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Theorem vitalilem4 21050
Description: Lemma for vitali 21052. (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 5688 . . . . . . . . 9  |-  ( n  =  m  ->  ( G `  n )  =  ( G `  m ) )
21oveq2d 6106 . . . . . . . 8  |-  ( n  =  m  ->  (
s  -  ( G `
 n ) )  =  ( s  -  ( G `  m ) ) )
32eleq1d 2507 . . . . . . 7  |-  ( n  =  m  ->  (
( s  -  ( G `  n )
)  e.  ran  F  <->  ( s  -  ( G `
 m ) )  e.  ran  F ) )
43rabbidv 2962 . . . . . 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 9369 . . . . . . 7  |-  RR  e.  _V
76rabex 4440 . . . . . 6  |-  { s  e.  RR  |  ( s  -  ( G `
 m ) )  e.  ran  F }  e.  _V
84, 5, 7fvmpt 5771 . . . . 5  |-  ( m  e.  NN  ->  ( T `  m )  =  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } )
98adantl 463 . . . 4  |-  ( (
ph  /\  m  e.  NN )  ->  ( T `
 m )  =  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } )
109fveq2d 5692 . . 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 21048 . . . . . . 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 995 . . . . . 6  |-  ( ph  ->  ran  F  C_  (
0 [,] 1 ) )
19 unitssre 11428 . . . . . 6  |-  ( 0 [,] 1 )  C_  RR
2018, 19syl6ss 3365 . . . . 5  |-  ( ph  ->  ran  F  C_  RR )
2120adantr 462 . . . 4  |-  ( (
ph  /\  m  e.  NN )  ->  ran  F  C_  RR )
22 neg1rr 10422 . . . . . 6  |-  -u 1  e.  RR
23 1re 9381 . . . . . 6  |-  1  e.  RR
24 iccssre 11373 . . . . . 6  |-  ( (
-u 1  e.  RR  /\  1  e.  RR )  ->  ( -u 1 [,] 1 )  C_  RR )
2522, 23, 24mp2an 667 . . . . 5  |-  ( -u
1 [,] 1 ) 
C_  RR
26 inss2 3568 . . . . . 6  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  ( -u 1 [,] 1
)
27 f1of 5638 . . . . . . . 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 5840 . . . . . 6  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  ( QQ  i^i  ( -u 1 [,] 1 ) ) )
3026, 29sseldi 3351 . . . . 5  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  ( -u 1 [,] 1 ) )
3125, 30sseldi 3351 . . . 4  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  RR )
32 eqidd 2442 . . . 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 20953 . . 3  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol* `  ran  F )  =  ( vol* `  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } ) )
3410, 33eqtr4d 2476 . 2  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol* `  ( T `  m ) )  =  ( vol* `  ran  F ) )
35 3re 10391 . . . . . . . 8  |-  3  e.  RR
3635rexri 9432 . . . . . . 7  |-  3  e.  RR*
3736a1i 11 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  3  e.  RR* )
38 3nn 10476 . . . . . . . . . . . . . 14  |-  3  e.  NN
39 nnrp 10996 . . . . . . . . . . . . . 14  |-  ( 3  e.  NN  ->  3  e.  RR+ )
4038, 39ax-mp 5 . . . . . . . . . . . . 13  |-  3  e.  RR+
41 0re 9382 . . . . . . . . . . . . . . . . . . . 20  |-  0  e.  RR
42 0le1 9859 . . . . . . . . . . . . . . . . . . . 20  |-  0  <_  1
43 ovolicc 20965 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  0  <_  1 )  ->  ( vol* `  ( 0 [,] 1 ) )  =  ( 1  -  0 ) )
4441, 23, 42, 43mp3an 1309 . . . . . . . . . . . . . . . . . . 19  |-  ( vol* `  ( 0 [,] 1 ) )  =  ( 1  -  0 )
45 1m0e1 10428 . . . . . . . . . . . . . . . . . . 19  |-  ( 1  -  0 )  =  1
4644, 45eqtri 2461 . . . . . . . . . . . . . . . . . 18  |-  ( vol* `  ( 0 [,] 1 ) )  =  1
4746, 23eqeltri 2511 . . . . . . . . . . . . . . . . 17  |-  ( vol* `  ( 0 [,] 1 ) )  e.  RR
48 ovolsscl 20928 . . . . . . . . . . . . . . . . 17  |-  ( ( ran  F  C_  (
0 [,] 1 )  /\  ( 0 [,] 1 )  C_  RR  /\  ( vol* `  ( 0 [,] 1
) )  e.  RR )  ->  ( vol* `  ran  F )  e.  RR )
4919, 47, 48mp3an23 1301 . . . . . . . . . . . . . . . 16  |-  ( ran 
F  C_  ( 0 [,] 1 )  -> 
( vol* `  ran  F )  e.  RR )
