( m e. NN |-> ( vol ` ( T ` m ) ) ) ) = seq 1 ( + , ( NN X. { ( vol* ` ran F ) } ) ) )
116	115	fveq1d 6193	. . . . . . . . . 10 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ( seq 1 ( + , ( m e. NN |-> ( vol ` ( T ` m ) ) ) ) ` ( ( |_ ` ( 3 / ( vol* ` ran F ) ) ) + 1 ) ) = ( seq 1 ( + , ( NN X. { ( vol* ` ran F ) } ) ) ` ( ( |_ ` ( 3 / ( vol* ` ran F ) ) ) + 1 ) ) )
117	51	recnd 10068	. . . . . . . . . . 11 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ( vol* ` ran F ) e. CC )
118		ser1const 12857	. . . . . . . . . . 11 |- ( ( ( vol* ` ran F ) e. CC /\ ( ( |_ ` ( 3 / ( vol* ` ran F ) ) ) + 1 ) e. NN ) -> ( seq 1 ( + , ( NN X. { ( vol* ` ran F ) } ) ) ` ( ( |_ ` ( 3 / ( vol* ` ran F ) ) ) + 1 ) ) = ( ( ( |_ ` ( 3 / ( vol* ` ran F ) ) ) + 1 ) x. ( vol* ` ran F ) ) )
119	117, 61, 118	syl2anc 693	. . . . . . . . . 10 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ( seq 1 ( + , ( NN X. { ( vol* ` ran F ) } ) ) ` ( ( |_ ` ( 3 / ( vol* ` ran F ) ) ) + 1 ) ) = ( ( ( |_ ` ( 3 / ( vol* ` ran F ) ) ) + 1 ) x. ( vol* ` ran F ) ) )
120	116, 119	eqtrd 2656	. . . . . . . . 9 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ( seq 1 ( + , ( m e. NN |-> ( vol ` ( T ` m ) ) ) ) ` ( ( |_ ` ( 3 / ( vol* ` ran F ) ) ) + 1 ) ) = ( ( ( |_ ` ( 3 / ( vol* ` ran F ) ) ) + 1 ) x. ( vol* ` ran F ) ) )
121		ffn 6045	. . . . . . . . . . 11 |- ( seq 1 ( + , ( m e. NN |-> ( vol ` ( T ` m ) ) ) ) : NN --> RR -> seq 1 ( + , ( m e. NN |-> ( vol ` ( T ` m ) ) ) ) Fn NN )
122	106, 121	syl 17	. . . . . . . . . 10 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> seq 1 ( + , ( m e. NN |-> ( vol ` ( T ` m ) ) ) ) Fn NN )
123		fnfvelrn 6356	. . . . . . . . . 10 |- ( ( seq 1 ( + , ( m e. NN |-> ( vol ` ( T ` m ) ) ) ) Fn NN /\ ( ( |_ ` ( 3 / ( vol* ` ran F ) ) ) + 1 ) e. NN ) -> ( seq 1 ( + , ( m e. NN |-> ( vol ` ( T ` m ) ) ) ) ` ( ( |_ ` ( 3 / ( vol* ` ran F ) ) ) + 1 ) ) e. ran seq 1 ( + , ( m e. NN |-> ( vol ` ( T ` m ) ) ) ) )
124	122, 61, 123	syl2anc 693	. . . . . . . . 9 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ( seq 1 ( + , ( m e. NN |-> ( vol ` ( T ` m ) ) ) ) ` ( ( |_ ` ( 3 / ( vol* ` ran F ) ) ) + 1 ) ) e. ran seq 1 ( + , ( m e. NN |-> ( vol ` ( T ` m ) ) ) ) )
125	120, 124	eqeltrrd 2702	. . . . . . . 8 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ( ( ( |_ ` ( 3 / ( vol* ` ran F ) ) ) + 1 ) x. ( vol* ` ran F ) ) e. ran seq 1 ( + , ( m e. NN |-> ( vol ` ( T ` m ) ) ) ) )
126		supxrub 12154	. . . . . . . 8 |- ( ( ran seq 1 ( + , ( m e. NN |-> ( vol ` ( T ` m ) ) ) ) C_ RR* /\ ( ( ( |_ ` ( 3 / ( vol* ` ran F ) ) ) + 1 ) x. ( vol* ` ran F ) ) e. ran seq 1 ( + , ( m e. NN |-> ( vol ` ( T ` m ) ) ) ) ) -> ( ( ( |_ ` ( 3 / ( vol* ` ran F ) ) ) + 1 ) x. ( vol* ` ran F ) ) <_ sup ( ran seq 1 ( + , ( m e. NN |-> ( vol ` ( T ` m ) ) ) ) , RR* , < ) )
127	110, 125, 126	syl2anc 693	. . . . . . 7 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ( ( ( |_ ` ( 3 / ( vol* ` ran F ) ) ) + 1 ) x. ( vol* ` ran F ) ) <_ sup ( ran seq 1 ( + , ( m e. NN |-> ( vol ` ( T ` m ) ) ) ) , RR* , < ) )
