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Theorem vitalilem4 22146
Description: Lemma for vitali 22148. (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 5872 . . . . . . . . 9  |-  ( n  =  m  ->  ( G `  n )  =  ( G `  m ) )
21oveq2d 6312 . . . . . . . 8  |-  ( n  =  m  ->  (
s  -  ( G `
 n ) )  =  ( s  -  ( G `  m ) ) )
32eleq1d 2526 . . . . . . 7  |-  ( n  =  m  ->  (
( s  -  ( G `  n )
)  e.  ran  F  <->  ( s  -  ( G `
 m ) )  e.  ran  F ) )
43rabbidv 3101 . . . . . 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 9600 . . . . . . 7  |-  RR  e.  _V
76rabex 4607 . . . . . 6  |-  { s  e.  RR  |  ( s  -  ( G `
 m ) )  e.  ran  F }  e.  _V
84, 5, 7fvmpt 5956 . . . . 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 5876 . . 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 22144 . . . . . . 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 11692 . . . . . 6  |-  ( 0 [,] 1 )  C_  RR
2018, 19syl6ss 3511 . . . . 5  |-  ( ph  ->  ran  F  C_  RR )
2120adantr 465 . . . 4  |-  ( (
ph  /\  m  e.  NN )  ->  ran  F  C_  RR )
22 neg1rr 10661 . . . . . 6  |-  -u 1  e.  RR
23 1re 9612 . . . . . 6  |-  1  e.  RR
24 iccssre 11631 . . . . . 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 3715 . . . . . 6  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  ( -u 1 [,] 1
)
27 f1of 5822 . . . . . . . 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 6032 . . . . . 6  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  ( QQ  i^i  ( -u 1 [,] 1 ) ) )
3026, 29sseldi 3497 . . . . 5  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  ( -u 1 [,] 1 ) )
3125, 30sseldi 3497 . . . 4  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  RR )
32 eqidd 2458 . . . 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 22048 . . 3  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol* `  ran  F )  =  ( vol* `  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } ) )
3410, 33eqtr4d 2501 . 2  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol* `  ( T `  m ) )  =  ( vol* `  ran  F ) )
35 3re 10630 . . . . . . . 8  |-  3  e.  RR
3635rexri 9663 . . . . . . 7  |-  3  e.  RR*
3736a1i 11 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  3  e.  RR* )
38 3nn 10715 . . . . . . . . . . . . . 14  |-  3  e.  NN
39 nnrp 11254 . . . . . . . . . . . . . 14  |-  ( 3  e.  NN  ->  3  e.  RR+ )
4038, 39ax-mp 5 . . . . . . . . . . . . 13  |-  3  e.  RR+
41 0re 9613 . . . . . . . . . . . . . . . . . . . 20  |-  0  e.  RR
42 0le1 10097 . . . . . . . . . . . . . . . . . . . 20  |-  0  <_  1
43 ovolicc 22060 . . . . . . . . . . . . . . . . . . . 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 10667 . . . . . . . . . . . . . . . . . . 19  |-  ( 1  -  0 )  =  1
4644, 45eqtri 2486 . . . . . . . . . . . . . . . . . 18  |-  ( vol* `  ( 0 [,] 1 ) )  =  1
4746, 23eqeltri 2541 . . . . . . . . . . . . . . . . 17  |-  ( vol* `  ( 0 [,] 1 ) )  e.  RR
48 ovolsscl 22023 . . . . . . . . . . . . . . . . 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 11279 . . . . . . . . . . . . 13  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( vol* `  ran  F
)  e.  RR+ )
54 rpdivcl 11267 . . . . . . . . . . . . 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 11281 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
3  /  ( vol* `  ran  F ) )  e.  RR )
5755rpge0d 11285 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  0  <_  ( 3  /  ( vol* `  ran  F
) ) )
58 flge0nn0 11957 . . . . . . . . . . 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 10856 . . . . . . . . . 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 10571 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
( |_ `  (
3  /  ( vol* `  ran  F ) ) )  +  1 )  e.  RR )
6362, 51remulcld 9641 . . . . . . 7  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
( ( |_ `  ( 3  /  ( vol* `  ran  F
) ) )  +  1 )  x.  ( vol* `  ran  F
) )  e.  RR )
