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Theorem vitalilem4 19460
Description: Lemma for vitali 19462. (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 5691 . . . . . . . . 9  |-  ( n  =  m  ->  ( G `  n )  =  ( G `  m ) )
21oveq2d 6060 . . . . . . . 8  |-  ( n  =  m  ->  (
s  -  ( G `
 n ) )  =  ( s  -  ( G `  m ) ) )
32eleq1d 2474 . . . . . . 7  |-  ( n  =  m  ->  (
( s  -  ( G `  n )
)  e.  ran  F  <->  ( s  -  ( G `
 m ) )  e.  ran  F ) )
43rabbidv 2912 . . . . . 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 9041 . . . . . . 7  |-  RR  e.  _V
76rabex 4318 . . . . . 6  |-  { s  e.  RR  |  ( s  -  ( G `
 m ) )  e.  ran  F }  e.  _V
84, 5, 7fvmpt 5769 . . . . 5  |-  ( m  e.  NN  ->  ( T `  m )  =  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } )
98adantl 453 . . . 4  |-  ( (
ph  /\  m  e.  NN )  ->  ( T `
 m )  =  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } )
109fveq2d 5695 . . 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 19458 . . . . . . 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 969 . . . . . 6  |-  ( ph  ->  ran  F  C_  (
0 [,] 1 ) )
19 unitssre 11002 . . . . . 6  |-  ( 0 [,] 1 )  C_  RR
2018, 19syl6ss 3324 . . . . 5  |-  ( ph  ->  ran  F  C_  RR )
2120adantr 452 . . . 4  |-  ( (
ph  /\  m  e.  NN )  ->  ran  F  C_  RR )
22 1re 9050 . . . . . . 7  |-  1  e.  RR
2322renegcli 9322 . . . . . 6  |-  -u 1  e.  RR
24 iccssre 10952 . . . . . 6  |-  ( (
-u 1  e.  RR  /\  1  e.  RR )  ->  ( -u 1 [,] 1 )  C_  RR )
2523, 22, 24mp2an 654 . . . . 5  |-  ( -u
1 [,] 1 ) 
C_  RR
26 inss2 3526 . . . . . 6  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  ( -u 1 [,] 1
)
27 f1of 5637 . . . . . . . 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 5833 . . . . . 6  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  ( QQ  i^i  ( -u 1 [,] 1 ) ) )
3026, 29sseldi 3310 . . . . 5  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  ( -u 1 [,] 1 ) )
3125, 30sseldi 3310 . . . 4  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  RR )
32 eqidd 2409 . . . 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 19364 . . 3  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol
* `  ran  F )  =  ( vol * `  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } ) )
3410, 33eqtr4d 2443 . 2  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol
* `  ( T `  m ) )  =  ( vol * `  ran  F ) )
35 3re 10031 . . . . . . . 8  |-  3  e.  RR
3635rexri 9097 . . . . . . 7  |-  3  e.  RR*
3736a1i 11 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  3  e.  RR* )
38 3nn 10094 . . . . . . . . . . . . . 14  |-  3  e.  NN
39 nnrp 10581 . . . . . . . . . . . . . 14  |-  ( 3  e.  NN  ->  3  e.  RR+ )
4038, 39ax-mp 8 . . . . . . . . . . . . 13  |-  3  e.  RR+
41 0re 9051 . . . . . . . . . . . . . . . . . . . 20  |-  0  e.  RR
42 0le1 9511 . . . . . . . . . . . . . . . . . . . 20  |-  0  <_  1
43 ovolicc 19376 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  0  <_  1 )  ->  ( vol * `  ( 0 [,] 1 ) )  =  ( 1  -  0 ) )
4441, 22, 42, 43mp3an 1279 . . . . . . . . . . . . . . . . . . 19  |-  ( vol
