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Theorem vitalilem4 21783
Description: Lemma for vitali 21785. (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.
StepHypRef Expression
1 fveq2 5866 . . . . . . . . 9  |-  ( n  =  m  ->  ( G `  n )  =  ( G `  m ) )
21oveq2d 6300 . . . . . . . 8  |-  ( n  =  m  ->  (
s  -  ( G `
 n ) )  =  ( s  -  ( G `  m ) ) )
32eleq1d 2536 . . . . . . 7  |-  ( n  =  m  ->  (
( s  -  ( G `  n )
)  e.  ran  F  <->  ( s  -  ( G `
 m ) )  e.  ran  F ) )
43rabbidv 3105 . . . . . 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 9583 . . . . . . 7  |-  RR  e.  _V
76rabex 4598 . . . . . 6  |-  { s  e.  RR  |  ( s  -  ( G `
 m ) )  e.  ran  F }  e.  _V
84, 5, 7fvmpt 5950 . . . . 5  |-  ( m  e.  NN  ->  ( T `  m )  =  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } )
98adantl 466 . . . 4  |-  ( (
ph  /\  m  e.  NN )  ->  ( T `
 m )  =  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } )
109fveq2d 5870 . . 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 ) )
1711, 12, 13, 14, 15, 5, 16vitalilem2 21781 . . . . . . 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 ) ) )
1817simp1d 1008 . . . . . 6  |-  ( ph  ->  ran  F  C_  (
0 [,] 1 ) )
19 unitssre 11667 . . . . . 6  |-  ( 0 [,] 1 )  C_  RR
2018, 19syl6ss 3516 . . . . 5  |-  ( ph  ->  ran  F  C_  RR )
2120adantr 465 . . . 4  |-  ( (
ph  /\  m  e.  NN )  ->  ran  F  C_  RR )
22 neg1rr 10640 . . . . . 6  |-  -u 1  e.  RR
23 1re 9595 . . . . . 6  |-  1  e.  RR
24 iccssre 11606 . . . . . 6  |-  ( (
-u 1  e.  RR  /\  1  e.  RR )  ->  ( -u 1 [,] 1 )  C_  RR )
2522, 23, 24mp2an 672 . . . . 5  |-  ( -u
1 [,] 1 ) 
C_  RR
26 inss2 3719 . . . . . 6  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  ( -u 1 [,] 1
)
27 f1of 5816 . . . . . . . 8  |-  ( G : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) )  ->  G : NN
--> ( QQ  i^i  ( -u 1 [,] 1 ) ) )
2815, 27syl 16 . . . . . . 7  |-  ( ph  ->  G : NN --> ( QQ 
i^i  ( -u 1 [,] 1 ) ) )
2928ffvelrnda 6021 . . . . . 6  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  ( QQ  i^i  ( -u 1 [,] 1 ) ) )
3026, 29sseldi 3502 . . . . 5  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  ( -u 1 [,] 1 ) )
3125, 30sseldi 3502 . . . 4  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  RR )
32 eqidd 2468 . . . 4  |-  ( (
ph  /\  m  e.  NN )  ->  { s  e.  RR  |  ( s  -  ( G `
 m ) )  e.  ran  F }  =  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } )
3321, 31, 32ovolshft 21685 . . 3  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol* `  ran  F )  =  ( vol* `  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } ) )
3410, 33eqtr4d 2511 . 2  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol* `  ( T `  m ) )  =  ( vol* `  ran  F ) )
35 3re 10609 . . . . . . . 8  |-  3  e.  RR
3635rexri 9646 . . . . . . 7  |-  3  e.  RR*
3736a1i 11 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  3  e.  RR* )
38 3nn 10694 . . . . . . . . . . . . . 14  |-  3  e.  NN
39 nnrp 11229 . . . . . . . . . . . . . 14  |-  ( 3  e.  NN  ->  3  e.  RR+ )
4038, 39ax-mp 5 . . . . . . . . . . . . 13  |-  3  e.  RR+
41 0re 9596 . . . . . . . . . . . . . . . . . . . 20  |-  0  e.  RR
42 0le1 10076 . . . . . . . . . . . . . . . . . . . 20  |-  0  <_  1
