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Theorem vitalilem4 21091
Description: Lemma for vitali 21093. (Contributed by Mario Carneiro, 16-Jun-2014.)
Hypotheses
Ref Expression
vitali.1  |-  .~  =  { <. x ,  y
>.  |  ( (
x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,] 1 ) )  /\  ( x  -  y )  e.  QQ ) }
vitali.2  |-  S  =  ( ( 0 [,] 1 ) /.  .~  )
vitali.3  |-  ( ph  ->  F  Fn  S )
vitali.4  |-  ( ph  ->  A. z  e.  S  ( z  =/=  (/)  ->  ( F `  z )  e.  z ) )
vitali.5  |-  ( ph  ->  G : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) ) )
vitali.6  |-  T  =  ( n  e.  NN  |->  { s  e.  RR  |  ( s  -  ( G `  n ) )  e.  ran  F } )
vitali.7  |-  ( ph  ->  -.  ran  F  e.  ( ~P RR  \  dom  vol ) )
Assertion
Ref Expression
vitalilem4  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol* `  ( T `  m ) )  =  0 )
Distinct variable groups:    m, n, s, x, y, z, G    ph, m, n, x, z   
z, S    T, m, x    m, F, n, s, x, y, z    .~ , m, n, s, x, y, z
Allowed substitution hints:    ph( y, s)    S( x, y, m, n, s)    T( y, z, n, s)

Proof of Theorem vitalilem4
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 fveq2 5691 . . . . . . . . 9  |-  ( n  =  m  ->  ( G `  n )  =  ( G `  m ) )
21oveq2d 6107 . . . . . . . 8  |-  ( n  =  m  ->  (
s  -  ( G `
 n ) )  =  ( s  -  ( G `  m ) ) )
32eleq1d 2509 . . . . . . 7  |-  ( n  =  m  ->  (
( s  -  ( G `  n )
)  e.  ran  F  <->  ( s  -  ( G `
 m ) )  e.  ran  F ) )
43rabbidv 2964 . . . . . 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 9373 . . . . . . 7  |-  RR  e.  _V
76rabex 4443 . . . . . 6  |-  { s  e.  RR  |  ( s  -  ( G `
 m ) )  e.  ran  F }  e.  _V
84, 5, 7fvmpt 5774 . . . . 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 5695 . . 3  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol* `  ( T `  m ) )  =  ( vol* `  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } ) )
11 vitali.1 . . . . . . . 8  |-  .~  =  { <. x ,  y
>.  |  ( (
x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,] 1 ) )  /\  ( x  -  y )  e.  QQ ) }
12 vitali.2 . . . . . . . 8  |-  S  =  ( ( 0 [,] 1 ) /.  .~  )
13 vitali.3 . . . . . . . 8  |-  ( ph  ->  F  Fn  S )
14 vitali.4 . . . . . . . 8  |-  ( ph  ->  A. z  e.  S  ( z  =/=  (/)  ->  ( F `  z )  e.  z ) )
15 vitali.5 . . . . . . . 8  |-  ( ph  ->  G : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) ) )
16 vitali.7 . . . . . . . 8  |-  ( ph  ->  -.  ran  F  e.  ( ~P RR  \  dom  vol ) )
1711, 12, 13, 14, 15, 5, 16vitalilem2 21089 . . . . . . 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 1000 . . . . . 6  |-  ( ph  ->  ran  F  C_  (
0 [,] 1 ) )
19 unitssre 11432 . . . . . 6  |-  ( 0 [,] 1 )  C_  RR
2018, 19syl6ss 3368 . . . . 5  |-  ( ph  ->  ran  F  C_  RR )
2120adantr 465 . . . 4  |-  ( (
ph  /\  m  e.  NN )  ->  ran  F  C_  RR )
22 neg1rr 10426 . . . . . 6  |-  -u 1  e.  RR
23 1re 9385 . . . . . 6  |-  1  e.  RR
24 iccssre 11377 . . . . . 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 3571 . . . . . 6  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  ( -u 1 [,] 1
)
27 f1of 5641 . . . . . . . 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 5843 . . . . . 6  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  ( QQ  i^i  ( -u 1 [,] 1 ) ) )
3026, 29sseldi 3354 . . . . 5  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  ( -u 1 [,] 1 ) )
3125, 30sseldi 3354 . . . 4  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  RR )
