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Theorem vitalilem4 22511
Description: Lemma for vitali 22513. (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 5825 . . . . . . . . 9  |-  ( n  =  m  ->  ( G `  n )  =  ( G `  m ) )
21oveq2d 6265 . . . . . . . 8  |-  ( n  =  m  ->  (
s  -  ( G `
 n ) )  =  ( s  -  ( G `  m ) ) )
32eleq1d 2490 . . . . . . 7  |-  ( n  =  m  ->  (
( s  -  ( G `  n )
)  e.  ran  F  <->  ( s  -  ( G `
 m ) )  e.  ran  F ) )
43rabbidv 3013 . . . . . 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 9581 . . . . . . 7  |-  RR  e.  _V
76rabex 4518 . . . . . 6  |-  { s  e.  RR  |  ( s  -  ( G `
 m ) )  e.  ran  F }  e.  _V
84, 5, 7fvmpt 5908 . . . . 5  |-  ( m  e.  NN  ->  ( T `  m )  =  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } )
98adantl 467 . . . 4  |-  ( (
ph  /\  m  e.  NN )  ->  ( T `
 m )  =  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } )
109fveq2d 5829 . . 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 22509 . . . . . . 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 1017 . . . . . 6  |-  ( ph  ->  ran  F  C_  (
0 [,] 1 ) )
19 unitssre 11730 . . . . . 6  |-  ( 0 [,] 1 )  C_  RR
2018, 19syl6ss 3419 . . . . 5  |-  ( ph  ->  ran  F  C_  RR )
2120adantr 466 . . . 4  |-  ( (
ph  /\  m  e.  NN )  ->  ran  F  C_  RR )
22 neg1rr 10665 . . . . . 6  |-  -u 1  e.  RR
23 1re 9593 . . . . . 6  |-  1  e.  RR
24 iccssre 11667 . . . . . 6  |-  ( (
-u 1  e.  RR  /\  1  e.  RR )  ->  ( -u 1 [,] 1 )  C_  RR )
2522, 23, 24mp2an 676 . . . . 5  |-  ( -u
1 [,] 1 ) 
C_  RR
26 inss2 3626 . . . . . 6  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  ( -u 1 [,] 1
)
27 f1of 5774 . . . . . . . 8  |-  ( G : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) )  ->  G : NN
--> ( QQ  i^i  ( -u 1 [,] 1 ) ) )
2815, 27syl 17 . . . . . . 7  |-  ( ph  ->  G : NN --> ( QQ 
i^i  ( -u 1 [,] 1 ) ) )
2928ffvelrnda 5981 . . . . . 6  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  ( QQ  i^i  ( -u 1 [,] 1 ) ) )
3026, 29sseldi 3405 . . . . 5  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  ( -u 1 [,] 1 ) )
3125, 30sseldi 3405 . . . 4  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  RR )
32 eqidd 2429 . . . 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 22406 . . 3  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol* `  ran  F )  =  ( vol* `  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } ) )
3410, 33eqtr4d 2465 . 2  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol* `  ( T `  m ) )  =  ( vol* `  ran  F ) )
35 3re 10634 . . . . . . . 8  |-  3  e.  RR
3635rexri 9644 . . . . . . 7  |-  3  e.  RR*
3736a1i 11 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  3  e.  RR* )
38 3nn 10719 . . . . . . . . . . . . . 14  |-  3  e.  NN
39 nnrp 11262 . . . . . . . . . . . . . 14  |-  ( 3  e.  NN  ->  3  e.  RR+ )
4038, 39ax-mp 5 . . . . . . . . . . . . 13  |-  3  e.  RR+
41 0re 9594 . . . . . . . . . . . . . . . . . . . 20  |-  0  e.  RR
42 0le1 10088 . . . . . . . . . . . . . . . . . . . 20  |-  0  <_  1
43 ovolicc 22419 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  0  <_  1 )  ->  ( vol* `  ( 0 [,] 1 ) )  =  ( 1  -  0 ) )