43 ovolicc 21697 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  0  <_  1 )  ->  ( vol* `  ( 0 [,] 1 ) )  =  ( 1  -  0 ) )
4441, 23, 42, 43mp3an 1324 . . . . . . . . . . . . . . . . . . 19  |-  ( vol* `  ( 0 [,] 1 ) )  =  ( 1  -  0 )
45 1m0e1 10646 . . . . . . . . . . . . . . . . . . 19  |-  ( 1  -  0 )  =  1
4644, 45eqtri 2496 . . . . . . . . . . . . . . . . . 18  |-  ( vol* `  ( 0 [,] 1 ) )  =  1
4746, 23eqeltri 2551 . . . . . . . . . . . . . . . . 17  |-  ( vol* `  ( 0 [,] 1 ) )  e.  RR
48 ovolsscl 21660 . . . . . . . . . . . . . . . . 17  |-  ( ( ran  F  C_  (
0 [,] 1 )  /\  ( 0 [,] 1 )  C_  RR  /\  ( vol* `  ( 0 [,] 1
) )  e.  RR )  ->  ( vol* `  ran  F )  e.  RR )
4919, 47, 48mp3an23 1316 . . . . . . . . . . . . . . . 16  |-  ( ran 
F  C_  ( 0 [,] 1 )  -> 
( vol* `  ran  F )  e.  RR )
5018, 49syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( vol* `  ran  F )  e.  RR )
5150adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( vol* `  ran  F
)  e.  RR )
52 simpr 461 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  0  <  ( vol* `  ran  F ) )
5351, 52elrpd 11254 . . . . . . . . . . . . 13  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( vol* `  ran  F
)  e.  RR+ )
54 rpdivcl 11242 . . . . . . . . . . . . 13  |-  ( ( 3  e.  RR+  /\  ( vol* `  ran  F
)  e.  RR+ )  ->  ( 3  /  ( vol* `  ran  F
) )  e.  RR+ )
5540, 53, 54sylancr 663 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
3  /  ( vol* `  ran  F ) )  e.  RR+ )
5655rpred 11256 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
3  /  ( vol* `  ran  F ) )  e.  RR )
5755rpge0d 11260 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  0  <_  ( 3  /  ( vol* `  ran  F
) ) )
58 flge0nn0 11922 . . . . . . . . . . 11  |-  ( ( ( 3  /  ( vol* `  ran  F
) )  e.  RR  /\  0  <_  ( 3  /  ( vol* `  ran  F ) ) )  ->  ( |_ `  ( 3  /  ( vol* `  ran  F
) ) )  e. 
NN0 )
5956, 57, 58syl2anc 661 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( |_ `  ( 3  / 
( vol* `  ran  F ) ) )  e.  NN0 )
60 nn0p1nn 10835 . . . . . . . . . 10  |-  ( ( |_ `  ( 3  /  ( vol* `  ran  F ) ) )  e.  NN0  ->  ( ( |_ `  (
3  /  ( vol* `  ran  F ) ) )  +  1 )  e.  NN )
6159, 60syl 16 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
( |_ `  (
3  /  ( vol* `  ran  F ) ) )  +  1 )  e.  NN )
6261nnred 10551 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
( |_ `  (
3  /  ( vol* `  ran  F ) ) )  +  1 )  e.  RR )
6362, 51remulcld 9624 . . . . . . 7  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
( ( |_ `  ( 3  /  ( vol* `  ran  F
) ) )  +  1 )  x.  ( vol* `  ran  F
) )  e.  RR )
6463rexrd 9643 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
( ( |_ `  ( 3  /  ( vol* `  ran  F
) ) )  +  1 )  x.  ( vol* `  ran  F
) )  e.  RR* )
656elpw2 4611 . . . . . . . . . . . . . . . . . . 19  |-  ( ran 
F  e.  ~P RR  <->  ran 
F  C_  RR )
6620, 65sylibr 212 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ran  F  e.  ~P RR )
6766anim1i 568 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  -.  ran  F  e.  dom  vol )  -> 
( ran  F  e.  ~P RR  /\  -.  ran  F  e.  dom  vol )
)
68 eldif 3486 . . . . . . . . . . . . . . . . 17  |-  ( ran 
F  e.  ( ~P RR  \  dom  vol ) 
<->  ( ran  F  e. 