32 eqidd 2444 . . . 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 20994 . . 3  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol* `  ran  F )  =  ( vol* `  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } ) )
3410, 33eqtr4d 2478 . 2  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol* `  ( T `  m ) )  =  ( vol* `  ran  F ) )
35 3re 10395 . . . . . . . 8  |-  3  e.  RR
3635rexri 9436 . . . . . . 7  |-  3  e.  RR*
3736a1i 11 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  3  e.  RR* )
38 3nn 10480 . . . . . . . . . . . . . 14  |-  3  e.  NN
39 nnrp 11000 . . . . . . . . . . . . . 14  |-  ( 3  e.  NN  ->  3  e.  RR+ )
4038, 39ax-mp 5 . . . . . . . . . . . . 13  |-  3  e.  RR+
41 0re 9386 . . . . . . . . . . . . . . . . . . . 20  |-  0  e.  RR
42 0le1 9863 . . . . . . . . . . . . . . . . . . . 20  |-  0  <_  1
43 ovolicc 21006 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  0  <_  1 )  ->  ( vol* `  ( 0 [,] 1 ) )  =  ( 1  -  0 ) )
4441, 23, 42, 43mp3an 1314 . . . . . . . . . . . . . . . . . . 19  |-  ( vol* `  ( 0 [,] 1 ) )  =  ( 1  -  0 )
45 1m0e1 10432 . . . . . . . . . . . . . . . . . . 19  |-  ( 1  -  0 )  =  1
4644, 45eqtri 2463 . . . . . . . . . . . . . . . . . 18  |-  ( vol* `  ( 0 [,] 1 ) )  =  1
4746, 23eqeltri 2513 . . . . . . . . . . . . . . . . 17  |-  ( vol* `  ( 0 [,] 1 ) )  e.  RR
48 ovolsscl 20969 . . . . . . . . . . . . . . . . 17  |-  ( ( ran  F  C_  (
0 [,] 1 )  /\  ( 0 [,] 1 )  C_  RR  /\  ( vol* `  ( 0 [,] 1
) )  e.  RR )  ->  ( vol* `  ran  F )  e.  RR )
4919, 47, 48mp3an23 1306 . . . . . . . . . . . . . . . 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 11025 . . . . . . . . . . . . 13  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( vol* `  ran  F
)  e.  RR+ )
54 rpdivcl 11013 . . . . . . . . . . . . 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 11027 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
3  /  ( vol* `  ran  F ) )  e.  RR )
5755rpge0d 11031 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  0  <_  ( 3  /  ( vol* `  ran  F
) ) )
58 flge0nn0 11666 . . . . . . . . . . 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 10619 . . . . . . . . . 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 10337 . . . . . . . 8  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
( |_ `  (
3  /  ( vol* `  ran  F ) ) )  +  1 )  e.  RR )
6362, 51remulcld 9414 . . . . . . 7  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
( ( |_ `  ( 3  /  ( vol* `  ran  F
) ) )  +  1 )  x.  ( vol* `  ran  F
) )  e.  RR )
6463rexrd 9433 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
( ( |_ `  ( 3  /  ( vol* `  ran  F
) ) )  +  1 )  x.  ( vol* `  ran  F
) )  e.  RR* )
656elpw2 4456 . . . . . . . . . . . . . . . . . . 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 3338 . . . . . . . . . . . . . . . . 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 3570 . . . . . . . . . . . . . . . 16  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  QQ
74 qssre 10963 . . . . . . . . . . . . . . . 16  |-  QQ  C_  RR
7573, 74sstri 3365 . . . . . . . . . . . . . . 15  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  RR
76 fss 5567 . . . . . . . . . . . . . . 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 5843 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  NN )  ->  ( G `
 n )  e.  RR )
79 shftmbl 21020 . . . . . . . . . . . . 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 5867 . . . . . . . . . . 11  |-  ( ph  ->  T : NN --> dom  vol )
8281ffvelrnda 5843 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN )  ->  ( T `
 m )  e. 