* `  ( 0 [,] 1 ) )  =  ( 1  -  0 )
45 ax-1cn 9008 . . . . . . . . . . . . . . . . . . . 20  |-  1  e.  CC
4645subid1i 9332 . . . . . . . . . . . . . . . . . . 19  |-  ( 1  -  0 )  =  1
4744, 46eqtri 2428 . . . . . . . . . . . . . . . . . 18  |-  ( vol
* `  ( 0 [,] 1 ) )  =  1
4847, 22eqeltri 2478 . . . . . . . . . . . . . . . . 17  |-  ( vol
* `  ( 0 [,] 1 ) )  e.  RR
49 ovolsscl 19339 . . . . . . . . . . . . . . . . 17  |-  ( ( ran  F  C_  (
0 [,] 1 )  /\  ( 0 [,] 1 )  C_  RR  /\  ( vol * `  ( 0 [,] 1
) )  e.  RR )  ->  ( vol * `  ran  F )  e.  RR )
5019, 48, 49mp3an23 1271 . . . . . . . . . . . . . . . 16  |-  ( ran 
F  C_  ( 0 [,] 1 )  -> 
( vol * `  ran  F )  e.  RR )
5118, 50syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( vol * `  ran  F )  e.  RR )
5251adantr 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  ( vol * `  ran  F
)  e.  RR )
53 simpr 448 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  0  <  ( vol * `  ran  F ) )
5452, 53elrpd 10606 . . . . . . . . . . . . 13  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  ( vol * `  ran  F
)  e.  RR+ )
55 rpdivcl 10594 . . . . . . . . . . . . 13  |-  ( ( 3  e.  RR+  /\  ( vol * `  ran  F
)  e.  RR+ )  ->  ( 3  /  ( vol * `  ran  F
) )  e.  RR+ )
5640, 54, 55sylancr 645 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  (
3  /  ( vol
* `  ran  F ) )  e.  RR+ )
5756rpred 10608 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  (
3  /  ( vol
* `  ran  F ) )  e.  RR )
5856rpge0d 10612 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  0  <_  ( 3  /  ( vol * `  ran  F
) ) )
59 flge0nn0 11184 . . . . . . . . . . 11  |-  ( ( ( 3  /  ( vol * `  ran  F
) )  e.  RR  /\  0  <_  ( 3  /  ( vol * `  ran  F ) ) )  ->  ( |_ `  ( 3  /  ( vol * `  ran  F
) ) )  e. 
NN0 )
6057, 58, 59syl2anc 643 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  ( |_ `  ( 3  / 
( vol * `  ran  F ) ) )  e.  NN0 )
61 nn0p1nn 10219 . . . . . . . . . 10  |-  ( ( |_ `  ( 3  /  ( vol * `  ran  F ) ) )  e.  NN0  ->  ( ( |_ `  (
3  /  ( vol
* `  ran  F ) ) )  +  1 )  e.  NN )
6260, 61syl 16 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  (
( |_ `  (
3  /  ( vol
* `  ran  F ) ) )  +  1 )  e.  NN )
6362nnred 9975 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  (
( |_ `  (
3  /  ( vol
* `  ran  F ) ) )  +  1 )  e.  RR )
6463, 52remulcld 9076 . . . . . . 7  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  (
( ( |_ `  ( 3  /  ( vol * `  ran  F
) ) )  +  1 )  x.  ( vol * `  ran  F
) )  e.  RR )
6564rexrd 9094 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  (
( ( |_ `  ( 3  /  ( vol * `  ran  F
) ) )  +  1 )  x.  ( vol * `  ran  F
) )  e.  RR* )
666elpw2 4328 . . . . . . . . . . . . . . . . . . 19  |-  ( ran 
F  e.  ~P RR  <->  ran 
F  C_  RR )
6720, 66sylibr 204 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ran  F  e.  ~P RR )
6867anim1i 552 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  -.  ran  F  e.  dom  vol )  -> 
( ran  F  e.  ~P RR  /\  -.  ran  F  e.  dom  vol )
)
69 eldif 3294 . . . . . . . . . . . . . . . . 17  |-  ( ran 
F  e.  ( ~P RR  \  dom  vol ) 
<->  ( ran  F  e. 