4441, 23, 42, 43mp3an 1360 . . . . . . . . . . . . . . . . . . 19  |-  ( vol* `  ( 0 [,] 1 ) )  =  ( 1  -  0 )
45 1m0e1 10671 . . . . . . . . . . . . . . . . . . 19  |-  ( 1  -  0 )  =  1
4644, 45eqtri 2450 . . . . . . . . . . . . . . . . . 18  |-  ( vol* `  ( 0 [,] 1 ) )  =  1
4746, 23eqeltri 2502 . . . . . . . . . . . . . . . . 17  |-  ( vol* `  ( 0 [,] 1 ) )  e.  RR
48 ovolsscl 22381 . . . . . . . . . . . . . . . . 17  |-  ( ( ran  F  C_  (
0 [,] 1 )  /\  ( 0 [,] 1 )  C_  RR  /\  ( vol* `  ( 0 [,] 1
) )  e.  RR )  ->  ( vol* `  ran  F )  e.  RR )
4919, 47, 48mp3an23 1352 . . . . . . . . . . . . . . . 16  |-  ( ran 
F  C_  ( 0 [,] 1 )  -> 
( vol* `  ran  F )  e.  RR )
5018, 49syl 17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( vol* `  ran  F )  e.  RR )
5150adantr 466 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( vol* `  ran  F
)  e.  RR )
52 simpr 462 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  0  <  ( vol* `  ran  F ) )
5351, 52elrpd 11289 . . . . . . . . . . . . 13  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( vol* `  ran  F
)  e.  RR+ )
54 rpdivcl 11276 . . . . . . . . . . . . 13  |-  ( ( 3  e.  RR+  /\  ( vol* `  ran  F
)  e.  RR+ )  ->  ( 3  /  ( vol* `  ran  F
) )  e.  RR+ )
5540, 53, 54sylancr 667 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
3  /  ( vol* `  ran  F ) )  e.  RR+ )
5655rpred 11292 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
3  /  ( vol* `  ran  F ) )  e.  RR )
5755rpge0d 11296 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  0  <_  ( 3  /  ( vol* `  ran  F
) ) )
58 flge0nn0 12004 . . . . . . . . . . 11  |-  ( ( ( 3  /  ( vol* `  ran  F
) )  e.  RR  /\  0  <_  ( 3  /  ( vol* `  ran  F ) ) )  ->  ( |_ `  ( 3  /  ( vol* `  ran  F
) ) )  e. 
NN0 )
5956, 57, 58syl2anc 665 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( |_ `  ( 3  / 
( vol* `  ran  F ) ) )  e.  NN0 )
60 nn0p1nn 10860 . . . . . . . . . 10  |-  ( ( |_ `  ( 3  /  ( vol* `  ran  F ) ) )  e.  NN0  ->  ( ( |_ `  (
3  /  ( vol* `  ran  F ) ) )  +  1 )  e.  NN )
6159, 60syl 17 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
( |_ `  (
3  /  ( vol* `  ran  F ) ) )  +  1 )  e.  NN )
6261nnred 10575 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
( |_ `  (
3  /  ( vol* `  ran  F ) ) )  +  1 )  e.  RR )
6362, 51remulcld 9622 . . . . . . 7  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
( ( |_ `  ( 3  /  ( vol* `  ran  F
) ) )  +  1 )  x.  ( vol* `  ran  F
) )  e.  RR )
6463rexrd 9641 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
( ( |_ `  ( 3  /  ( vol* `  ran  F
) ) )  +  1 )  x.  ( vol* `  ran  F
) )  e.  RR* )
656elpw2 4531 . . . . . . . . . . . . . . . . . . 19  |-  ( ran 
F  e.  ~P RR  <->  ran 
F  C_  RR )
6620, 65sylibr 215 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ran  F  e.  ~P RR )
6766anim1i 570 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  -.  ran  F  e.  dom  vol )  -> 
( ran  F  e.  ~P RR  /\  -.  ran  F  e.  dom  vol )
)
68 eldif 3389 . . . . . . . . . . . . . . . . 17  |-  ( ran 
F  e.  ( ~P RR  \  dom  vol ) 
<->  ( ran  F  e. 
