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Theorem vitalilem2 19454
Description: Lemma for vitali 19458. (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
vitalilem2  |-  ( 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 ) ) )
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 vitalilem2
Dummy variables  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vitali.3 . . . 4  |-  ( ph  ->  F  Fn  S )
2 vitali.4 . . . . 5  |-  ( ph  ->  A. z  e.  S  ( z  =/=  (/)  ->  ( F `  z )  e.  z ) )
3 vitali.2 . . . . . . . . 9  |-  S  =  ( ( 0 [,] 1 ) /.  .~  )
4 neeq1 2575 . . . . . . . . 9  |-  ( [ v ]  .~  =  z  ->  ( [ v ]  .~  =/=  (/)  <->  z  =/=  (/) ) )
5 vitali.1 . . . . . . . . . . . . . 14  |-  .~  =  { <. x ,  y
>.  |  ( (
x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,] 1 ) )  /\  ( x  -  y )  e.  QQ ) }
65vitalilem1 19453 . . . . . . . . . . . . 13  |-  .~  Er  ( 0 [,] 1
)
7 erdm 6874 . . . . . . . . . . . . 13  |-  (  .~  Er  ( 0 [,] 1
)  ->  dom  .~  =  ( 0 [,] 1
) )
86, 7ax-mp 8 . . . . . . . . . . . 12  |-  dom  .~  =  ( 0 [,] 1 )
98eleq2i 2468 . . . . . . . . . . 11  |-  ( v  e.  dom  .~  <->  v  e.  ( 0 [,] 1
) )
10 ecdmn0 6906 . . . . . . . . . . 11  |-  ( v  e.  dom  .~  <->  [ v ]  .~  =/=  (/) )
119, 10bitr3i 243 . . . . . . . . . 10  |-  ( v  e.  ( 0 [,] 1 )  <->  [ v ]  .~  =/=  (/) )
1211biimpi 187 . . . . . . . . 9  |-  ( v  e.  ( 0 [,] 1 )  ->  [ v ]  .~  =/=  (/) )
133, 4, 12ectocl 6931 . . . . . . . 8  |-  ( z  e.  S  ->  z  =/=  (/) )
1413adantl 453 . . . . . . 7  |-  ( (
ph  /\  z  e.  S )  ->  z  =/=  (/) )
15 sseq1 3329 . . . . . . . . . 10  |-  ( [ w ]  .~  =  z  ->  ( [ w ]  .~  C_  ( 0 [,] 1 )  <->  z  C_  ( 0 [,] 1
) ) )
166a1i 11 . . . . . . . . . . 11  |-  ( w  e.  ( 0 [,] 1 )  ->  .~  Er  ( 0 [,] 1
) )
1716ecss 6905 . . . . . . . . . 10  |-  ( w  e.  ( 0 [,] 1 )  ->  [ w ]  .~  C_  ( 0 [,] 1 ) )
183, 15, 17ectocl 6931 . . . . . . . . 9  |-  ( z  e.  S  ->  z  C_  ( 0 [,] 1
) )
1918adantl 453 . . . . . . . 8  |-  ( (
ph  /\  z  e.  S )  ->  z  C_  ( 0 [,] 1
) )
2019sseld 3307 . . . . . . 7  |-  ( (
ph  /\  z  e.  S )  ->  (
( F `  z
)  e.  z  -> 
( F `  z
)  e.  ( 0 [,] 1 ) ) )
2114, 20embantd 52 . . . . . 6  |-  ( (
ph  /\  z  e.  S )  ->  (
( z  =/=  (/)  ->  ( F `  z )  e.  z )  ->  ( F `  z )  e.  ( 0 [,] 1
) ) )
2221ralimdva 2744 . . . . 5  |-  ( ph  ->  ( A. z  e.  S  ( z  =/=  (/)  ->  ( F `  z )  e.  z )  ->  A. z  e.  S  ( F `  z )  e.  ( 0 [,] 1 ) ) )
232, 22mpd 15 . . . 4  |-  ( ph  ->  A. z  e.  S  ( F `  z )  e.  ( 0 [,] 1 ) )
24 ffnfv 5853 . . . 4  |-  ( F : S --> ( 0 [,] 1 )  <->  ( F  Fn  S  /\  A. z  e.  S  ( F `  z )  e.  ( 0 [,] 1 ) ) )
251, 23, 24sylanbrc 646 . . 3  |-  ( ph  ->  F : S --> ( 0 [,] 1 ) )