5018, 49syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( vol* `  ran  F )  e.  RR )
5150adantr 462 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( vol* `  ran  F
)  e.  RR )
52 simpr 458 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  0  <  ( vol* `  ran  F ) )
5351, 52elrpd 11021 . . . . . . . . . . . . 13  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( vol* `  ran  F
)  e.  RR+ )
54 rpdivcl 11009 . . . . . . . . . . . . 13  |-  ( ( 3  e.  RR+  /\  ( vol* `  ran  F
)  e.  RR+ )  ->  ( 3  /  ( vol* `  ran  F
) )  e.  RR+ )
5540, 53, 54sylancr 658 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
3  /  ( vol* `  ran  F ) )  e.  RR+ )
5655rpred 11023 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
3  /  ( vol* `  ran  F ) )  e.  RR )
5755rpge0d 11027 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  0  <_  ( 3  /  ( vol* `  ran  F
) ) )
58 flge0nn0 11662 . . . . . . . . . . 11  |-  ( ( ( 3  /  ( vol* `  ran  F
) )  e.  RR  /\  0  <_  ( 3  /  ( vol* `  ran  F ) ) )  ->  ( |_ `  ( 3  /  ( vol* `  ran  F
) ) )  e. 
NN0 )
5956, 57, 58syl2anc 656 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( |_ `  ( 3  / 
( vol* `  ran  F ) ) )  e.  NN0 )
60 nn0p1nn 10615 . . . . . . . . . 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 10333 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
( |_ `  (
3  /  ( vol* `  ran  F ) ) )  +  1 )  e.  RR )
6362, 51remulcld 9410 . . . . . . 7  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
( ( |_ `  ( 3  /  ( vol* `  ran  F
) ) )  +  1 )  x.  ( vol* `  ran  F
) )  e.  RR )
6463rexrd 9429 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
( ( |_ `  ( 3  /  ( vol* `  ran  F
) ) )  +  1 )  x.  ( vol* `  ran  F
) )  e.  RR* )
656elpw2 4453 . . . . . . . . . . . . . . . . . . 19  |-  ( ran 
F  e.  ~P RR  <->  ran 
F  C_  RR )
6620, 65sylibr 212 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ran  F  e.  ~P RR )
6766anim1i 565 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  -.  ran  F  e.  dom  vol )  -> 
( ran  F  e.  ~P RR  /\  -.  ran  F  e.  dom  vol )
)
68 eldif 3335 . . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  ran  F  e.  dom  vol )
73 inss1 3567 . . . . . . . . . . . . . . . 16  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  QQ
74 qssre 10959 . . . . . . . . . . . . . . . 16  |-  QQ  C_  RR
7573, 74sstri 3362 . . . . . . . . . . . . . . 15  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  RR
76 fss 5564 . . . . . . . . . . . . . . 15  |-  ( ( G : NN --> ( QQ 
i^i  ( -u 1 [,] 1 ) )  /\  ( QQ  i^i  ( -u 1 [,] 1 ) )  C_  RR )  ->  G : NN --> RR )
7728, 75, 76sylancl 657 . . . . . . . . . . . . . 14  |-  ( ph  ->  G : NN --> RR )
7877ffvelrnda 5840 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 n )  e.  RR )
79 shftmbl 20979 . . . . . . . . . . . . 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 656 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  { s  e.  RR  |  ( s  -  ( G `
 n ) )  e.  ran  F }  e.  dom  vol )
8180, 5fmptd 5864 . . . . . . . . . . 11  |-  ( ph  ->  T : NN --> dom  vol )
8281ffvelrnda 5840 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN )  ->  ( T `
 m )  e. 
dom  vol )
8382ralrimiva 2797 . . . . . . . . 9  |-  ( ph  ->  A. m  e.  NN  ( T `  m )  e.  dom  vol )
84 iunmbl 20993 . . . . . . . . 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 20973 . . . . . . . 8  |-  ( U_ m  e.  NN  ( T `  m )  e.  dom  vol  ->  U_ m  e.  NN  ( T `  m )  C_  RR )
87 ovolcl 20920 . . . . . . . 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 462 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( vol* `  U_ m  e.  NN  ( T `  m ) )  e. 