~P RR  /\  -.  ran  F  e.  dom  vol ) )
6967, 68sylibr 215 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  -.  ran  F  e.  dom  vol )  ->  ran  F  e.  ( ~P RR  \  dom  vol ) )
7069ex 435 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( -.  ran  F  e.  dom  vol  ->  ran  F  e.  ( ~P RR  \  dom  vol ) ) )
7116, 70mt3d 128 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  F  e.  dom  vol )
7271adantr 466 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  ran  F  e.  dom  vol )
73 inss1 3625 . . . . . . . . . . . . . . . 16  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  QQ
74 qssre 11225 . . . . . . . . . . . . . . . 16  |-  QQ  C_  RR
7573, 74sstri 3416 . . . . . . . . . . . . . . 15  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  RR
76 fss 5697 . . . . . . . . . . . . . . 15  |-  ( ( G : NN --> ( QQ 
i^i  ( -u 1 [,] 1 ) )  /\  ( QQ  i^i  ( -u 1 [,] 1 ) )  C_  RR )  ->  G : NN --> RR )
7728, 75, 76sylancl 666 . . . . . . . . . . . . . 14  |-  ( ph  ->  G : NN --> RR )
7877ffvelrnda 5981 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 n )  e.  RR )
79 shftmbl 22434 . . . . . . . . . . . . 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 665 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  { s  e.  RR  |  ( s  -  ( G `
 n ) )  e.  ran  F }  e.  dom  vol )
8180, 5fmptd 6005 . . . . . . . . . . 11  |-  ( ph  ->  T : NN --> dom  vol )
8281ffvelrnda 5981 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN )  ->  ( T `
 m )  e. 
dom  vol )
8382ralrimiva 2779 . . . . . . . . 9  |-  ( ph  ->  A. m  e.  NN  ( T `  m )  e.  dom  vol )
84 iunmbl 22448 . . . . . . . . 9  |-  ( A. m  e.  NN  ( T `  m )  e.  dom  vol  ->  U_ m  e.  NN  ( T `  m )  e.  dom  vol )
8583, 84syl 17 . . . . . . . 8  |-  ( ph  ->  U_ m  e.  NN  ( T `  m )  e.  dom  vol )
86 mblss 22427 . . . . . . . 8  |-  ( U_ m  e.  NN  ( T `  m )  e.  dom  vol  ->  U_ m  e.  NN  ( T `  m )  C_  RR )
87 ovolcl 22373 . . . . . . . 8  |-  ( U_ m  e.  NN  ( T `  m )  C_  RR  ->  ( vol* `  U_ m  e.  NN  ( T `  m ) )  e. 
RR* )
8885, 86, 873syl 18 . . . . . . 7  |-  ( ph  ->  ( vol* `  U_ m  e.  NN  ( T `  m )
)  e.  RR* )
8988adantr 466 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( vol* `  U_ m  e.  NN  ( T `  m ) )  e. 