dom  vol )
8382ralrimiva 2799 . . . . . . . . 9  |-  ( ph  ->  A. m  e.  NN  ( T `  m )  e.  dom  vol )
84 iunmbl 21034 . . . . . . . . 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 21014 . . . . . . . 8  |-  ( U_ m  e.  NN  ( T `  m )  e.  dom  vol  ->  U_ m  e.  NN  ( T `  m )  C_  RR )
87 ovolcl 20961 . . . . . . . 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 11650 . . . . . . . 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 11075 . . . . . . 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 10896 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
96 1zzd 10677 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  1  e.  ZZ )
97 mblvol 21013 . . . . . . . . . . . . . . . . 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 2475 . . . . . . . . . . . . . . 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 2517 . . . . . . . . . . . . . 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 2443 . . . . . . . . . . . . 13  |-  ( m  e.  NN  |->  ( vol `  ( T `  m
) ) )  =  ( m  e.  NN  |->  ( vol `  ( T `
 m ) ) )
104102, 103fmptd 5867 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) : NN --> RR )
105104ffvelrnda 5843 . . . . . . . . . . 11  |-  ( ( ( ph  /\  0  <  ( vol* `  ran  F ) )  /\  k  e.  NN )  ->  ( ( m  e.  NN  |->  ( vol `  ( T `  m )
) ) `  k
)  e.  RR )
10695, 96, 105serfre 11835 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  seq 1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) ) : NN --> RR )
107 frn 5565 . . . . . . . . . 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 9427 . . . . . . . . 9  |-  RR  C_  RR*
110108, 109syl6ss 3368 . . . . . . . 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 4378 . . . . . . . . . . . . 13  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
m  e.  NN  |->  ( vol `  ( T `
 m ) ) )  =  ( m  e.  NN  |->  ( vol* `  ran  F ) ) )
113 fconstmpt 4882 . . . . . . . . . . . . 13  |-  ( NN 
X.  { ( vol* `  ran  F ) } )  =  ( m  e.  NN  |->  ( vol* `  ran  F ) )
114112, 113syl6eqr 2493 . . . . . . . . . . . 12  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (
m  e.  NN  |->  ( vol `  ( T `
 m ) ) )  =  ( NN 
X.  { ( vol* `  ran  F ) } ) )
115114seqeq3d 11814 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  seq 1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )  =  seq 1 (  +  , 
( NN  X.  {
( vol* `  ran  F ) } ) ) )
116115fveq1d 5693 . . . . . . . . . 10  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  (  seq 1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) ) `  (
( |_ `  (
3  /  ( vol* `  ran  F ) ) )  +  1 ) )  =  (  seq 1 (  +  ,  ( NN  X.  { ( vol* `  ran  F ) } ) ) `  (
( |_ `  (
3  /  ( vol* `  ran  F ) ) )  +  1 ) ) )
11751recnd 9412 . . . . . . . . . . 11  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( vol* `  ran  F
)  e.  CC )
118 ser1const 11862 . . . . . . . . . . 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 2475 . . . . . . . . 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 5559 . . . . . . . . . . 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 5840 . . . . . . . . . 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 2518 . . . . . . . 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 11287 . . . . . . . 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 21013 . . . . . . . . 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 2799 . . . . . . . . . 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 21090 . . . . . . . . . 10  |-  ( ph  -> Disj  m  e.  NN  ( T `  m )
)
135134adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  -> Disj  m  e.  NN  ( T `  m ) )
136 eqid 2443 . . . . . . . . . 10  |-  seq 1
(  +  ,  ( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )  =  seq 1 (  +  , 
( m  e.  NN  |->  ( vol `  ( T `