~P RR  /\  -.  ran  F  e.  dom  vol ) )
6967, 68sylibr 212 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  -.  ran  F  e.  dom  vol )  ->  ran  F  e.  ( ~P RR  \  dom  vol ) )
7069ex 434 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( -.  ran  F  e.  dom  vol  ->  ran  F  e.  ( ~P RR  \  dom  vol ) ) )
7116, 70mt3d 125 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  F  e.  dom  vol )
7271adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  ran  F  e.  dom  vol )
73 inss1 3718 . . . . . . . . . . . . . . . 16  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  QQ
74 qssre 11192 . . . . . . . . . . . . . . . 16  |-  QQ  C_  RR
7573, 74sstri 3513 . . . . . . . . . . . . . . 15  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  RR
76 fss 5739 . . . . . . . . . . . . . . 15  |-  ( ( G : NN --> ( QQ 
i^i  ( -u 1 [,] 1 ) )  /\  ( QQ  i^i  ( -u 1 [,] 1 ) )  C_  RR )  ->  G : NN --> RR )
7728, 75, 76sylancl 662 . . . . . . . . . . . . . 14  |-  ( ph  ->  G : NN --> RR )
7877ffvelrnda 6021 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 n )  e.  RR )
79 shftmbl 21712 . . . . . . . . . . . . 13  |-  ( ( ran  F  e.  dom  vol 
/\  ( G `  n )  e.  RR )  ->  { s  e.  RR  |  ( s  -  ( G `  n ) )  e. 
ran  F }  e.  dom  vol )
8072, 78, 79syl2anc 661 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  { s  e.  RR  |  ( s  -  ( G `
 n ) )  e.  ran  F }  e.  dom  vol )
8180, 5fmptd 6045 . . . . . . . . . . 11  |-  ( ph  ->  T : NN --> dom  vol )
8281ffvelrnda 6021 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN )  ->  ( T `
 m )  e. 
dom  vol )
8382ralrimiva 2878 . . . . . . . . 9  |-  ( ph  ->  A. m  e.  NN  ( T `  m )  e.  dom  vol )
84 iunmbl 21726 . . . . . . . . 9  |-  ( A. m  e.  NN  ( T `  m )  e.  dom  vol  ->  U_ m  e.  NN  ( T `  m )  e.  dom  vol )
8583, 84syl 16 . . . . . . . 8  |-  ( ph  ->  U_ m  e.  NN  ( T `  m )  e.  dom  vol )
86 mblss 21705 . . . . . . . 8  |-  ( U_ m  e.  NN  ( T `  m )  e.  dom  vol  ->  U_ m  e.  NN  ( T `  m )  C_  RR )
87 ovolcl 21652 . . . . . . . 8  |-  ( U_ m  e.  NN  ( T `  m )  C_  RR  ->  ( vol* `  U_ m  e.  NN  ( T `  m ) )  e. 
RR* )
8885, 86, 873syl 20 . . . . . . 7  |-  ( ph  ->  ( vol* `  U_ m  e.  NN  ( T `  m )
)  e.  RR* )
8988adantr 465 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( vol* `  U_ m  e.  NN  ( T `  m ) )  e. 