RR* )
90 flltp1 11986 . . . . . . . 8  |-  ( ( 3  /  ( vol* `  ran  F ) )  e.  RR  ->  ( 3  /  ( vol* `  ran  F ) )  <  ( ( |_ `  ( 3  /  ( vol* `  ran  F ) ) )  +  1 ) )
9156, 90syl 17 . . . . . . 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 11341 . . . . . . 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 213 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  3  <  ( ( ( |_
`  ( 3  / 
( vol* `  ran  F ) ) )  +  1 )  x.  ( vol* `  ran  F ) ) )
95 nnuz 11145 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
96 1zzd 10919 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  1  e.  ZZ )
97 mblvol 22426 . . . . . . . . . . . . . . . . 17  |-  ( ( T `  m )  e.  dom  vol  ->  ( vol `  ( T `
 m ) )  =  ( vol* `  ( T `  m
) ) )
9882, 97syl 17 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol `  ( T `  m
) )  =  ( vol* `  ( T `  m )
) )
9998, 34eqtrd 2462 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol `  ( T `  m
) )  =  ( vol* `  ran  F ) )
10050adantr 466 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol* `  ran  F )  e.  RR )
10199, 100eqeltrd 2506 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol `  ( T `  m
) )  e.  RR )
102101adantlr 719 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  0  <  ( vol* `  ran  F ) )  /\  m  e.  NN )  ->  ( vol `  ( T `  m )
)  e.  RR )
103 eqid 2428 . . . . . . . . . . . . 13  |-  ( m  e.  NN  |->  ( vol `  ( T `  m
) ) )  =  ( m  e.  NN  |->  ( vol `  ( T `
 m ) ) )
104102, 103fmptd 6005 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) : NN --> RR )
105104ffvelrnda 5981 . . . . . . . . . . 11  |-  ( ( ( ph  /\  0  <  ( vol* `  ran  F ) )  /\  k  e.  NN )  ->  ( ( m  e.  NN  |->  ( vol `  ( T `  m )
) ) `  k
)  e.  RR )
10695, 96, 105serfre 12192 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  seq 1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) ) : NN --> RR )
107 frn 5695 . . . . . . . . . 10  |-  (  seq 1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) ) : NN --> RR  ->  ran  seq 1
(  +  ,  ( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )  C_  RR )
108106, 107syl 17 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ran  seq 1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )  C_  RR )
109 ressxr 9635 . . . . . . . . 9  |-  RR  C_  RR*
110108, 109syl6ss 3419 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ran  seq 1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )  C_  RR* )
11199adantlr 719 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  0  <  ( vol* `  ran  F ) )  /\  m  e.  NN )  ->  ( vol `  ( T `  m )
)  =  ( vol* `  ran  F ) )
112111mpteq2dva 4453 . . . . . . . . . . . . 13  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
m  e.  NN  |->  ( vol `  ( T `
 m ) ) )  =  ( m  e.  NN  |->  ( vol* `  ran  F ) ) )
113 fconstmpt 4840 . . . . . . . . . . . . 13  |-  ( NN 
X.  { ( vol* `  ran  F ) } )  =  ( m  e.  NN  |->  ( vol* `  ran  F ) )
114112, 113syl6eqr 2480 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
m  e.  NN  |->  ( vol `  ( T `
 m ) ) )  =  ( NN 
X.  { ( vol* `  ran  F ) } ) )
115114seqeq3d 12171 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  seq 1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )  =  seq 1 (  +  , 
( NN  X.  {
( vol* `  ran  F ) } ) ) )
116115fveq1d 5827 . . . . . . . . . 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 9620 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( vol* `  ran  F
)  e.  CC )
118 ser1const 12219 . . . . . . . . . . 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 665 . . . . . . . . . 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 2462 . . . . . . . . 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 5689 . . . . . . . . . . 11  |-  (  seq 1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) ) : NN --> RR  ->  seq 1 (  +  ,  ( m  e.  NN  |->  ( vol `  ( T `  m )
