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Theorem vitalilem4 19503
Description: Lemma for vitali 19505. (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 5728 . . . . . . . . 9  |-  ( n  =  m  ->  ( G `  n )  =  ( G `  m ) )
21oveq2d 6097 . . . . . . . 8  |-  ( n  =  m  ->  (
s  -  ( G `
 n ) )  =  ( s  -  ( G `  m ) ) )
32eleq1d 2502 . . . . . . 7  |-  ( n  =  m  ->  (
( s  -  ( G `  n )
)  e.  ran  F  <->  ( s  -  ( G `
 m ) )  e.  ran  F ) )
43rabbidv 2948 . . . . . 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 9081 . . . . . . 7  |-  RR  e.  _V
76rabex 4354 . . . . . 6  |-  { s  e.  RR  |  ( s  -  ( G `
 m ) )  e.  ran  F }  e.  _V
84, 5, 7fvmpt 5806 . . . . 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 5732 . . 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 19501 . . . . . . 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 11042 . . . . . 6  |-  ( 0 [,] 1 )  C_  RR
2018, 19syl6ss 3360 . . . . 5  |-  ( ph  ->  ran  F  C_  RR )
2120adantr 452 . . . 4  |-  ( (
ph  /\  m  e.  NN )  ->  ran  F  C_  RR )
22 1re 9090 . . . . . . 7  |-  1  e.  RR
2322renegcli 9362 . . . . . 6  |-  -u 1  e.  RR
24 iccssre 10992 . . . . . 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 3562 . . . . . 6  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  ( -u 1 [,] 1
)
27 f1of 5674 . . . . . . . 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 5870 . . . . . 6  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  ( QQ  i^i  ( -u 1 [,] 1 ) ) )
3026, 29sseldi 3346 . . . . 5  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  ( -u 1 [,] 1 ) )
3125, 30sseldi 3346 . . . 4  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  RR )
32 eqidd 2437 . . . 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 19407 . . 3  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol
* `  ran  F )  =  ( vol * `  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } ) )
3410, 33eqtr4d 2471 . 2  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol
* `  ( T `  m ) )  =  ( vol * `  ran  F ) )
35 3re 10071 . . . . . . . 8  |-  3  e.  RR
3635rexri 9137 . . . . . . 7  |-  3  e.  RR*
3736a1i 11 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  3  e.  RR* )
38 3nn 10134 . . . . . . . . . . . . . 14  |-  3  e.  NN
39 nnrp 10621 . . . . . . . . . . . . . 14  |-  ( 3  e.  NN  ->  3  e.  RR+ )
4038, 39ax-mp 8 . . . . . . . . . . . . 13  |-  3  e.  RR+
41 0re 9091 . . . . . . . . . . . . . . . . . . . 20  |-  0  e.  RR
42 0le1 9551 . . . . . . . . . . . . . . . . . . . 20  |-  0  <_  1
43 ovolicc 19419 . . . . . . . . . . . . . . . . . . . 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 9048 . . . . . . . . . . . . . . . . . . . 20  |-  1  e.  CC
4645subid1i 9372 . . . . . . . . . . . . . . . . . . 19  |-  ( 1  -  0 )  =  1
4744, 46eqtri 2456 . . . . . . . . . . . . . . . . . 18  |-  ( vol
* `  ( 0 [,] 1 ) )  =  1
4847, 22eqeltri 2506 . . . . . . . . . . . . . . . . 17  |-  ( vol
* `  ( 0 [,] 1 ) )  e.  RR
49 ovolsscl 19382 . . . . . . . . . . . . . . . . 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 10646 . . . . . . . . . . . . 13  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  ( vol * `  ran  F
)  e.  RR+ )
55 rpdivcl 10634 . . . . . . . . . . . . 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 10648 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  (
3  /  ( vol
* `  ran  F ) )  e.  RR )
5856rpge0d 10652 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  0  <_  ( 3  /  ( vol * `  ran  F
) ) )
59 flge0nn0 11225 . . . . . . . . . . 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 10259 . . . . . . . . . 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 10015 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  (
( |_ `  (
3  /  ( vol
* `  ran  F ) ) )  +  1 )  e.  RR )
6463, 52remulcld 9116 . . . . . . 7  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  (
( ( |_ `  ( 3  /  ( vol * `  ran  F
) ) )  +  1 )  x.  ( vol * `  ran  F
) )  e.  RR )
6564rexrd 9134 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  (