26 frn 5556 . . 3  |-  ( F : S --> ( 0 [,] 1 )  ->  ran  F  C_  ( 0 [,] 1 ) )
2725, 26syl 16 . 2  |-  ( ph  ->  ran  F  C_  (
0 [,] 1 ) )
28 vitali.5 . . . . . . . . 9  |-  ( ph  ->  G : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) ) )
2928adantr 452 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  G : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) ) )
30 f1ocnv 5646 . . . . . . . 8  |-  ( G : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) )  ->  `' G : ( QQ  i^i  ( -u 1 [,] 1
) ) -1-1-onto-> NN )
31 f1of 5633 . . . . . . . 8  |-  ( `' G : ( QQ 
i^i  ( -u 1 [,] 1 ) ) -1-1-onto-> NN  ->  `' G : ( QQ 
i^i  ( -u 1 [,] 1 ) ) --> NN )
3229, 30, 313syl 19 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  `' G : ( QQ  i^i  ( -u 1 [,] 1
) ) --> NN )
33 ovex 6065 . . . . . . . . . . . . . . 15  |-  ( 0 [,] 1 )  e. 
_V
34 erex 6888 . . . . . . . . . . . . . . 15  |-  (  .~  Er  ( 0 [,] 1
)  ->  ( (
0 [,] 1 )  e.  _V  ->  .~  e.  _V ) )
356, 33, 34mp2 9 . . . . . . . . . . . . . 14  |-  .~  e.  _V
3635ecelqsi 6919 . . . . . . . . . . . . 13  |-  ( v  e.  ( 0 [,] 1 )  ->  [ v ]  .~  e.  ( ( 0 [,] 1
) /.  .~  )
)
3736adantl 453 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  [ v ]  .~  e.  ( ( 0 [,] 1
) /.  .~  )
)
3837, 3syl6eleqr 2495 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  [ v ]  .~  e.  S
)
392adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  A. z  e.  S  ( z  =/=  (/)  ->  ( F `  z )  e.  z ) )
40 simpr 448 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  v  e.  ( 0 [,] 1
) )
4140, 11sylib 189 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  [ v ]  .~  =/=  (/) )
42 neeq1 2575 . . . . . . . . . . . . 13  |-  ( z  =  [ v ]  .~  ->  ( z  =/=  (/)  <->  [ v ]  .~  =/=  (/) ) )
43 fveq2 5687 . . . . . . . . . . . . . 14  |-  ( z  =  [ v ]  .~  ->  ( F `  z )  =  ( F `  [ v ]  .~  ) )
44 id 20 . . . . . . . . . . . . . 14  |-  ( z  =  [ v ]  .~  ->  z  =  [ v ]  .~  )
4543, 44eleq12d 2472 . . . . . . . . . . . . 13  |-  ( z  =  [ v ]  .~  ->  ( ( F `  z )  e.  z  <->  ( F `  [ v ]  .~  )  e.  [ v ]  .~  ) )
4642, 45imbi12d 312 . . . . . . . . . . . 12  |-  ( z  =  [ v ]  .~  ->  ( (
z  =/=  (/)  ->  ( F `  z )  e.  z )  <->  ( [
v ]  .~  =/=  (/) 
->  ( F `  [
v ]  .~  )  e.  [ v ]  .~  ) ) )
4746rspcv 3008 . . . . . . . . . . 11  |-  ( [ v ]  .~  e.  S  ->  ( A. z  e.  S  ( z  =/=  (/)  ->  ( F `  z )  e.  z )  ->  ( [
v ]  .~  =/=  (/) 
->  ( F `  [
v ]  .~  )  e.  [ v ]  .~  ) ) )
4838, 39, 41, 47syl3c 59 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( F `  [ v ]  .~  )  e.  [
v ]  .~  )
49 fvex 5701 . . . . . . . . . . . 12  |-  ( F `
 [ v ]  .~  )  e.  _V
50 vex 2919 . . . . . . . . . . . 12  |-  v  e. 