) ) )  Fn  NN )
122106, 121syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  seq 1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )  Fn  NN )
123 fnfvelrn 5978 . . . . . . . . . 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 665 . . . . . . . . 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 2507 . . . . . . . 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 11561 . . . . . . . 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 665 . . . . . . 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 466 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  U_ m  e.  NN  ( T `  m )  e.  dom  vol )
129 mblvol 22426 . . . . . . . . 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 17 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( vol `  U_ m  e.  NN  ( T `  m ) )  =  ( vol* `  U_ m  e.  NN  ( T `  m )
) )
13182, 101jca 534 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN )  ->  ( ( T `  m )  e.  dom  vol  /\  ( vol `  ( T `
 m ) )  e.  RR ) )
132131ralrimiva 2779 . . . . . . . . . 10  |-  ( ph  ->  A. m  e.  NN  ( ( T `  m )  e.  dom  vol 
/\  ( vol `  ( T `  m )
)  e.  RR ) )
133132adantr 466 . . . . . . . . 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 22510 . . . . . . . . . 10  |-  ( ph  -> Disj  m  e.  NN  ( T `  m )
)
135134adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  -> Disj  m  e.  NN  ( T `  m ) )
136 eqid 2428 . . . . . . . . . 10  |-  seq 1
(  +  ,  ( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )  =  seq 1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )
137136, 103voliun 22449 . . . . . . . . 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 665 . . . . . . . 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 2464 . . . . . . 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 4393 . . . . . 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 11409 . . . . 5  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  3  <  ( vol* `  U_ m  e.  NN  ( T `  m )
) )
14217simp3d 1019 . . . . . . . . 9  |-  ( ph  ->  U_ m  e.  NN  ( T `  m ) 
C_  ( -u 1 [,] 2 ) )
143142adantr 466 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  U_ m  e.  NN  ( T `  m )  C_  ( -u 1 [,] 2 ) )
144 2re 10630 . . . . . . . . 9  |-  2  e.  RR
145 iccssre 11667 . . . . . . . . 9  |-  ( (
-u 1  e.  RR  /\  2  e.  RR )  ->  ( -u 1 [,] 2 )  C_  RR )
14622, 144, 145mp2an 676 . . . . . . . 8  |-  ( -u
1 [,] 2 ) 
C_  RR
147 ovolss 22380 . . . . . . . 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 666 . . . . . . 7  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( vol* `  U_ m  e.  NN  ( T `  m ) )  <_ 
( vol* `  ( -u 1 [,] 2
) ) )
149 2cn 10631 . . . . . . . . 9  |-  2  e.  CC
150 ax-1cn 9548 . . . . . . . . 9  |-  1  e.  CC
151149, 150subnegi 9904 . . . . . . . 8  |-  ( 2  -  -u 1 )  =  ( 2  +  1 )
152 neg1lt0 10667 . . . . . . . . . . 11  |-  -u 1  <  0
153 2pos 10652 . . . . . . . . . . 11  |-  0  <  2
15422, 41, 144lttri 9711 . . . . . . . . . . 11  |-  ( (
-u 1  <  0  /\  0  <  2
)  ->  -u 1  <  2 )
155152, 153, 154mp2an 676 . . . . . . . . . 10  |-  -u 1  <  2
15622, 144, 155ltleii 9708 . . . . . . . . 9  |-  -u 1  <_  2
157 ovolicc 22419 . . . . . . . . 9  |-  ( (
-u 1  e.  RR  /\  2  e.  RR  /\  -u 1  <_  2 )  ->  ( vol* `  ( -u 1 [,] 2 ) )  =  ( 2  -  -u 1
) )
15822, 144, 156, 157mp3an 1360 . . . . . . . 8  |-  ( vol* `  ( -u 1 [,] 2 ) )  =  ( 2  -  -u 1
)
159 df-3 10620 . . . . . . . 8  |-  3  =  ( 2  +  1 )
160151, 158, 1593eqtr4i 2460 . . . . . . 7  |-  ( vol* `  ( -u 1 [,] 2 ) )  =  3
161148, 160syl6breq 4406 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( vol* `  U_ m  e.  NN  ( T `  m ) )  <_ 