RR* )
90 flltp1 11905 . . . . . . . 8  |-  ( ( 3  /  ( vol* `  ran  F ) )  e.  RR  ->  ( 3  /  ( vol* `  ran  F ) )  <  ( ( |_ `  ( 3  /  ( vol* `  ran  F ) ) )  +  1 ) )
9156, 90syl 16 . . . . . . 7  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
3  /  ( vol* `  ran  F ) )  <  ( ( |_ `  ( 3  /  ( vol* `  ran  F ) ) )  +  1 ) )
9235a1i 11 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  3  e.  RR )
9392, 62, 53ltdivmul2d 11304 . . . . . . 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
) ) ) )
9491, 93mpbid 210 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  3  <  ( ( ( |_
`  ( 3  / 
( vol* `  ran  F ) ) )  +  1 )  x.  ( vol* `  ran  F ) ) )
95 nnuz 11117 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
96 1zzd 10895 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  1  e.  ZZ )
97 mblvol 21704 . . . . . . . . . . . . . . . . 17  |-  ( ( T `  m )  e.  dom  vol  ->  ( vol `  ( T `
 m ) )  =  ( vol* `  ( T `  m
) ) )
9882, 97syl 16 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol `  ( T `  m
) )  =  ( vol* `  ( T `  m )
) )
9998, 34eqtrd 2508 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol `  ( T `  m
) )  =  ( vol* `  ran  F ) )
10050adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol* `  ran  F )  e.  RR )
10199, 100eqeltrd 2555 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol `  ( T `  m
) )  e.  RR )
102101adantlr 714 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  0  <  ( vol* `  ran  F ) )  /\  m  e.  NN )  ->  ( vol `  ( T `  m )
)  e.  RR )
103 eqid 2467 . . . . . . . . . . . . 13  |-  ( m  e.  NN  |->  ( vol `  ( T `  m
) ) )  =  ( m  e.  NN  |->  ( vol `  ( T `
 m ) ) )
104102, 103fmptd 6045 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) : NN --> RR )
105104ffvelrnda 6021 . . . . . . . . . . 11  |-  ( ( ( ph  /\  0  <  ( vol* `  ran  F ) )  /\  k  e.  NN )  ->  ( ( m  e.  NN  |->  ( vol `  ( T `  m )
) ) `  k
)  e.  RR )
10695, 96, 105serfre 12104 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  seq 1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) ) : NN --> RR )
107 frn 5737 . . . . . . . . . 10  |-  (  seq 1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) ) : NN --> RR  ->  ran  seq 1
(  +  ,  ( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )  C_  RR )
108106, 107syl 16 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ran  seq 1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )  C_  RR )
109 ressxr 9637 . . . . . . . . 9  |-  RR  C_  RR*
110108, 109syl6ss 3516 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ran  seq 1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )  C_  RR* )
11199adantlr 714 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  0  <  ( vol* `  ran  F ) )  /\  m  e.  NN )  ->  ( vol `  ( T `  m )
)  =  ( vol* `  ran  F ) )
112111mpteq2dva 4533 . . . . . . . . . . . . 13  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
m  e.  NN  |->  ( vol `  ( T `
 m ) ) )  =  ( m  e.  NN  |->  ( vol* `  ran  F ) ) )
113 fconstmpt 5043 . . . . . . . . . . . . 13  |-  ( NN 
X.  { ( vol* `  ran  F ) } )  =  ( m  e.  NN  |->  ( vol* `  ran  F ) )
114112, 113syl6eqr 2526 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
m  e.  NN  |->  ( vol `  ( T `
 m ) ) )  =  ( NN 
X.  { ( vol* `  ran  F ) } ) )
115114seqeq3d 12083 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  seq 1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )  =  seq 1 (  +  , 
( NN  X.  {
( vol* `  ran  F ) } ) ) )
116115fveq1d 5868 . . . . . . . . . 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 ) ) )
11751recnd 9622 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( vol* `  ran  F
)  e.  CC )
118 ser1const 12131 . . . . . . . . . . 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 ) ) )
119117, 61, 118syl2anc 661 . . . . . . . . . 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 ) ) )
120116, 119eqtrd 2508 . . . . . . . . 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 5731 . . . . . . . . . . 11  |-  (  seq 1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) ) : NN --> RR  ->  seq 1 (  +  ,  ( m  e.  NN  |->  ( vol `  ( T `  m )
) ) )  Fn  NN )
122106, 121syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  seq 1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )  Fn  NN )
123 fnfvelrn 6018 . . . . . . . . . 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 )
) ) ) )
124122, 61, 123syl2anc 661 . . . . . . . . 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 ) ) ) ) )
125120, 124eqeltrrd 2556 . . . . . . . 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 11516 . . . . . . . 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* ,  <  ) )
127110, 125, 126syl2anc 661 . . . . . . 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* ,  <  )
)
12885adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  U_ m  e.  NN  ( T `  m )  e.  dom  vol )
129 mblvol 21704 . . . . . . . . 9  |-  ( U_ m  e.  NN  ( T `  m )  e.  dom  vol  ->  ( vol `  U_ m  e.  NN  ( T `  m ) )  =  ( vol* `  U_ m  e.  NN  ( T `  m ) ) )
130128, 129syl 16 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( vol `  U_ m  e.  NN  ( T `  m ) )  =  ( vol* `  U_ m  e.  NN  ( T `  m )
) )
13182, 101jca 532 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN )  ->  ( ( T `  m )  e.  dom  vol  /\  ( vol `  ( T `
 m ) )  e.  RR ) )
132131ralrimiva 2878 . . . . . . . . . 10  |-  ( ph  ->  A. m  e.  NN  ( ( T `  m )  e.  dom  vol 
/\  ( vol `  ( T `  m )
)  e.  RR ) )
133132adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  A. m  e.  NN  ( ( T `
 m )  e. 