~P RR  /\  -.  ran  F  e.  dom  vol ) )
7068, 69sylibr 204 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  -.  ran  F  e.  dom  vol )  ->  ran  F  e.  ( ~P RR  \  dom  vol ) )
7170ex 424 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( -.  ran  F  e.  dom  vol  ->  ran  F  e.  ( ~P RR  \  dom  vol ) ) )
7216, 71mt3d 119 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  F  e.  dom  vol )
7372adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  ran  F  e.  dom  vol )
74 inss1 3525 . . . . . . . . . . . . . . . 16  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  QQ
75 qssre 10544 . . . . . . . . . . . . . . . 16  |-  QQ  C_  RR
7674, 75sstri 3321 . . . . . . . . . . . . . . 15  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  RR
77 fss 5562 . . . . . . . . . . . . . . 15  |-  ( ( G : NN --> ( QQ 
i^i  ( -u 1 [,] 1 ) )  /\  ( QQ  i^i  ( -u 1 [,] 1 ) )  C_  RR )  ->  G : NN --> RR )
7828, 76, 77sylancl 644 . . . . . . . . . . . . . 14  |-  ( ph  ->  G : NN --> RR )
7978ffvelrnda 5833 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 n )  e.  RR )
80 shftmbl 19390 . . . . . . . . . . . . 13  |-  ( ( ran  F  e.  dom  vol 
/\  ( G `  n )  e.  RR )  ->  { s  e.  RR  |  ( s  -  ( G `  n ) )  e. 
ran  F }  e.  dom  vol )
8173, 79, 80syl2anc 643 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  { s  e.  RR  |  ( s  -  ( G `
 n ) )  e.  ran  F }  e.  dom  vol )
8281, 5fmptd 5856 . . . . . . . . . . 11  |-  ( ph  ->  T : NN --> dom  vol )
8382ffvelrnda 5833 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN )  ->  ( T `
 m )  e. 
dom  vol )
8483ralrimiva 2753 . . . . . . . . 9  |-  ( ph  ->  A. m  e.  NN  ( T `  m )  e.  dom  vol )
85 iunmbl 19404 . . . . . . . . 9  |-  ( A. m  e.  NN  ( T `  m )  e.  dom  vol  ->  U_ m  e.  NN  ( T `  m )  e.  dom  vol )
8684, 85syl 16 . . . . . . . 8  |-  ( ph  ->  U_ m  e.  NN  ( T `  m )  e.  dom  vol )
87 mblss 19384 . . . . . . . 8  |-  ( U_ m  e.  NN  ( T `  m )  e.  dom  vol  ->  U_ m  e.  NN  ( T `  m )  C_  RR )
88 ovolcl 19331 . . . . . . . 8  |-  ( U_ m  e.  NN  ( T `  m )  C_  RR  ->  ( vol * `
 U_ m  e.  NN  ( T `  m ) )  e.  RR* )
8986, 87, 883syl 19 . . . . . . 7  |-  ( ph  ->  ( vol * `  U_ m  e.  NN  ( T `  m )
)  e.  RR* )
9089adantr 452 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  ( vol * `  U_ m  e.  NN  ( T `  m ) )  e. 
RR* )
91 flltp1 11168 . . . . . . . 8  |-  ( ( 3  /  ( vol
* `  ran  F ) )  e.  RR  ->  ( 3  /  ( vol
* `  ran  F ) )  <  ( ( |_ `  ( 3  /  ( vol * `  ran  F ) ) )  +  1 ) )
9257, 91syl 16 . . . . . . 7  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  (
3  /  ( vol
* `  ran  F ) )  <  ( ( |_ `  ( 3  /  ( vol * `  ran  F ) ) )  +  1 ) )
9335a1i 11 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  3  e.  RR )
9493, 63, 54ltdivmul2d 10656 . . . . . . 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
) ) ) )
9592, 94mpbid 202 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  3  <  ( ( ( |_
`  ( 3  / 
( vol * `  ran  F ) ) )  +  1 )  x.  ( vol * `  ran  F ) ) )
96 nnuz 10481 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
97 1z 10271 . . . . . . . . . . . 12  |-  1  e.  ZZ
9897a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  1  e.  ZZ )
99 mblvol 19383 . . . . . . . . . . . . . . . . 17  |-  ( ( T `  m )  e.  dom  vol  ->  ( vol `  ( T `
 m ) )  =  ( vol * `  ( T `  m
) ) )
10083, 99syl 16 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol `  ( T `  m
) )  =  ( vol * `  ( T `  m )
) )
101100, 34eqtrd 2440 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol `  ( T `  m
) )  =  ( vol * `  ran  F ) )
10251adantr 452 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol
* `  ran  F )  e.  RR )
103101, 102eqeltrd 2482 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol `  ( T `  m
) )  e.  RR )
104103adantlr 696 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  0  <  ( vol * `  ran  F ) )  /\  m  e.  NN )  ->  ( vol `  ( T `  m )
)  e.  RR )
105 eqid 2408 . . . . . . . . . . . . 13  |-  ( m  e.  NN  |->  ( vol `  ( T `  m
