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Theorem vitalilem2 19506
Description: Lemma for vitali 19510. (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 2611 . . . . . . . . 9  |-  ( [ v ]  .~  =  z  ->  ( [ v ]  .~  =/=  (/)  <->  z  =/=  (/) ) )
5 vitali.1 . . . . . . . . . . . . . 14  |-  .~  =  { <. x ,  y
>.  |  ( (
x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,] 1 ) )  /\  ( x  -  y )  e.  QQ ) }
65vitalilem1 19505 . . . . . . . . . . . . 13  |-  .~  Er  ( 0 [,] 1
)
7 erdm 6918 . . . . . . . . . . . . 13  |-  (  .~  Er  ( 0 [,] 1
)  ->  dom  .~  =  ( 0 [,] 1
) )
86, 7ax-mp 5 . . . . . . . . . . . 12  |-  dom  .~  =  ( 0 [,] 1 )
98eleq2i 2502 . . . . . . . . . . 11  |-  ( v  e.  dom  .~  <->  v  e.  ( 0 [,] 1
) )
10 ecdmn0 6950 . . . . . . . . . . 11  |-  ( v  e.  dom  .~  <->  [ v ]  .~  =/=  (/) )
119, 10bitr3i 244 . . . . . . . . . 10  |-  ( v  e.  ( 0 [,] 1 )  <->  [ v ]  .~  =/=  (/) )
1211biimpi 188 . . . . . . . . 9  |-  ( v  e.  ( 0 [,] 1 )  ->  [ v ]  .~  =/=  (/) )
133, 4, 12ectocl 6975 . . . . . . . 8  |-  ( z  e.  S  ->  z  =/=  (/) )
1413adantl 454 . . . . . . 7  |-  ( (
ph  /\  z  e.  S )  ->  z  =/=  (/) )
15 sseq1 3371 . . . . . . . . . 10  |-  ( [ w ]  .~  =  z  ->  ( [ w ]  .~  C_  ( 0 [,] 1 )  <->  z  C_  ( 0 [,] 1
) ) )
166a1i 11 . . . . . . . . . . 11  |-  ( w  e.  ( 0 [,] 1 )  ->  .~  Er  ( 0 [,] 1
) )
1716ecss 6949 . . . . . . . . . 10  |-  ( w  e.  ( 0 [,] 1 )  ->  [ w ]  .~  C_  ( 0 [,] 1 ) )
183, 15, 17ectocl 6975 . . . . . . . . 9  |-  ( z  e.  S  ->  z  C_  ( 0 [,] 1
) )
1918adantl 454 . . . . . . . 8  |-  ( (
ph  /\  z  e.  S )  ->  z  C_  ( 0 [,] 1
) )
2019sseld 3349 . . . . . . 7  |-  ( (
ph  /\  z  e.  S )  ->  (
( F `  z
)  e.  z  -> 
( F `  z
)  e.  ( 0 [,] 1 ) ) )
2114, 20embantd 53 . . . . . 6  |-  ( (
ph  /\  z  e.  S )  ->  (
( z  =/=  (/)  ->  ( F `  z )  e.  z )  ->  ( F `  z )  e.  ( 0 [,] 1
) ) )
2221ralimdva 2786 . . . . 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 5897 . . . 4  |-  ( F : S --> ( 0 [,] 1 )  <->  ( F  Fn  S  /\  A. z  e.  S  ( F `  z )  e.  ( 0 [,] 1 ) ) )
251, 23, 24sylanbrc 647 . . 3  |-  ( ph  ->  F : S --> ( 0 [,] 1 ) )
26 frn 5600 . . 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 453 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  G : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) ) )
30 f1ocnv 5690 . . . . . . . 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 5677 . . . . . . . 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 6109 . . . . . . . . . . . . . . 15  |-  ( 0 [,] 1 )  e. 