6463rexrd 9660 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
( ( |_ `  ( 3  /  ( vol* `  ran  F
) ) )  +  1 )  x.  ( vol* `  ran  F
) )  e.  RR* )
656elpw2 4620 . . . . . . . . . . . . . . . . . . 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 3481 . . . . . . . . . . . . . . . . 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 3714 . . . . . . . . . . . . . . . 16  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  QQ
74 qssre 11217 . . . . . . . . . . . . . . . 16  |-  QQ  C_  RR
7573, 74sstri 3508 . . . . . . . . . . . . . . 15  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  RR
76 fss 5745 . . . . . . . . . . . . . . 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 6032 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 n )  e.  RR )
79 shftmbl 22075 . . . . . . . . . . . . 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 6056 . . . . . . . . . . 11  |-  ( ph  ->  T : NN --> dom  vol )
8281ffvelrnda 6032 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN )  ->  ( T `
 m )  e. 
dom  vol )
8382ralrimiva 2871 . . . . . . . . 9  |-  ( ph  ->  A. m  e.  NN  ( T `  m )  e.  dom  vol )
84 iunmbl 22089 . . . . . . . . 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 22068 . . . . . . . 8  |-  ( U_ m  e.  NN  ( T `  m )  e.  dom  vol  ->  U_ m  e.  NN  ( T `  m )  C_  RR )
87 ovolcl 22015 . . . . . . . 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 11940 . . . . . . . 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 11329 . . . . . . 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 11141 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
96 1zzd 10916 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  1  e.  ZZ )
97 mblvol 22067 . . . . . . . . . . . . . . . . 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 2498 . . . . . . . . . . . . . . 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 2545 . . . . . . . . . . . . . 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 2457 . . . . . . . . . . . . 13  |-  ( m  e.  NN  |->  ( vol `  ( T `  m
) ) )  =  ( m  e.  NN  |->  ( vol `  ( T `
 m ) ) )
104102, 103fmptd 6056 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) : NN --> RR )
105104ffvelrnda 6032 . . . . . . . . . . 11  |-  ( ( ( ph  /\  0  <  ( vol* `  ran  F ) )  /\  k  e.  NN )  ->  ( ( m  e.  NN  |->  ( vol `  ( T `  m )
) ) `  k
)  e.  RR )
10695, 96, 105serfre 12139 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  seq 1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) ) : NN --> RR )
107 frn 5743 . . . . . . . . . 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 9654 . . . . . . . . 9  |-  RR  C_  RR*
110108, 109syl6ss 3511 . . . . . . . 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 4543 . . . . . . . . . . . . 13  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
m  e.  NN  |->  ( vol `  ( T `
 m ) ) )  =  ( m  e.  NN  |->  ( vol* `  ran  F ) ) )
113 fconstmpt 5052 . . . . . . . . . . . . 13  |-  ( NN 
X.  { ( vol* `  ran  F ) } )  =  ( m  e.  NN  |->  ( vol* `  ran  F ) )
114112, 113syl6eqr 2516 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
m  e.  NN  |->  ( vol `  ( T `
 m ) ) )  =  ( NN 
X.  { ( vol* `  ran  F ) } ) )
115114seqeq3d 12118 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  seq 1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )  =  seq 1 (  +  , 
( NN  X.  {
( vol* `  ran  F ) } ) ) )
116115fveq1d 5874 . . . . . . . . . 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 9639 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( vol* `  ran  F
)  e.  CC )