dom  vol  /\  ( vol `  ( T `  m
) )  e.  RR ) )
13411, 12, 13, 14, 15, 5, 16vitalilem3 21782 . . . . . . . . . 10  |-  ( ph  -> Disj  m  e.  NN  ( T `  m )
)
135134adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  -> Disj  m  e.  NN  ( T `  m ) )
136 eqid 2467 . . . . . . . . . 10  |-  seq 1
(  +  ,  ( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )  =  seq 1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )
137136, 103voliun 21727 . . . . . . . . 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* ,  <  ) )
138133, 135, 137syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( vol `  U_ m  e.  NN  ( T `  m ) )  =  sup ( ran  seq 1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) ) ,  RR* ,  <  ) )
139130, 138eqtr3d 2510 . . . . . . 7  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( vol* `  U_ m  e.  NN  ( T `  m ) )  =  sup ( ran  seq 1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) ) ,  RR* ,  <  ) )
140127, 139breqtrrd 4473 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
( ( |_ `  ( 3  /  ( vol* `  ran  F
) ) )  +  1 )  x.  ( vol* `  ran  F
) )  <_  ( vol* `  U_ m  e.  NN  ( T `  m ) ) )
14137, 64, 89, 94, 140xrltletrd 11364 . . . . 5  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  3  <  ( vol* `  U_ m  e.  NN  ( T `  m )
) )
14217simp3d 1010 . . . . . . . . 9  |-  ( ph  ->  U_ m  e.  NN  ( T `  m ) 
C_  ( -u 1 [,] 2 ) )
143142adantr 465 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  U_ m  e.  NN  ( T `  m )  C_  ( -u 1 [,] 2 ) )
144 2re 10605 . . . . . . . . 9  |-  2  e.  RR
145 iccssre 11606 . . . . . . . . 9  |-  ( (
-u 1  e.  RR  /\  2  e.  RR )  ->  ( -u 1 [,] 2 )  C_  RR )
14622, 144, 145mp2an 672 . . . . . . . 8  |-  ( -u
1 [,] 2 ) 
C_  RR
147 ovolss 21659 . . . . . . . 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 ) ) )
148143, 146, 147sylancl 662 . . . . . . 7  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( vol* `  U_ m  e.  NN  ( T `  m ) )  <_ 
( vol* `  ( -u 1 [,] 2
) ) )
149 2cn 10606 . . . . . . . . 9  |-  2  e.  CC
150 ax-1cn 9550 . . . . . . . . 9  |-  1  e.  CC
151149, 150subnegi 9898 . . . . . . . 8  |-  ( 2  -  -u 1 )  =  ( 2  +  1 )
152 neg1lt0 10642 . . . . . . . . . . 11  |-  -u 1  <  0
153 2pos 10627 . . . . . . . . . . 11  |-  0  <  2
15422, 41, 144lttri 9710 . . . . . . . . . . 11  |-  ( (
-u 1  <  0  /\  0  <  2
)  ->  -u 1  <  2 )
155152, 153, 154mp2an 672 . . . . . . . . . 10  |-  -u 1  <  2
15622, 144, 155ltleii 9707 . . . . . . . . 9  |-  -u 1  <_  2
157 ovolicc 21697 . . . . . . . . 9  |-  ( (
-u 1  e.  RR  /\  2  e.  RR  /\  -u 1  <_  2 )  ->  ( vol* `  ( -u 1 [,] 2 ) )  =  ( 2  -  -u 1
) )
15822, 144, 156, 157mp3an 1324 . . . . . . . 8  |-  ( vol* `  ( -u 1 [,] 2 ) )  =  ( 2  -  -u 1
)
159 df-3 10595 . . . . . . . 8  |-  3  =  ( 2  +  1 )
160151, 158, 1593eqtr4i 2506 . . . . . . 7  |-  ( vol* `  ( -u 1 [,] 2 ) )  =  3
161148, 160syl6breq 4486 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( vol* `  U_ m  e.  NN  ( T `  m ) )  <_ 
3 )
162 xrlenlt 9652 . . . . . . 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 )
) ) )