128	85	adantr 481	. . . . . . . . 9 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> U_ m e. NN ( T ` m ) e. dom vol )
129		mblvol 23298	. . . . . . . . 9 |- ( U_ m e. NN ( T ` m ) e. dom vol -> ( vol ` U_ m e. NN ( T ` m ) ) = ( vol* ` U_ m e. NN ( T ` m ) ) )
130	128, 129	syl 17	. . . . . . . 8 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ( vol ` U_ m e. NN ( T ` m ) ) = ( vol* ` U_ m e. NN ( T ` m ) ) )
131	82, 101	jca 554	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN ) -> ( ( T ` m ) e. dom vol /\ ( vol ` ( T ` m ) ) e. RR ) )
132	131	ralrimiva 2966	. . . . . . . . . 10 |- ( ph -> A. m e. NN ( ( T ` m ) e. dom vol /\ ( vol ` ( T ` m ) ) e. RR ) )
133	132	adantr 481	. . . . . . . . 9 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> A. m e. NN ( ( T ` m ) e. dom vol /\ ( vol ` ( T ` m ) ) e. RR ) )
134	11, 12, 13, 14, 15, 5, 16	vitalilem3 23379	. . . . . . . . . 10 |- ( ph -> Disj m e. NN ( T ` m ) )
135	134	adantr 481	. . . . . . . . 9 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> Disj m e. NN ( T ` m ) )
136		eqid 2622	. . . . . . . . . 10 |- seq 1 ( + , ( m e. NN |-> ( vol ` ( T ` m ) ) ) ) = seq 1 ( + , ( m e. NN |-> ( vol ` ( T ` m ) ) ) )
137	136, 103	voliun 23322	. . . . . . . . 9 |- ( ( A. m e. NN ( ( T ` m ) e. dom vol /\ ( vol ` ( T ` m ) ) e. RR ) /\ Disj m e. NN ( T ` m ) ) -> ( vol ` U_ m e. NN ( T ` m ) ) = sup ( ran seq 1 ( + , ( m e. NN |-> ( vol ` ( T ` m ) ) ) ) , RR* , < ) )
138	133, 135, 137	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ( vol ` U_ m e. NN ( T ` m ) ) = sup ( ran seq 1 ( + , ( m e. NN |-> ( vol ` ( T ` m ) ) ) ) , RR* , < ) )
139	130, 138	eqtr3d 2658	. . . . . . 7 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ( vol* ` U_ m e. NN ( T ` m ) ) = sup ( ran seq 1 ( + , ( m e. NN |-> ( vol ` ( T ` m ) ) ) ) , RR* , < ) )
140	127, 139	breqtrrd 4681	. . . . . 6 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ( ( ( |_ ` ( 3 / ( vol* ` ran F ) ) ) + 1 ) x. ( vol* ` ran F ) ) <_ ( vol* ` U_ m e. NN ( T ` m ) ) )
141	37, 64, 89, 94, 140	xrltletrd 11992	. . . . 5 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> 3 < ( vol* ` U_ m e. NN ( T ` m ) ) )
142	17	simp3d 1075	. . . . . . . . 9 |- ( ph -> U_ m e. NN ( T ` m ) C_ ( -u 1 [,] 2 ) )
143	142	adantr 481	. . . . . . . 8 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> U_ m e. NN ( T ` m ) C_ ( -u 1 [,] 2 ) )
144		2re 11090	. . . . . . . . 9 |- 2 e. RR
145		iccssre 12255	. . . . . . . . 9 |- ( ( -u 1 e. RR /\ 2 e. RR ) -> ( -u 1 [,] 2 ) C_ RR )
146	22, 144, 145	mp2an 708	. . . . . . . 8 |- ( -u 1 [,] 2 ) C_ RR
147		ovolss 23253	. . . . . . . 8 |- ( ( U_ m e. NN ( T ` m ) C_ ( -u 1 [,] 2 ) /\ ( -u 1 [,] 2 ) C_ RR ) -> ( vol* ` U_ m e. NN ( T ` m ) ) <_ ( vol* ` ( -u 1 [,] 2 ) ) )
148	143, 146, 147	sylancl 694	. . . . . . 7 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ( vol* ` U_ m e. NN ( T ` m ) ) <_ ( vol* ` ( -u 1 [,] 2 ) ) )
149		2cn 11091	. . . . . . . . 9 |- 2 e. CC
150		ax-1cn 9994	. . . . . . . . 9 |- 1 e. CC
151	149, 150	subnegi 10360	. . . . . . . 8 |- ( 2 - -u 1 ) = ( 2 + 1 )