_V
5149, 50elec 6903 . . . . . . . . . . 11  |-  ( ( F `  [ v ]  .~  )  e. 
[ v ]  .~  <->  v  .~  ( F `  [ v ]  .~  ) )
52 oveq12 6049 . . . . . . . . . . . . 13  |-  ( ( x  =  v  /\  y  =  ( F `  [ v ]  .~  ) )  ->  (
x  -  y )  =  ( v  -  ( F `  [ v ]  .~  ) ) )
5352eleq1d 2470 . . . . . . . . . . . 12  |-  ( ( x  =  v  /\  y  =  ( F `  [ v ]  .~  ) )  ->  (
( x  -  y
)  e.  QQ  <->  ( v  -  ( F `  [ v ]  .~  ) )  e.  QQ ) )
5453, 5brab2ga 4910 . . . . . . . . . . 11  |-  ( v  .~  ( F `  [ v ]  .~  ) 
<->  ( ( v  e.  ( 0 [,] 1
)  /\  ( F `  [ v ]  .~  )  e.  ( 0 [,] 1 ) )  /\  ( v  -  ( F `  [ v ]  .~  ) )  e.  QQ ) )
5551, 54bitri 241 . . . . . . . . . 10  |-  ( ( F `  [ v ]  .~  )  e. 
[ v ]  .~  <->  ( ( v  e.  ( 0 [,] 1 )  /\  ( F `  [ v ]  .~  )  e.  ( 0 [,] 1 ) )  /\  ( v  -  ( F `  [ v ]  .~  ) )  e.  QQ ) )
5648, 55sylib 189 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
( v  e.  ( 0 [,] 1 )  /\  ( F `  [ v ]  .~  )  e.  ( 0 [,] 1 ) )  /\  ( v  -  ( F `  [ v ]  .~  ) )  e.  QQ ) )
5756simprd 450 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( F `
 [ v ]  .~  ) )  e.  QQ )
58 0re 9047 . . . . . . . . . . . . 13  |-  0  e.  RR
59 1re 9046 . . . . . . . . . . . . 13  |-  1  e.  RR
6058, 59elicc2i 10932 . . . . . . . . . . . 12  |-  ( v  e.  ( 0 [,] 1 )  <->  ( v  e.  RR  /\  0  <_ 
v  /\  v  <_  1 ) )
6140, 60sylib 189 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  e.  RR  /\  0  <_  v  /\  v  <_  1 ) )
6261simp1d 969 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  v  e.  RR )
6356simpld 446 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  e.  ( 0 [,] 1 )  /\  ( F `  [ v ]  .~  )  e.  ( 0 [,] 1
) ) )
6463simprd 450 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( F `  [ v ]  .~  )  e.  ( 0 [,] 1 ) )
6558, 59elicc2i 10932 . . . . . . . . . . . 12  |-  ( ( F `  [ v ]  .~  )  e.  ( 0 [,] 1
)  <->  ( ( F `
 [ v ]  .~  )  e.  RR  /\  0  <_  ( F `  [ v ]  .~  )  /\  ( F `  [ v ]  .~  )  <_  1 ) )
6664, 65sylib 189 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
( F `  [
v ]  .~  )  e.  RR  /\  0  <_ 
( F `  [
v ]  .~  )  /\  ( F `  [
v ]  .~  )  <_  1 ) )
6766simp1d 969 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( F `  [ v ]  .~  )  e.  RR )
6862, 67resubcld 9421 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( F `
 [ v ]  .~  ) )  e.  RR )
6967, 62resubcld 9421 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
( F `  [
v ]  .~  )  -  v )  e.  RR )
7059a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  1  e.  RR )
7161simp2d 970 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  0  <_  v )
7267, 62subge02d 9574 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
0  <_  v  <->  ( ( F `  [ v ]  .~  )  -  v
)  <_  ( F `  [ v ]  .~  ) ) )
7371, 72mpbid 202 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
( F `  [
v ]  .~  )  -  v )  <_ 
( F `  [
v ]  .~  )
)
7466simp3d 971 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( F `  [ v ]  .~  )  <_  1
)
7569, 67, 70, 73, 74letrd 9183 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
( F `  [
v ]  .~  )  -  v )  <_ 
1 )
7669, 70lenegd 9561 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
( ( F `  [ v ]  .~  )  -  v )  <_  1  <->  -u 1  <_  -u (
( F `  [
v ]  .~  )  -  v ) ) )
7775, 76mpbid 202 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  -u 1  <_ 
-u ( ( F `
 [ v ]  .~  )  -  v
) )
7867recnd 9070 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( F `  [ v ]  .~  )  e.  CC )
7962recnd 9070 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  v  e.  CC )
8078, 79negsubdi2d 9383 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  -u (
( F `  [
v ]  .~  )  -  v )  =  ( v  -  ( F `  [ v ]  .~  ) ) )
8177, 80breqtrd 4196 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  -u 1  <_  ( v  -  ( F `  [ v ]  .~  ) ) )