 m ) ) ) )
137136, 103voliun 21035 . . . . . . . . 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 2477 . . . . . . 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 4318 . . . . . 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 11135 . . . . 5  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  3  <  ( vol* `  U_ m  e.  NN  ( T `  m )
) )
14217simp3d 1002 . . . . . . . . 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 10391 . . . . . . . . 9  |-  2  e.  RR
145 iccssre 11377 . . . . . . . . 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 20968 . . . . . . . 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 10392 . . . . . . . . 9  |-  2  e.  CC
150 ax-1cn 9340 . . . . . . . . 9  |-  1  e.  CC
151149, 150subnegi 9687 . . . . . . . 8  |-  ( 2  -  -u 1 )  =  ( 2  +  1 )
152 neg1lt0 10428 . . . . . . . . . . 11  |-  -u 1  <  0
153 2pos 10413 . . . . . . . . . . 11  |-  0  <  2
15422, 41, 144lttri 9500 . . . . . . . . . . 11  |-  ( (
-u 1  <  0  /\  0  <  2
)  ->  -u 1  <  2 )
155152, 153, 154mp2an 672 . . . . . . . . . 10  |-  -u 1  <  2
15622, 144, 155ltleii 9497 . . . . . . . . 9  |-  -u 1  <_  2
157 ovolicc 21006 . . . . . . . . 9  |-  ( (
-u 1  e.  RR  /\  2  e.  RR  /\  -u 1  <_  2 )  ->  ( vol* `  ( -u 1 [,] 2 ) )  =  ( 2  -  -u 1
) )
15822, 144, 156, 157mp3an 1314 . . . . . . . 8  |-  ( vol* `  ( -u 1 [,] 2 ) )  =  ( 2  -  -u 1
)
159 df-3 10381 . . . . . . . 8  |-  3  =  ( 2  +  1 )
160151, 158, 1593eqtr4i 2473 . . . . . . 7  |-  ( vol* `  ( -u 1 [,] 2 ) )  =  3
161148, 160syl6breq 4331 . . . . . 6  |-  ( (
ph  /\  0  <  ( vol* `  ran  F ) )  ->  ( vol* `  U_ m  e.  NN  ( T `  m ) )  <_ 
3 )
162 xrlenlt 9442 . . . . . . 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 20964 . . . . . . 7  |-  ( ran 
F  C_  RR  ->  0  <_  ( vol* `  ran  F ) )
16720, 166syl 16 . . . . . 6  |-  ( ph  ->  0  <_  ( vol* `  ran  F ) )
168 0xr 9430 . . . . . . 7  |-  0  e.  RR*
169 ovolcl 20961 . . . . . . . 8  |-  ( ran 
F  C_  RR  ->  ( vol* `  ran  F )  e.  RR* )
17020, 169syl 16 . . . . . . 7  |-  ( ph  ->  ( vol* `  ran  F )  e.  RR* )
171 xrleloe 11121 . . . . . . 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 2478 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 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   {crab 2719    \ cdif 3325    i^i cin 3327    C_ wss 3328   (/)c0 3637   ~Pcpw 3860   {csn 3877   U_ciun 4171  Disj wdisj 4262   class class class wbr 4292   {copab 4349    e. cmpt 4350    X. cxp 4838   dom cdm 4840   ran crn 4841    Fn wfn 5413   -->wf 5414   -1-1-onto->wf1o 5417   ` cfv 5418  (class class class)co 6091   /.cqs 7100   supcsup 7690   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283    + caddc 9285    x. cmul 9287   RR*cxr 9417    < clt 9418    <_ cle 9419    - cmin 9595   -ucneg 9596    / cdiv 9993   NNcn 10322   2c2 10371   3c3 10372   NN0cn0 10579   QQcq 10953   RR+crp 10991   [,]cicc 11303   |_cfl 11640    seqcseq 11806   vol*covol 20946   volcvol 20947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cc 8604  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-disj 4263  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-ec 7103  df-qs 7107  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fi 7661  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ioo 11304  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-fl 11642  df-seq 11807  df-exp 11866  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-clim 12966  df-rlim 12967  df-sum 13164  df-rest 14361  df-topgen 14382  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-top 18503  df-bases 18505  df-topon 18506  df-cmp 18990  df-ovol 20948  df-vol 20949
This theorem is referenced by:  vitalilem5  21092
  Copyright terms: Public domain W3C validator