RR* )
90 flltp1 11646 . . . . . . . 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 11071 . . . . . . 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 10892 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
96 1zzd 10673 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  1  e.  ZZ )
97 mblvol 20972 . . . . . . . . . . . . . . . . 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 2473 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol `  ( T `  m
) )  =  ( vol* `  ran  F ) )
10050adantr 462 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol* `  ran  F )  e.  RR )
10199, 100eqeltrd 2515 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol `  ( T `  m
) )  e.  RR )
102101adantlr 709 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  0  <  ( vol* `  ran  F ) )  /\  m  e.  NN )  ->  ( vol `  ( T `  m )
)  e.  RR )
103 eqid 2441 . . . . . . . . . . . . 13  |-  ( m  e.  NN  |->  ( vol `  ( T `  m
) ) )  =  ( m  e.  NN  |->  ( vol `  ( T `
 m ) ) )
104102, 103fmptd 5864 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) : NN --> RR )
105104ffvelrnda 5840 . . . . . . . . . . 11  |-  ( ( ( ph  /\  0  <  ( vol* `  ran  F ) )  /\  k  e.  NN )  ->  ( ( m  e.  NN  |->  ( vol `  ( T `  m )
) ) `  k
)  e.  RR )
10695, 96, 105serfre 11831 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  seq 1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) ) : NN --> RR )
107 frn 5562 . . . . . . . . . 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 9423 . . . . . . . . 9  |-  RR  C_  RR*
110108, 109syl6ss 3365 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ran  seq 1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )  C_  RR* )
11199adantlr 709 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  0  <  ( vol* `  ran  F ) )  /\  m  e.  NN )  ->  ( vol `  ( T `  m )
)  =  ( vol* `  ran  F ) )
112111mpteq2dva 4375 . . . . . . . . . . . . 13  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
m  e.  NN  |->  ( vol `  ( T `
 m ) ) )  =  ( m  e.  NN  |->  ( vol* `  ran  F ) ) )
113 fconstmpt 4878 . . . . . . . . . . . . 13  |-  ( NN 
X.  { ( vol* `  ran  F ) } )  =  ( m  e.  NN  |->  ( vol* `  ran  F ) )
114112, 113syl6eqr 2491 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
m  e.  NN  |->  ( vol `  ( T `
 m ) ) )  =  ( NN 
X.  { ( vol* `  ran  F ) } ) )
115114seqeq3d 11810 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  seq 1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )  =  seq 1 (  +  , 
( NN  X.  {
( vol* `  ran  F ) } ) ) )
116115fveq1d 5690 . . . . . . . . . 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 9408 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( vol* `  ran  F
)  e.  CC )
118 ser1const 11858 . . . . . . . . . . 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 656 . . . . . . . . . 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 2473 . . . . . . . . 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 5556 . . . . . . . . . . 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 5837 . . . . . . . . . 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 656 . . . . . . . . 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 2516 . . . . . . . 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 11283 . . . . . . . 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 656 . . . . . . 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 462 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  U_ m  e.  NN  ( T `  m )  e.  dom  vol )
129 mblvol 20972 . . . . . . . . 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 529 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN )  ->  ( ( T `  m )  e.  dom  vol  /\  ( vol `  ( T `
 m ) )  e.  RR ) )
132131ralrimiva 2797 . . . . . . . . . 10  |-  ( ph  ->  A. m  e.  NN  ( ( T `  m )  e.  dom  vol 
/\  ( vol `  ( T `  m )
)  e.  RR ) )
133132adantr 462 . . . . . . . . 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 21049 . . . . . . . . . 10  |-  ( ph  -> Disj  m  e.  NN  ( T `  m )
)
135134adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  -> Disj  m  e.  NN  ( T `  m ) )
136 eqid 2441 . . . . . . . . . 10  |-  seq 1
(  +  ,  ( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )  =  seq 1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )
137136, 103voliun 20994 . . . . . . . . 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 656 . . . . . . . 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 2475 . . . . . . 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 4315 . . . . . 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 11131 . . . . 5  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  3  <  ( vol* `  U_ m  e.  NN  ( T `  m )
) )
14217simp3d 997 . . . . . . . . 9  |-  ( ph  ->  U_ m  e.  NN  ( T `  m ) 
C_  ( -u 1 [,] 2 ) )
143142adantr 462 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  U_ m  e.  NN  ( T `  m )  C_  ( -u 1 [,] 2 ) )
144 2re 10387 . . . . . . . . 9  |-  2  e.  RR
145 iccssre 11373 . . . . . . . . 9  |-  ( (
-u 1  e.  RR  /\  2  e.  RR )  ->  ( -u 1 [,] 2 )  C_  RR )
14622, 144, 145mp2an 667 . . . . . . . 8  |-  ( -u
1 [,] 2 ) 
C_  RR
147 ovolss 20927 . . . . . . . 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 657 . . . . . . 7  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( vol* `  U_ m  e.  NN  ( T `  m ) )  <_ 
( vol* `  ( -u 1 [,] 2
) ) )
149 2cn 10388 . . . . . . . . 9  |-  2  e.  CC
150 ax-1cn 9336 . . . . . . . . 9  |-  1  e.  CC
151149, 150subnegi 9683 . . . . . . . 8  |-  ( 2  -  -u 1 )  =  ( 2  +  1 )
152 neg1lt0 10424 . . . . . . . . . . 11  |-  -u 1  <  0
153 2pos 10409 . . . . . . . . . . 11  |-  0  <  2
15422, 41, 144lttri 9496 . . . . . . . . . . 11  |-  ( (
-u 1  <  0  /\  0  <  2
)  ->  -u 1  <  2 )
155152, 153, 154mp2an 667 . . . . . . . . . 10  |-  -u 1  <  2
15622, 144, 155ltleii 9493 . . . . . . . . 9  |-  -u 1  <_  2
157 ovolicc 20965 . . . . . . . . 9  |-  ( (
-u 1  e.  RR  /\  2  e.  RR  /\  -u 1  <_  2 )  ->  ( vol* `  ( -u 1 [,] 2 ) )  =  ( 2  -  -u 1
) )
15822, 144, 156, 157mp3an 1309 . . . . . . . 8  |-  ( vol* `  ( -u 1 [,] 2 ) )  =  ( 2  -  -u 1
)
159 df-3 10377 . . . . . . . 8  |-  3  =  ( 2  +  1 )
160151, 158, 1593eqtr4i 2471 . . . . . . 7  |-  ( vol* `  ( -u 1 [,] 2 ) )  =  3
161148, 160syl6breq 4328 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( vol* `  U_ m  e.  NN  ( T `  m ) )  <_ 
3 )
162 xrlenlt 9438 . . . . . . 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 657 . . . . . 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 573 . . . 4  |-  ( ph  ->  -.  0  <  ( vol* `  ran  F
) )
166 ovolge0 20923 . . . . . . 7  |-  ( ran 
F  C_  RR  ->  0  <_  ( vol* `  ran  F ) )
16720, 166syl 16 . . . . . 6  |-  ( ph  ->  0  <_  ( vol* `  ran  F ) )
168 0xr 9426 . . . . . . 7  |-  0  e.  RR*
169 ovolcl 20920 . . . . . . . 8  |-  ( ran 
F  C_  RR  ->  ( vol* `  ran  F )  e.  RR* )
17020, 169syl 16 . . . . . . 7  |-  ( ph  ->  ( vol* `  ran  F )  e.  RR* )
171 xrleloe 11117 . . . . . . 7  |-  ( ( 0  e.  RR*  /\  ( vol* `  ran  F
)  e.  RR* )  ->  ( 0  <_  ( vol* `  ran  F
)  <->  ( 0  < 
( vol* `  ran  F )  \/  0  =  ( vol* `  ran  F ) ) ) )
172168, 170, 171sylancr 658 . . . . . 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 462 . 2  |-  ( (
ph  /\  m  e.  NN )  ->  0  =  ( vol* `  ran  F ) )
17734, 176eqtr4d 2476 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 1364    e. wcel 1761    =/= wne 2604   A.wral 2713   {crab 2717    \ cdif 3322    i^i cin 3324    C_ wss 3325   (/)c0 3634   ~Pcpw 3857   {csn 3874   U_ciun 4168  Disj wdisj 4259   class class class wbr 4289   {copab 4346    e. cmpt 4347    X. cxp 4834   dom cdm 4836   ran crn 4837    Fn wfn 5410   -->wf 5411   -1-1-onto->wf1o 5414   ` cfv 5415  (class class class)co 6090   /.cqs 7096   supcsup 7686   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279    + caddc 9281    x. cmul 9283   RR*cxr 9413    < clt 9414    <_ cle 9415    - cmin 9591   -ucneg 9592    / cdiv 9989   NNcn 10318   2c2 10367   3c3 10368   NN0cn0 10575   QQcq 10949   RR+crp 10987   [,]cicc 11299   |_cfl 11636    seqcseq 11802   vol*covol 20905   volcvol 20906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cc 8600  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-disj 4260  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-ec 7099  df-qs 7103  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-rlim 12963  df-sum 13160  df-rest 14357  df-topgen 14378  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-top 18462  df-bases 18464  df-topon 18465  df-cmp 18949  df-ovol 20907  df-vol 20908
This theorem is referenced by:  vitalilem5  21051
  Copyright terms: Public domain W3C validator