8266simp2d 970 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  0  <_  ( F `  [
v ]  .~  )
)
8362, 67subge02d 9574 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
0  <_  ( F `  [ v ]  .~  ) 
<->  ( v  -  ( F `  [ v ]  .~  ) )  <_ 
v ) )
8482, 83mpbid 202 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( F `
 [ v ]  .~  ) )  <_ 
v )
8561simp3d 971 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  v  <_  1 )
8668, 62, 70, 84, 85letrd 9183 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( F `
 [ v ]  .~  ) )  <_ 
1 )
8759renegcli 9318 . . . . . . . . . 10  |-  -u 1  e.  RR
8887, 59elicc2i 10932 . . . . . . . . 9  |-  ( ( v  -  ( F `
 [ v ]  .~  ) )  e.  ( -u 1 [,] 1 )  <->  ( (
v  -  ( F `
 [ v ]  .~  ) )  e.  RR  /\  -u 1  <_  ( v  -  ( F `  [ v ]  .~  ) )  /\  ( v  -  ( F `  [ v ]  .~  ) )  <_ 
1 ) )
8968, 81, 86, 88syl3anbrc 1138 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( F `
 [ v ]  .~  ) )  e.  ( -u 1 [,] 1 ) )
90 elin 3490 . . . . . . . 8  |-  ( ( v  -  ( F `
 [ v ]  .~  ) )  e.  ( QQ  i^i  ( -u 1 [,] 1 ) )  <->  ( ( v  -  ( F `  [ v ]  .~  ) )  e.  QQ  /\  ( v  -  ( F `  [ v ]  .~  ) )  e.  ( -u 1 [,] 1 ) ) )
9157, 89, 90sylanbrc 646 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( F `
 [ v ]  .~  ) )  e.  ( QQ  i^i  ( -u 1 [,] 1 ) ) )
9232, 91ffvelrnd 5830 . . . . . 6  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) )  e.  NN )
93 f1ocnvfv2 5974 . . . . . . . . . . . 12  |-  ( ( G : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) )  /\  ( v  -  ( F `  [ v ]  .~  ) )  e.  ( QQ  i^i  ( -u
1 [,] 1 ) ) )  ->  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) )  =  ( v  -  ( F `  [ v ]  .~  ) ) )
9429, 91, 93syl2anc 643 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) )  =  ( v  -  ( F `  [ v ]  .~  ) ) )
9594oveq2d 6056 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  =  ( v  -  ( v  -  ( F `  [ v ]  .~  ) ) ) )
9679, 78nncand 9372 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( v  -  ( F `  [ v ]  .~  ) ) )  =  ( F `  [
v ]  .~  )
)
9795, 96eqtrd 2436 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  =  ( F `  [ v ]  .~  ) )
981adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  F  Fn  S )
99 fnfvelrn 5826 . . . . . . . . . 10  |-  ( ( F  Fn  S  /\  [ v ]  .~  e.  S )  ->  ( F `  [ v ]  .~  )  e.  ran  F )
10098, 38, 99syl2anc 643 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( F `  [ v ]  .~  )  e.  ran  F )
10197, 100eqeltrd 2478 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F )
102 oveq1 6047 . . . . . . . . . 10  |-  ( s  =  v  ->  (
s  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  =  ( v  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) ) )
103102eleq1d 2470 . . . . . . . . 9  |-  ( s  =  v  ->  (
( s  -  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F  <->  ( v  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F ) )
104103elrab 3052 . . . . . . . 8  |-  ( v  e.  { s  e.  RR  |  ( s  -  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F } 
<->  ( v  e.  RR  /\  ( v  -  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F
) )
10562, 101, 104sylanbrc 646 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  v  e.  { s  e.  RR  |  ( s  -  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F } )
106 fveq2 5687 . . . . . . . . . . . 12  |-  ( n  =  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) )  ->  ( G `  n )  =  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )
107106oveq2d 6056 . . . . . . . . . . 11  |-  ( n  =  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) )  ->  ( s  -  ( G `  n ) )  =  ( s  -  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) ) )
108107eleq1d 2470 . . . . . . . . . 10  |-  ( n  =  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) )  ->  ( ( s  -  ( G `  n ) )  e. 