( ( |_ `  ( 3  /  ( vol * `  ran  F
) ) )  +  1 )  x.  ( vol * `  ran  F
) )  e.  RR* )
666elpw2 4364 . . . . . . . . . . . . . . . . . . 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 3330 . . . . . . . . . . . . . . . . 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 3561 . . . . . . . . . . . . . . . 16  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  QQ
75 qssre 10584 . . . . . . . . . . . . . . . 16  |-  QQ  C_  RR
7674, 75sstri 3357 . . . . . . . . . . . . . . 15  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  RR
77 fss 5599 . . . . . . . . . . . . . . 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 5870 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 n )  e.  RR )
80 shftmbl 19433 . . . . . . . . . . . . 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 5893 . . . . . . . . . . 11  |-  ( ph  ->  T : NN --> dom  vol )
8382ffvelrnda 5870 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN )  ->  ( T `
 m )  e. 
dom  vol )
8483ralrimiva 2789 . . . . . . . . 9  |-  ( ph  ->  A. m  e.  NN  ( T `  m )  e.  dom  vol )
85 iunmbl 19447 . . . . . . . . 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 19427 . . . . . . . 8  |-  ( U_ m  e.  NN  ( T `  m )  e.  dom  vol  ->  U_ m  e.  NN  ( T `  m )  C_  RR )
88 ovolcl 19374 . . . . . . . 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 11209 . . . . . . . 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 10696 . . . . . . 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 10521 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
97 1z 10311 . . . . . . . . . . . 12  |-  1  e.  ZZ
9897a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  1  e.  ZZ )
99 mblvol 19426 . . . . . . . . . . . . . . . . 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 2468 . . . . . . . . . . . . . . 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 2510 . . . . . . . . . . . . . 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 2436 . . . . . . . . . . . . 13  |-  ( m  e.  NN  |->  ( vol `  ( T `  m
) ) )  =  ( m  e.  NN  |->  ( vol `  ( T `
 m ) ) )
106104, 105fmptd 5893 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  (
m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) : NN --> RR )
107106ffvelrnda 5870 . . . . . . . . . . 11  |-  ( ( ( ph  /\  0  <  ( vol * `  ran  F ) )  /\  k  e.  NN )  ->  ( ( m  e.  NN  |->  ( vol `  ( T `  m )
) ) `  k
)  e.  RR )
10896, 98, 107serfre 11352 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  seq  1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) ) : NN --> RR )
109 frn 5597 . . . . . . . . . 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 9129 . . . . . . . . 9  |-  RR  C_  RR*
112110, 111syl6ss 3360 . . . . . . . 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 4295 . . . . . . . . . . . . 13  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  (
m  e.  NN  |->  ( vol `  ( T `
 m ) ) )  =  ( m  e.  NN  |->  ( vol
* `  ran  F ) ) )
115 fconstmpt 4921 . . . . . . . . . . . . 13  |-  ( NN 
X.  { ( vol
* `  ran  F ) } )  =  ( m  e.  NN  |->  ( vol * `  ran  F ) )
116114, 115syl6eqr 2486 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  (
m  e.  NN  |->  ( vol `  ( T `
 m ) ) )  =  ( NN 
X.  { ( vol
* `  ran  F ) } ) )
117116seqeq3d 11331 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  seq  1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )  =  seq  1 (  +  , 
( NN  X.  {
( vol * `  ran  F ) } ) ) )
118117fveq1d 5730 . . . . . . . . . 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 9114 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  ( vol * `  ran  F
)  e.  CC )
120 ser1const 11379 . . . . . . . . . . 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 2468 . . . . . . . . 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 5591 . . . . . . . . . . 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 5867 . . . . . . . . . 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 2511 . . . . . . . 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 10903 . . . . . . . 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 19426 . . . . . . . . 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 2789 . . . . . . . . . 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 19502 . . . . . . . . . 10  |-  ( ph  -> Disj  m  e.  NN ( T `  m )