118 ser1const 12166 . . . . . . . . . . 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 2498 . . . . . . . . 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 5737 . . . . . . . . . . 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 6029 . . . . . . . . . 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 2546 . . . . . . . 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 11541 . . . . . . . 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 22067 . . . . . . . . 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 2871 . . . . . . . . . 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 22145 . . . . . . . . . 10  |-  ( ph  -> Disj  m  e.  NN  ( T `  m )
)
135134adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  -> Disj  m  e.  NN  ( T `  m ) )
136 eqid 2457 . . . . . . . . . 10  |-  seq 1
(  +  ,  ( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )  =  seq 1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )
137136, 103voliun 22090 . . . . . . . . 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 2500 . . . . . . 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 4482 . . . . . 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 11389 . . . . 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 10626 . . . . . . . . 9  |-  2  e.  RR
145 iccssre 11631 . . . . . . . . 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 22022 . . . . . . . 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 10627 . . . . . . . . 9  |-  2  e.  CC
150 ax-1cn 9567 . . . . . . . . 9  |-  1  e.  CC
151149, 150subnegi 9917 . . . . . . . 8  |-  ( 2  -  -u 1 )  =  ( 2  +  1 )
152 neg1lt0 10663 . . . . . . . . . . 11  |-  -u 1  <  0
153 2pos 10648 . . . . . . . . . . 11  |-  0  <  2
15422, 41, 144lttri 9727 . . . . . . . . . . 11  |-  ( (
-u 1  <  0  /\  0  <  2
)  ->  -u 1  <  2 )
155152, 153, 154mp2an 672 . . . . . . . . . 10  |-  -u 1  <  2
15622, 144, 155ltleii 9724 . . . . . . . . 9  |-  -u 1  <_  2
157 ovolicc 22060 . . . . . . . . 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 10616 . . . . . . . 8  |-  3  =  ( 2  +  1 )
160151, 158, 1593eqtr4i 2496 . . . . . . 7  |-  ( vol* `  ( -u 1 [,] 2 ) )  =  3
161148, 160syl6breq 4495 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( vol* `  U_ m  e.  NN  ( T `  m ) )  <_ 
3 )
162 xrlenlt 9669 . . . . . . 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 22018 . . . . . . 7  |-  ( ran 
F  C_  RR  ->  0  <_  ( vol* `  ran  F ) )
16720, 166syl 16 . . . . . 6  |-  ( ph  ->  0  <_  ( vol* `  ran  F ) )
168 0xr 9657 . . . . . . 7  |-  0  e.  RR*
169 ovolcl 22015 . . . . . . . 8  |-  ( ran 
F  C_  RR  ->  ( vol* `  ran  F )  e.  RR* )
17020, 169syl 16 . . . . . . 7  |-  ( ph  ->  ( vol* `  ran  F )  e.  RR* )
171 xrleloe 11375 . . . . . . 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 2501 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 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   {crab 2811    \ cdif 3468    i^i cin 3470    C_ wss 3471   (/)c0 3793   ~Pcpw 4015   {csn 4032   U_ciun 4332  Disj wdisj 4427   class class class wbr 4456   {copab 4514    |-> cmpt 4515    X. cxp 5006   dom cdm 5008   ran crn 5009    Fn wfn 5589   -->wf 5590   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296   /.cqs 7328   supcsup 7918   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514   RR*cxr 9644    < clt 9645    <_ cle 9646    - cmin 9824   -ucneg 9825    / cdiv 10227   NNcn 10556   2c2 10606   3c3 10607   NN0cn0 10816   QQcq 11207   RR+crp 11245   [,]cicc 11557   |_cfl 11930    seqcseq 12110   vol*covol 22000   volcvol 22001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cc 8832  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-disj 4428  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-ec 7331  df-qs 7335  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-fl 11932  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323  df-rlim 13324  df-sum 13521  df-rest 14840  df-topgen 14861  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-top 19526  df-bases 19528  df-topon 19529  df-cmp 20014  df-ovol 22002  df-vol 22003
This theorem is referenced by:  vitalilem5  22147
  Copyright terms: Public domain W3C validator