ran  F  <->  ( s  -  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F
) )
109108rabbidv 2908 . . . . . . . . 9  |-  ( n  =  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) )  ->  { s  e.  RR  |  ( s  -  ( G `  n ) )  e. 
ran  F }  =  { s  e.  RR  |  ( s  -  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F } )
110 vitali.6 . . . . . . . . 9  |-  T  =  ( n  e.  NN  |->  { s  e.  RR  |  ( s  -  ( G `  n ) )  e.  ran  F } )
111 reex 9037 . . . . . . . . . 10  |-  RR  e.  _V
112111rabex 4314 . . . . . . . . 9  |-  { s  e.  RR  |  ( s  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F }  e.  _V
113109, 110, 112fvmpt 5765 . . . . . . . 8  |-  ( ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) )  e.  NN  ->  ( T `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) )  =  { s  e.  RR  |  ( s  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F } )
11492, 113syl 16 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( T `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) )  =  { s  e.  RR  |  ( s  -  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F } )
115105, 114eleqtrrd 2481 . . . . . 6  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  v  e.  ( T `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )
116 fveq2 5687 . . . . . . . 8  |-  ( m  =  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) )  ->  ( T `  m )  =  ( T `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )
117116eleq2d 2471 . . . . . . 7  |-  ( m  =  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) )  ->  ( v  e.  ( T `  m
)  <->  v  e.  ( T `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) ) )
118117rspcev 3012 . . . . . 6  |-  ( ( ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) )  e.  NN  /\  v  e.  ( T `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  ->  E. m  e.  NN  v  e.  ( T `  m ) )
11992, 115, 118syl2anc 643 . . . . 5  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  E. m  e.  NN  v  e.  ( T `  m ) )
120 eliun 4057 . . . . 5  |-  ( v  e.  U_ m  e.  NN  ( T `  m )  <->  E. m  e.  NN  v  e.  ( T `  m ) )
121119, 120sylibr 204 . . . 4  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  v  e.  U_ m  e.  NN  ( T `  m ) )
122121ex 424 . . 3  |-  ( ph  ->  ( v  e.  ( 0 [,] 1 )  ->  v  e.  U_ m  e.  NN  ( T `  m )
) )
123122ssrdv 3314 . 2  |-  ( ph  ->  ( 0 [,] 1
)  C_  U_ m  e.  NN  ( T `  m ) )
124 eliun 4057 . . . 4  |-  ( x  e.  U_ m  e.  NN  ( T `  m )  <->  E. m  e.  NN  x  e.  ( T `  m ) )
125 fveq2 5687 . . . . . . . . . . . . . . . 16  |-  ( n  =  m  ->  ( G `  n )  =  ( G `  m ) )
126125oveq2d 6056 . . . . . . . . . . . . . . 15  |-  ( n  =  m  ->  (
s  -  ( G `
 n ) )  =  ( s  -  ( G `  m ) ) )
127126eleq1d 2470 . . . . . . . . . . . . . 14  |-  ( n  =  m  ->  (
( s  -  ( G `  n )
)  e.  ran  F  <->  ( s  -  ( G `
 m ) )  e.  ran  F ) )
128127rabbidv 2908 . . . . . . . . . . . . 13  |-  ( n  =  m  ->  { s  e.  RR  |  ( s  -  ( G `
 n ) )  e.  ran  F }  =  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } )
129111rabex 4314 . . . . . . . . . . . . 13  |-  { s  e.  RR  |  ( s  -  ( G `
 m ) )  e.  ran  F }  e.  _V
130128, 110, 129fvmpt 5765 . . . . . . . . . . . 12  |-  ( m  e.  NN  ->  ( T `  m )  =  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } )
131130adantl 453 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN )  ->  ( T `
 m )  =  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } )
132131eleq2d 2471 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN )  ->  ( x  e.  ( T `  m )  <->  x  e.  { s  e.  RR  | 
( s  -  ( G `  m )
)  e.  ran  F } ) )
133132biimpa 471 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  x  e.  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } )
134 oveq1 6047 . . . . . . . . . . 11  |-  ( s  =  x  ->  (
s  -  ( G `
 m ) )  =  ( x  -  ( G `  m ) ) )
135134eleq1d 2470 . . . . . . . . . 10  |-  ( s  =  x  ->  (
( s  -  ( G `  m )
)  e.  ran  F  <->  ( x  -  ( G `
 m ) )  e.  ran  F ) )
136135elrab 3052 . . . . . . . . 9  |-  ( x  e.  { s  e.  RR  |  ( s  -  ( G `  m ) )  e. 
ran  F }  <->  ( x  e.  RR  /\  ( x  -  ( G `  m ) )  e. 
ran  F ) )
137133, 136sylib 189 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
x  e.  RR  /\  ( x  -  ( G `  m )
)  e.  ran  F
) )
138137simpld 446 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  x  e.  RR )
13987a1i 11 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  -u 1  e.  RR )
140 iccssre 10948 . . . . . . . . . . 11  |-  ( (
-u 1  e.  RR  /\  1  e.  RR )  ->  ( -u 1 [,] 1 )  C_  RR )
14187, 59, 140mp2an 654 . . . . . . . . . 10  |-  ( -u
1 [,] 1 ) 
C_  RR
142 inss2 3522 . . . . . . . . . . 11  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  ( -u 1 [,] 1
)
143 f1of 5633 . . . . . . . . . . . . 13  |-  ( G : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) )  ->  G : NN
--> ( QQ  i^i  ( -u 1 [,] 1 ) ) )
14428, 143syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  G : NN --> ( QQ 
i^i  ( -u 1 [,] 1 ) ) )
145144ffvelrnda 5829 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  ( QQ  i^i  ( -u 1 [,] 1 ) ) )
146142, 145sseldi 3306 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  ( -u 1 [,] 1 ) )
147141, 146sseldi 3306 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  RR )
148147adantr 452 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  ( G `  m )  e.  RR )
149146adantr 452 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  ( G `  m )  e.  ( -u 1 [,] 1 ) )
15087, 59elicc2i 10932 . . . . . . . . . 10  |-  ( ( G `  m )  e.  ( -u 1 [,] 1 )  <->  ( ( G `  m )  e.  RR  /\  -u 1  <_  ( G `  m
)  /\  ( G `  m )  <_  1
) )
151149, 150sylib 189 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
( G `  m
)  e.  RR  /\  -u 1  <_  ( G `  m )  /\  ( G `  m )  <_  1 ) )
152151simp2d 970 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  -u 1  <_  ( G `  m
) )
15327ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  ran  F 
C_  ( 0 [,] 1 ) )
154137simprd 450 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
x  -  ( G `
 m ) )  e.  ran  F )
155153, 154sseldd 3309 . . . . . . . . . . 11  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
x  -  ( G `
 m ) )  e.  ( 0 [,] 1 ) )
15658, 59elicc2i 10932 . . . . . . . . . . 11  |-  ( ( x  -  ( G `
 m ) )  e.  ( 0 [,] 1 )  <->  ( (
x  -  ( G `
 m ) )  e.  RR  /\  0  <_  ( x  -  ( G `  m )
)  /\  ( x  -  ( G `  m ) )  <_ 
1 ) )
157155, 156sylib 189 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
( x  -  ( G `  m )
)  e.  RR  /\  0  <_  ( x  -  ( G `  m ) )  /\  ( x  -  ( G `  m ) )  <_ 