152		neg1lt0 11127	. . . . . . . . . . 11 |- -u 1 < 0
153		2pos 11112	. . . . . . . . . . 11 |- 0 < 2
154	22, 41, 144	lttri 10163	. . . . . . . . . . 11 |- ( ( -u 1 < 0 /\ 0 < 2 ) -> -u 1 < 2 )
155	152, 153, 154	mp2an 708	. . . . . . . . . 10 |- -u 1 < 2
156	22, 144, 155	ltleii 10160	. . . . . . . . 9 |- -u 1 <_ 2
157		ovolicc 23291	. . . . . . . . 9 |- ( ( -u 1 e. RR /\ 2 e. RR /\ -u 1 <_ 2 ) -> ( vol* ` ( -u 1 [,] 2 ) ) = ( 2 - -u 1 ) )
158	22, 144, 156, 157	mp3an 1424	. . . . . . . 8 |- ( vol* ` ( -u 1 [,] 2 ) ) = ( 2 - -u 1 )
159		df-3 11080	. . . . . . . 8 |- 3 = ( 2 + 1 )
160	151, 158, 159	3eqtr4i 2654	. . . . . . 7 |- ( vol* ` ( -u 1 [,] 2 ) ) = 3
161	148, 160	syl6breq 4694	. . . . . 6 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ( vol* ` U_ m e. NN ( T ` m ) ) <_ 3 )
162		xrlenlt 10103	. . . . . . 7 |- ( ( ( vol* ` U_ m e. NN ( T ` m ) ) e. RR* /\ 3 e. RR* ) -> ( ( vol* ` U_ m e. NN ( T ` m ) ) <_ 3 <-> -. 3 < ( vol* ` U_ m e. NN ( T ` m ) ) ) )
163	89, 36, 162	sylancl 694	. . . . . 6 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> ( ( vol* ` U_ m e. NN ( T ` m ) ) <_ 3 <-> -. 3 < ( vol* ` U_ m e. NN ( T ` m ) ) ) )
164	161, 163	mpbid 222	. . . . 5 |- ( ( ph /\ 0 < ( vol* ` ran F ) ) -> -. 3 < ( vol* ` U_ m e. NN ( T ` m ) ) )
165	141, 164	pm2.65da 600	. . . 4 |- ( ph -> -. 0 < ( vol* ` ran F ) )
166		ovolge0 23249	. . . . . . 7 |- ( ran F C_ RR -> 0 <_ ( vol* ` ran F ) )
167	20, 166	syl 17	. . . . . 6 |- ( ph -> 0 <_ ( vol* ` ran F ) )
168		0xr 10086	. . . . . . 7 |- 0 e. RR*
169		ovolcl 23246	. . . . . . . 8 |- ( ran F C_ RR -> ( vol* ` ran F ) e. RR* )
170	20, 169	syl 17	. . . . . . 7 |- ( ph -> ( vol* ` ran F ) e. RR* )
171		xrleloe 11977	. . . . . . 7 |- ( ( 0 e. RR* /\ ( vol* ` ran F ) e. RR* ) -> ( 0 <_ ( vol* ` ran F ) <-> ( 0 < ( vol* ` ran F ) \/ 0 = ( vol* ` ran F ) ) ) )
172	168, 170, 171	sylancr 695	. . . . . 6 |- ( ph -> ( 0 <_ ( vol* ` ran F ) <-> ( 0 < ( vol* ` ran F ) \/ 0 = ( vol* ` ran F ) ) ) )
173	167, 172	mpbid 222	. . . . 5 |- ( ph -> ( 0 < ( vol* ` ran F ) \/ 0 = ( vol* ` ran F ) ) )
174	173	ord 392	. . . 4 |- ( ph -> ( -. 0 < ( vol* ` ran F ) -> 0 = ( vol* ` ran F ) ) )
175	165, 174	mpd 15	. . 3 |- ( ph -> 0 = ( vol* ` ran F ) )
176	175	adantr 481	. 2 |- ( ( ph /\ m e. NN ) -> 0 = ( vol* ` ran F ) )
177	34, 176	eqtr4d 2659	1 |- ( ( ph /\ m e. NN ) -> ( vol* ` ( T ` m ) ) = 0 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   U_ciun 4520  Disj wdisj 4620   class class class wbr 4653   {copab 4712   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   ran crn 5115   Fn wfn 5883   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   /.cqs 7741   supcsup 8346   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   3c3 11071   NN0cn0 11292   QQcq 11788   RR+crp 11832   [,]cicc 12178   |_cfl 12591   seqcseq 12801   vol*covol 23231   volcvol 23232

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cc 9257  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cmp 21190  df-ovol 23233  df-vol 23234

This theorem is referenced by:  vitalilem5  23381