)
137136adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  -> Disj  m  e.  NN ( T `  m ) )
138 eqid 2436 . . . . . . . . . 10  |-  seq  1
(  +  ,  ( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )  =  seq  1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )
139138, 105voliun 19448 . . . . . . . . 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 2470 . . . . . . 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 4238 . . . . . 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 10751 . . . . 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 10069 . . . . . . . . 9  |-  2  e.  RR
147 iccssre 10992 . . . . . . . . 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 19381 . . . . . . . 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 10070 . . . . . . . . 9  |-  2  e.  CC
152151, 45subnegi 9379 . . . . . . . 8  |-  ( 2  -  -u 1 )  =  ( 2  +  1 )
153 0lt1 9550 . . . . . . . . . . . 12  |-  0  <  1
154 lt0neg2 9535 . . . . . . . . . . . . 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 10082 . . . . . . . . . . 11  |-  0  <  2
15823, 41, 146lttri 9199 . . . . . . . . . . 11  |-  ( (
-u 1  <  0  /\  0  <  2
)  ->  -u 1  <  2 )
159156, 157, 158mp2an 654 . . . . . . . . . 10  |-  -u 1  <  2
16023, 146, 159ltleii 9196 . . . . . . . . 9  |-  -u 1  <_  2
161 ovolicc 19419 . . . . . . . . 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 10059 . . . . . . . 8  |-  3  =  ( 2  +  1 )
164152, 162, 1633eqtr4i 2466 . . . . . . 7  |-  ( vol
* `  ( -u 1 [,] 2 ) )  =  3
165150, 164syl6breq 4251 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol * `  ran  F ) )  ->  ( vol * `  U_ m  e.  NN  ( T `  m ) )  <_ 
3 )
166 xrlenlt 9143 . . . . . . 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 19377 . . . . . . 7  |-  ( ran 
F  C_  RR  ->  0  <_  ( vol * `  ran  F ) )
17120, 170syl 16 . . . . . 6  |-  ( ph  ->  0  <_  ( vol * `
 ran  F )
)
172 0xr 9131 . . . . . . 7  |-  0  e.  RR*
173 ovolcl 19374 . . . . . . . 8  |-  ( ran 
F  C_  RR  ->  ( vol * `  ran  F )  e.  RR* )
17420, 173syl 16 . . . . . . 7  |-  ( ph  ->  ( vol * `  ran  F )  e.  RR* )
175 xrleloe 10737 . . . . . . 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 2471 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 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   {crab 2709    \ cdif 3317    i^i cin 3319    C_ wss 3320   (/)c0 3628   ~Pcpw 3799   {csn 3814   U_ciun 4093  Disj wdisj 4182   class class class wbr 4212   {copab 4265    e. cmpt 4266    X. cxp 4876   dom cdm 4878   ran crn 4879    Fn wfn 5449   -->wf 5450   -1-1-onto->wf1o 5453   ` cfv 5454  (class class class)co 6081   /.cqs 6904   supcsup 7445   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995   RR*cxr 9119    < clt 9120    <_ cle 9121    - cmin 9291   -ucneg 9292    / cdiv 9677   NNcn 10000   2c2 10049   3c3 10050   NN0cn0 10221   ZZcz 10282   QQcq 10574   RR+crp 10612   [,]cicc 10919   |_cfl 11201    seq cseq 11323   vol
*covol 19359   volcvol 19360
This theorem is referenced by:  vitalilem5  19504
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cc 8315  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-disj 4183  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-ec 6907  df-qs 6911  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-fi 7416  df-sup 7446  df-oi 7479  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ioo 10920  df-ico 10922  df-icc 10923  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-rlim 12283  df-sum 12480  df-rest 13650  df-topgen 13667  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-top 16963  df-bases 16965  df-topon 16966  df-cmp 17450  df-ovol 19361  df-vol 19362
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