) ) )  =  ( m  e.  NN  |->  ( vol `  ( T `
 m ) ) )
106104, 105fmptd 5856 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  (
m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) : NN --> RR )
107106ffvelrnda 5833 . . . . . . . . . . 11  |-  ( ( ( ph  /\  0  <  ( vol * `  ran  F ) )  /\  k  e.  NN )  ->  ( ( m  e.  NN  |->  ( vol `  ( T `  m )
) ) `  k
)  e.  RR )
10896, 98, 107serfre 11311 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  seq  1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) ) : NN --> RR )
109 frn 5560 . . . . . . . . . 10  |-  (  seq  1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) ) : NN --> RR  ->  ran  seq  1
(  +  ,  ( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )  C_  RR )
110108, 109syl 16 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  ran  seq  1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )  C_  RR )
111 ressxr 9089 . . . . . . . . 9  |-  RR  C_  RR*
112110, 111syl6ss 3324 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  ran  seq  1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )  C_  RR* )
113101adantlr 696 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  0  <  ( vol * `  ran  F ) )  /\  m  e.  NN )  ->  ( vol `  ( T `  m )
)  =  ( vol
* `  ran  F ) )
114113mpteq2dva 4259 . . . . . . . . . . . . 13  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  (
m  e.  NN  |->  ( vol `  ( T `
 m ) ) )  =  ( m  e.  NN  |->  ( vol
* `  ran  F ) ) )
115 fconstmpt 4884 . . . . . . . . . . . . 13  |-  ( NN 
X.  { ( vol
* `  ran  F ) } )  =  ( m  e.  NN  |->  ( vol * `  ran  F ) )
116114, 115syl6eqr 2458 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  (
m  e.  NN  |->  ( vol `  ( T `
 m ) ) )  =  ( NN 
X.  { ( vol
* `  ran  F ) } ) )
117116seqeq3d 11290 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  seq  1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )  =  seq  1 (  +  , 
( NN  X.  {
( vol * `  ran  F ) } ) ) )
118117fveq1d 5693 . . . . . . . . . 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 ) ) )
11952recnd 9074 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  ( vol * `  ran  F
)  e.  CC )
120 ser1const 11338 . . . . . . . . . . 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 ) ) )
121119, 62, 120syl2anc 643 . . . . . . . . . 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 ) ) )
122118, 121eqtrd 2440 . . . . . . . . 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
) ) )
123 ffn 5554 . . . . . . . . . . 11  |-  (  seq  1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) ) : NN --> RR  ->  seq  1 (  +  ,  ( m  e.  NN  |->  ( vol `  ( T `  m
) ) ) )  Fn  NN )
124108, 123syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  seq  1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )  Fn  NN )
125 fnfvelrn 5830 . . . . . . . . . 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 ) ) ) ) )
126124, 62, 125syl2anc 643 . . . . . . . . 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 ) ) ) ) )
127122, 126eqeltrrd 2483 . . . . . . . 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 ) ) ) ) )
128 supxrub 10863 . . . . . . . 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* ,  <  ) )
129112, 127, 128syl2anc 643 . . . . . . 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* ,  <  )
)
13086adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  U_ m  e.  NN  ( T `  m )  e.  dom  vol )
131 mblvol 19383 . . . . . . . . 9  |-  ( U_ m  e.  NN  ( T `  m )  e.  dom  vol  ->  ( vol `  U_ m  e.  NN  ( T `  m ) )  =  ( vol
* `  U_ m  e.  NN  ( T `  m ) ) )
132130, 131syl 16 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  ( vol `  U_ m  e.  NN  ( T `  m ) )  =  ( vol * `  U_ m  e.  NN  ( T `  m )
) )
13383, 103jca 519 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN )  ->  ( ( T `  m )  e.  dom  vol  /\  ( vol `  ( T `
 m ) )  e.  RR ) )
134133ralrimiva 2753 . . . . . . . . . 10  |-  ( ph  ->  A. m  e.  NN  ( ( T `  m )  e.  dom  vol 
/\  ( vol `  ( T `  m )
)  e.  RR ) )
135134adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  A. m  e.  NN  ( ( T `
 m )  e. 