16389, 36, 162sylancl 662 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
( vol* `  U_ m  e.  NN  ( T `  m )
)  <_  3  <->  -.  3  <  ( vol* `  U_ m  e.  NN  ( T `  m )
) ) )
164161, 163mpbid 210 . . . . 5  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  -.  3  <  ( vol* `  U_ m  e.  NN  ( T `  m ) ) )
165141, 164pm2.65da 576 . . . 4  |-  ( ph  ->  -.  0  <  ( vol* `  ran  F
) )
166 ovolge0 21655 . . . . . . 7  |-  ( ran 
F  C_  RR  ->  0  <_  ( vol* `  ran  F ) )
16720, 166syl 16 . . . . . 6  |-  ( ph  ->  0  <_  ( vol* `  ran  F ) )
168 0xr 9640 . . . . . . 7  |-  0  e.  RR*
169 ovolcl 21652 . . . . . . . 8  |-  ( ran 
F  C_  RR  ->  ( vol* `  ran  F )  e.  RR* )
17020, 169syl 16 . . . . . . 7  |-  ( ph  ->  ( vol* `  ran  F )  e.  RR* )
171 xrleloe 11350 . . . . . . 7  |-  ( ( 0  e.  RR*  /\  ( vol* `  ran  F
)  e.  RR* )  ->  ( 0  <_  ( vol* `  ran  F
)  <->  ( 0  < 
( vol* `  ran  F )  \/  0  =  ( vol* `  ran  F ) ) ) )
172168, 170, 171sylancr 663 . . . . . 6  |-  ( ph  ->  ( 0  <_  ( vol* `  ran  F
)  <->  ( 0  < 
( vol* `  ran  F )  \/  0  =  ( vol* `  ran  F ) ) ) )
173167, 172mpbid 210 . . . . 5  |-  ( ph  ->  ( 0  <  ( vol* `  ran  F
)  \/  0  =  ( vol* `  ran  F ) ) )
174173ord 377 . . . 4  |-  ( ph  ->  ( -.  0  < 
( vol* `  ran  F )  ->  0  =  ( vol* `  ran  F ) ) )
175165, 174mpd 15 . . 3  |-  ( ph  ->  0  =  ( vol* `  ran  F ) )
176175adantr 465 . 2  |-  ( (
ph  /\  m  e.  NN )  ->  0  =  ( vol* `  ran  F ) )
17734, 176eqtr4d 2511 1  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol* `  ( T `  m ) )  =  0 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   {crab 2818    \ cdif 3473    i^i cin 3475    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   {csn 4027   U_ciun 4325  Disj wdisj 4417   class class class wbr 4447   {copab 4504    |-> cmpt 4505    X. cxp 4997   dom cdm 4999   ran crn 5000    Fn wfn 5583   -->wf 5584   -1-1-onto->wf1o 5587   ` cfv 5588  (class class class)co 6284   /.cqs 7310   supcsup 7900   CCcc 9490   RRcr 9491   0cc0 9492   1c1 9493    + caddc 9495    x. cmul 9497   RR*cxr 9627    < clt 9628    <_ cle 9629    - cmin 9805   -ucneg 9806    / cdiv 10206   NNcn 10536   2c2 10585   3c3 10586   NN0cn0 10795   QQcq 11182   RR+crp 11220   [,]cicc 11532   |_cfl 11895    seqcseq 12075   vol*covol 21637   volcvol 21638
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-cc 8815  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-disj 4418  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-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-ec 7313  df-qs 7317  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fi 7871  df-sup 7901  df-oi 7935  df-card 8320  df-cda 8548  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-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ioo 11533  df-ico 11535  df-icc 11536  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-rest 14678  df-topgen 14699  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-top 19194  df-bases 19196  df-topon 19197  df-cmp 19681  df-ovol 21639  df-vol 21640
This theorem is referenced by:  vitalilem5  21784
  Copyright terms: Public domain W3C validator