1 ) )
158157simp2d 970 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  0  <_  ( x  -  ( G `  m )
) )
159138, 148subge0d 9572 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
0  <_  ( x  -  ( G `  m ) )  <->  ( G `  m )  <_  x
) )
160158, 159mpbid 202 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  ( G `  m )  <_  x )
161139, 148, 138, 152, 160letrd 9183 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  -u 1  <_  x )
162 peano2re 9195 . . . . . . . . 9  |-  ( ( G `  m )  e.  RR  ->  (
( G `  m
)  +  1 )  e.  RR )
163148, 162syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
( G `  m
)  +  1 )  e.  RR )
164 2re 10025 . . . . . . . . 9  |-  2  e.  RR
165164a1i 11 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  2  e.  RR )
166157simp3d 971 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
x  -  ( G `
 m ) )  <_  1 )
16759a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  1  e.  RR )
168138, 148, 167lesubadd2d 9581 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
( x  -  ( G `  m )
)  <_  1  <->  x  <_  ( ( G `  m
)  +  1 ) ) )
169166, 168mpbid 202 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  x  <_  ( ( G `  m )  +  1 ) )
170151simp3d 971 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  ( G `  m )  <_  1 )
171148, 167, 167, 170leadd1dd 9596 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
( G `  m
)  +  1 )  <_  ( 1  +  1 ) )
172 df-2 10014 . . . . . . . . 9  |-  2  =  ( 1  +  1 )
173171, 172syl6breqr 4212 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
( G `  m
)  +  1 )  <_  2 )
174138, 163, 165, 169, 173letrd 9183 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  x  <_  2 )
17587, 164elicc2i 10932 . . . . . . 7  |-  ( x  e.  ( -u 1 [,] 2 )  <->  ( x  e.  RR  /\  -u 1  <_  x  /\  x  <_ 
2 ) )
176138, 161, 174, 175syl3anbrc 1138 . . . . . 6  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  x  e.  ( -u 1 [,] 2 ) )
177176ex 424 . . . . 5  |-  ( (
ph  /\  m  e.  NN )  ->  ( x  e.  ( T `  m )  ->  x  e.  ( -u 1 [,] 2 ) ) )
178177rexlimdva 2790 . . . 4  |-  ( ph  ->  ( E. m  e.  NN  x  e.  ( T `  m )  ->  x  e.  (
-u 1 [,] 2
) ) )
179124, 178syl5bi 209 . . 3  |-  ( ph  ->  ( x  e.  U_ m  e.  NN  ( T `  m )  ->  x  e.  ( -u
1 [,] 2 ) ) )
180179ssrdv 3314 . 2  |-  ( ph  ->  U_ m  e.  NN  ( T `  m ) 
C_  ( -u 1 [,] 2 ) )
18127, 123, 1803jca 1134 1  |-  ( 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 ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   {crab 2670   _Vcvv 2916    \ cdif 3277    i^i cin 3279    C_ wss 3280   (/)c0 3588   ~Pcpw 3759   U_ciun 4053   class class class wbr 4172   {copab 4225    e. cmpt 4226   `'ccnv 4836   dom cdm 4837   ran crn 4838    Fn wfn 5408   -->wf 5409   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040    Er wer 6861   [cec 6862   /.cqs 6863   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    <_ cle 9077    - cmin 9247   -ucneg 9248   NNcn 9956   2c2 10005   QQcq 10530   [,]cicc 10875   volcvol 19313
This theorem is referenced by:  vitalilem3  19455  vitalilem4  19456  vitalilem5  19457
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-ec 6866  df-qs 6870  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-q 10531  df-icc 10879
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