dom  vol  /\  ( vol `  ( T `  m
) )  e.  RR ) )
13611, 12, 13, 14, 15, 5, 16vitalilem3 19459 . . . . . . . . . 10  |-  ( ph  -> Disj  m  e.  NN ( T `  m )
)
137136adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  -> Disj  m  e.  NN ( T `  m ) )
138 eqid 2408 . . . . . . . . . 10  |-  seq  1
(  +  ,  ( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )  =  seq  1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )
139138, 105voliun 19405 . . . . . . . . 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* ,  <  ) )
140135, 137, 139syl2anc 643 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  ( vol `  U_ m  e.  NN  ( T `  m ) )  =  sup ( ran  seq  1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) ) ,  RR* ,  <  ) )
141132, 140eqtr3d 2442 . . . . . . 7  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  ( vol * `  U_ m  e.  NN  ( T `  m ) )  =  sup ( ran  seq  1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) ) ,  RR* ,  <  ) )
142129, 141breqtrrd 4202 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  (
( ( |_ `  ( 3  /  ( vol * `  ran  F
) ) )  +  1 )  x.  ( vol * `  ran  F
) )  <_  ( vol * `  U_ m  e.  NN  ( T `  m ) ) )
14337, 65, 90, 95, 142xrltletrd 10711 . . . . 5  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  3  <  ( vol * `  U_ m  e.  NN  ( T `  m )
) )
14417simp3d 971 . . . . . . . . 9  |-  ( ph  ->  U_ m  e.  NN  ( T `  m ) 
C_  ( -u 1 [,] 2 ) )
145144adantr 452 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  U_ m  e.  NN  ( T `  m )  C_  ( -u 1 [,] 2 ) )
146 2re 10029 . . . . . . . . 9  |-  2  e.  RR
147 iccssre 10952 . . . . . . . . 9  |-  ( (
-u 1  e.  RR  /\  2  e.  RR )  ->  ( -u 1 [,] 2 )  C_  RR )
14823, 146, 147mp2an 654 . . . . . . . 8  |-  ( -u
1 [,] 2 ) 
C_  RR
149 ovolss 19338 . . . . . . . 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 ) ) )
150145, 148, 149sylancl 644 . . . . . . 7  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  ( vol * `  U_ m  e.  NN  ( T `  m ) )  <_ 
( vol * `  ( -u 1 [,] 2
) ) )
151 2cn 10030 . . . . . . . . 9  |-  2  e.  CC
152151, 45subnegi 9339 . . . . . . . 8  |-  ( 2  -  -u 1 )  =  ( 2  +  1 )
153 0lt1 9510 . . . . . . . . . . . 12  |-  0  <  1
154 lt0neg2 9495 . . . . . . . . . . . . 13  |-  ( 1  e.  RR  ->  (
0  <  1  <->  -u 1  <  0 ) )
15522, 154ax-mp 8 . . . . . . . . . . . 12  |-  ( 0  <  1  <->  -u 1  <  0 )
156153, 155mpbi 200 . . . . . . . . . . 11  |-  -u 1  <  0
157 2pos 10042 . . . . . . . . . . 11  |-  0  <  2
15823, 41, 146lttri 9159 . . . . . . . . . . 11  |-  ( (
-u 1  <  0  /\  0  <  2
)  ->  -u 1  <  2 )
159156, 157, 158mp2an 654 . . . . . . . . . 10  |-  -u 1  <  2
16023, 146, 159ltleii 9156 . . . . . . . . 9  |-  -u 1  <_  2
161 ovolicc 19376 . . . . . . . . 9  |-  ( (
-u 1  e.  RR  /\  2  e.  RR  /\  -u 1  <_  2 )  ->  ( vol * `  ( -u 1 [,] 2 ) )  =  ( 2  -  -u 1
) )
16223, 146, 160, 161mp3an 1279 . . . . . . . 8  |-  ( vol
* `  ( -u 1 [,] 2 ) )  =  ( 2  -  -u 1
)
163 df-3 10019 . . . . . . . 8  |-  3  =  ( 2  +  1 )
164152, 162, 1633eqtr4i 2438 . . . . . . 7  |-  ( vol
* `  ( -u 1 [,] 2 ) )  =  3
165150, 164syl6breq 4215 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  ( vol * `  U_ m  e.  NN  ( T `  m ) )  <_ 
3 )
166 xrlenlt 9103 . . . . . . 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 )
) ) )
16790, 36, 166sylancl 644 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  (