3 )
162 xrlenlt 9650 . . . . . . 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 666 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
( vol* `  U_ m  e.  NN  ( T `  m )
)  <_  3  <->  -.  3  <  ( vol* `  U_ m  e.  NN  ( T `  m )
) ) )
164161, 163mpbid 213 . . . . 5  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  -.  3  <  ( vol* `  U_ m  e.  NN  ( T `  m ) ) )
165141, 164pm2.65da 578 . . . 4  |-  ( ph  ->  -.  0  <  ( vol* `  ran  F
) )
166 ovolge0 22376 . . . . . . 7  |-  ( ran 
F  C_  RR  ->  0  <_  ( vol* `  ran  F ) )
16720, 166syl 17 . . . . . 6  |-  ( ph  ->  0  <_  ( vol* `  ran  F ) )
168 0xr 9638 . . . . . . 7  |-  0  e.  RR*
169 ovolcl 22373 . . . . . . . 8  |-  ( ran 
F  C_  RR  ->  ( vol* `  ran  F )  e.  RR* )
17020, 169syl 17 . . . . . . 7  |-  ( ph  ->  ( vol* `  ran  F )  e.  RR* )
171 xrleloe 11394 . . . . . . 7  |-  ( ( 0  e.  RR*  /\  ( vol* `  ran  F
)  e.  RR* )  ->  ( 0  <_  ( vol* `  ran  F
)  <->  ( 0  < 
( vol* `  ran  F )  \/  0  =  ( vol* `  ran  F ) ) ) )
172168, 170, 171sylancr 667 . . . . . 6  |-  ( ph  ->  ( 0  <_  ( vol* `  ran  F
)  <->  ( 0  < 
( vol* `  ran  F )  \/  0  =  ( vol* `  ran  F ) ) ) )
173167, 172mpbid 213 . . . . 5  |-  ( ph  ->  ( 0  <  ( vol* `  ran  F
)  \/  0  =  ( vol* `  ran  F ) ) )
174173ord 378 . . . 4  |-  ( ph  ->  ( -.  0  < 
( vol* `  ran  F )  ->  0  =  ( vol* `  ran  F ) ) )
175165, 174mpd 15 . . 3  |-  ( ph  ->  0  =  ( vol* `  ran  F ) )
176175adantr 466 . 2  |-  ( (
ph  /\  m  e.  NN )  ->  0  =  ( vol* `  ran  F ) )
17734, 176eqtr4d 2465 1  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol* `  ( T `  m ) )  =  0 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2599   A.wral 2714   {crab 2718    \ cdif 3376    i^i cin 3378    C_ wss 3379   (/)c0 3704   ~Pcpw 3924   {csn 3941   U_ciun 4242  Disj wdisj 4337   class class class wbr 4366   {copab 4424    |-> cmpt 4425    X. cxp 4794   dom cdm 4796   ran crn 4797    Fn wfn 5539   -->wf 5540   -1-1-onto->wf1o 5543   ` cfv 5544  (class class class)co 6249   /.cqs 7317   supcsup 7907   CCcc 9488   RRcr 9489   0cc0 9490   1c1 9491    + caddc 9493    x. cmul 9495   RR*cxr 9625    < clt 9626    <_ cle 9627    - cmin 9811   -ucneg 9812    / cdiv 10220   NNcn 10560   2c2 10610   3c3 10611   NN0cn0 10820   QQcq 11215   RR+crp 11253   [,]cicc 11589   |_cfl 11976    seqcseq 12163   vol*covol 22355   volcvol 22357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-inf2 8099  ax-cc 8816  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-disj 4338  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-se 4756  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-isom 5553  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-of 6489  df-om 6651  df-1st 6751  df-2nd 6752  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-1o 7137  df-2o 7138  df-oadd 7141  df-er 7318  df-ec 7320  df-qs 7324  df-map 7429  df-pm 7430  df-en 7525  df-dom 7526  df-sdom 7527  df-fin 7528  df-fi 7878  df-sup 7909  df-inf 7910  df-oi 7978  df-card 8325  df-cda 8549  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-div 10221  df-nn 10561  df-2 10619  df-3 10620  df-n0 10821  df-z 10889  df-uz 11111  df-q 11216  df-rp 11254  df-xneg 11360  df-xadd 11361  df-xmul 11362  df-ioo 11590  df-ico 11592  df-icc 11593  df-fz 11736  df-fzo 11867  df-fl 11978  df-seq 12164  df-exp 12223  df-hash 12466  df-cj 13106  df-re 13107  df-im 13108  df-sqrt 13242  df-abs 13243  df-clim 13495  df-rlim 13496  df-sum 13696  df-rest 15264  df-topgen 15285  df-psmet 18905  df-xmet 18906  df-met 18907  df-bl 18908  df-mopn 18909  df-top 19863  df-bases 19864  df-topon 19865  df-cmp 20344  df-ovol 22358  df-vol 22360
This theorem is referenced by:  vitalilem5  22512
  Copyright terms: Public domain W3C validator