( vol * `  U_ m  e.  NN  ( T `  m )
)  <_  3  <->  -.  3  <  ( vol * `  U_ m  e.  NN  ( T `  m )
) ) )
168165, 167mpbid 202 . . . . 5  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  -.  3  <  ( vol * `  U_ m  e.  NN  ( T `  m ) ) )
169143, 168pm2.65da 560 . . . 4  |-  ( ph  ->  -.  0  <  ( vol * `  ran  F
) )
170 ovolge0 19334 . . . . . . 7  |-  ( ran 
F  C_  RR  ->  0  <_  ( vol * `  ran  F ) )
17120, 170syl 16 . . . . . 6  |-  ( ph  ->  0  <_  ( vol * `
 ran  F )
)
172 0xr 9091 . . . . . . 7  |-  0  e.  RR*
173 ovolcl 19331 . . . . . . . 8  |-  ( ran 
F  C_  RR  ->  ( vol * `  ran  F )  e.  RR* )
17420, 173syl 16 . . . . . . 7  |-  ( ph  ->  ( vol * `  ran  F )  e.  RR* )
175 xrleloe 10697 . . . . . . 7  |-  ( ( 0  e.  RR*  /\  ( vol * `  ran  F
)  e.  RR* )  ->  ( 0  <_  ( vol * `  ran  F
)  <->  ( 0  < 
( vol * `  ran  F )  \/  0  =  ( vol * `  ran  F ) ) ) )
176172, 174, 175sylancr 645 . . . . . 6  |-  ( ph  ->  ( 0  <_  ( vol * `  ran  F
)  <->  ( 0  < 
( vol * `  ran  F )  \/  0  =  ( vol * `  ran  F ) ) ) )
177171, 176mpbid 202 . . . . 5  |-  ( ph  ->  ( 0  <  ( vol * `  ran  F
)  \/  0  =  ( vol * `  ran  F ) ) )
178177ord 367 . . . 4  |-  ( ph  ->  ( -.  0  < 
( vol * `  ran  F )  ->  0  =  ( vol * `  ran  F ) ) )
179169, 178mpd 15 . . 3  |-  ( ph  ->  0  =  ( vol
* `  ran  F ) )
180179adantr 452 . 2  |-  ( (
ph  /\  m  e.  NN )  ->  0  =  ( vol * `  ran  F ) )
18134, 180eqtr4d 2443 1  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol
* `  ( T `  m ) )  =  0 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2571   A.wral 2670   {crab 2674    \ cdif 3281    i^i cin 3283    C_ wss 3284   (/)c0 3592   ~Pcpw 3763   {csn 3778   U_ciun 4057  Disj wdisj 4146   class class class wbr 4176   {copab 4229    e. cmpt 4230    X. cxp 4839   dom cdm 4841   ran crn 4842    Fn wfn 5412   -->wf 5413   -1-1-onto->wf1o 5416   ` cfv 5417  (class class class)co 6044   /.cqs 6867   supcsup 7407   CCcc 8948   RRcr 8949   0cc0 8950   1c1 8951    + caddc 8953    x. cmul 8955   RR*cxr 9079    < clt 9080    <_ cle 9081    - cmin 9251   -ucneg 9252    / cdiv 9637   NNcn 9960   2c2 10009   3c3 10010   NN0cn0 10181   ZZcz 10242   QQcq 10534   RR+crp 10572   [,]cicc 10879   |_cfl 11160    seq cseq 11282   vol
*covol 19316   volcvol 19317
This theorem is referenced by:  vitalilem5  19461
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556  ax-cc 8275  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-disj 4147  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-of 6268  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6868  df-ec 6870  df-qs 6874  df-map 6983  df-pm 6984  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-fi 7378  df-sup 7408  df-oi 7439  df-card 7786  df-cda 8008  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-n0 10182  df-z 10243  df-uz 10449  df-q 10535  df-rp 10573  df-xneg 10670  df-xadd 10671  df-xmul 10672  df-ioo 10880  df-ico 10882  df-icc 10883  df-fz 11004  df-fzo 11095  df-fl 11161  df-seq 11283  df-exp 11342  df-hash 11578  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-clim 12241  df-rlim 12242  df-sum 12439  df-rest 13609  df-topgen 13626  df-psmet 16653  df-xmet 16654  df-met 16655  df-bl 16656  df-mopn 16657  df-top 16922  df-bases 16924  df-topon 16925  df-cmp 17408  df-ovol 19318  df-vol 19319
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