_V
34 erex 6932 . . . . . . . . . . . . . . 15  |-  (  .~  Er  ( 0 [,] 1
)  ->  ( (
0 [,] 1 )  e.  _V  ->  .~  e.  _V ) )
356, 33, 34mp2 9 . . . . . . . . . . . . . 14  |-  .~  e.  _V
3635ecelqsi 6963 . . . . . . . . . . . . 13  |-  ( v  e.  ( 0 [,] 1 )  ->  [ v ]  .~  e.  ( ( 0 [,] 1
) /.  .~  )
)
3736adantl 454 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  [ v ]  .~  e.  ( ( 0 [,] 1
) /.  .~  )
)
3837, 3syl6eleqr 2529 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  [ v ]  .~  e.  S
)
392adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  A. z  e.  S  ( z  =/=  (/)  ->  ( F `  z )  e.  z ) )
40 simpr 449 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  v  e.  ( 0 [,] 1
) )
4140, 11sylib 190 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  [ v ]  .~  =/=  (/) )
42 neeq1 2611 . . . . . . . . . . . . 13  |-  ( z  =  [ v ]  .~  ->  ( z  =/=  (/)  <->  [ v ]  .~  =/=  (/) ) )
43 fveq2 5731 . . . . . . . . . . . . . 14  |-  ( z  =  [ v ]  .~  ->  ( F `  z )  =  ( F `  [ v ]  .~  ) )
44 id 21 . . . . . . . . . . . . . 14  |-  ( z  =  [ v ]  .~  ->  z  =  [ v ]  .~  )
4543, 44eleq12d 2506 . . . . . . . . . . . . 13  |-  ( z  =  [ v ]  .~  ->  ( ( F `  z )  e.  z  <->  ( F `  [ v ]  .~  )  e.  [ v ]  .~  ) )
4642, 45imbi12d 313 . . . . . . . . . . . 12  |-  ( z  =  [ v ]  .~  ->  ( (
z  =/=  (/)  ->  ( F `  z )  e.  z )  <->  ( [
v ]  .~  =/=  (/) 
->  ( F `  [
v ]  .~  )  e.  [ v ]  .~  ) ) )
4746rspcv 3050 . . . . . . . . . . 11  |-  ( [ v ]  .~  e.  S  ->  ( A. z  e.  S  ( z  =/=  (/)  ->  ( F `  z )  e.  z )  ->  ( [
v ]  .~  =/=  (/) 
->  ( F `  [
v ]  .~  )  e.  [ v ]  .~  ) ) )
4838, 39, 41, 47syl3c 60 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( F `  [ v ]  .~  )  e.  [
v ]  .~  )
49 fvex 5745 . . . . . . . . . . . 12  |-  ( F `
 [ v ]  .~  )  e.  _V
50 vex 2961 . . . . . . . . . . . 12  |-  v  e. 
_V
5149, 50elec 6947 . . . . . . . . . . 11  |-  ( ( F `  [ v ]  .~  )  e. 
[ v ]  .~  <->  v  .~  ( F `  [ v ]  .~  ) )
52 oveq12 6093 . . . . . . . . . . . . 13  |-  ( ( x  =  v  /\  y  =  ( F `  [ v ]  .~  ) )  ->  (
x  -  y )  =  ( v  -  ( F `  [ v ]  .~  ) ) )
5352eleq1d 2504 . . . . . . . . . . . 12  |-  ( ( x  =  v  /\  y  =  ( F `  [ v ]  .~  ) )  ->  (
( x  -  y
)  e.  QQ  <->  ( v  -  ( F `  [ v ]  .~  ) )  e.  QQ ) )
5453, 5brab2ga 4954 . . . . . . . . . . 11  |-  ( v  .~  ( F `  [ v ]  .~  ) 
<->  ( ( v  e.  ( 0 [,] 1
)  /\  ( F `  [ v ]  .~  )  e.  ( 0 [,] 1 ) )  /\  ( v  -  ( F `  [ v ]  .~  ) )  e.  QQ ) )
5551, 54bitri 242 . . . . . . . . . 10  |-  ( ( F `  [ v ]  .~  )  e. 
[ v ]  .~  <->  ( ( v  e.  ( 0 [,] 1 )  /\  ( F `  [ v ]  .~  )  e.  ( 0 [,] 1 ) )  /\  ( v  -  ( F `  [ v ]  .~  ) )  e.  QQ ) )
5648, 55sylib 190 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
( v  e.  ( 0 [,] 1 )  /\  ( F `  [ v ]  .~  )  e.  ( 0 [,] 1 ) )  /\  ( v  -  ( F `  [ v ]  .~  ) )  e.  QQ ) )
5756simprd 451 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( F `
 [ v ]  .~  ) )  e.  QQ )
58 0re 9096 . . . . . . . . . . . . 13  |-  0  e.  RR
59 1re 9095 . . . . . . . . . . . . 13  |-  1  e.  RR
6058, 59elicc2i 10981 . . . . . . . . . . . 12  |-  ( v  e.  ( 0 [,] 1 )  <->  ( v  e.  RR  /\  0  <_ 
v  /\  v  <_  1 ) )
6140, 60sylib 190 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  e.  RR  /\  0  <_  v  /\  v  <_  1 ) )
6261simp1d 970 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  v  e.  RR )
6356simpld 447 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  e.  ( 0 [,] 1 )  /\  ( F `  [ v ]  .~  )  e.  ( 0 [,] 1
) ) )
6463simprd 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( F `  [ v ]  .~  )  e.  ( 0 [,] 1 ) )
6558, 59elicc2i 10981 . . . . . . . . . . . 12  |-  ( ( F `  [ v ]  .~  )  e.  ( 0 [,] 1
)  <->  ( ( F `
 [ v ]  .~  )  e.  RR  /\  0  <_  ( F `  [ v ]  .~  )  /\  ( F `  [ v ]  .~  )  <_  1 ) )
6664, 65sylib 190 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
( F `  [
v ]  .~  )  e.  RR  /\  0  <_ 
( F `  [
v ]  .~  )  /\  ( F `  [
v ]  .~  )  <_  1 ) )
6766simp1d 970 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( F `  [ v ]  .~  )  e.  RR )
6862, 67resubcld 9470 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( F `
 [ v ]  .~  ) )  e.  RR )
6967, 62resubcld 9470 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
( F `  [
v ]  .~  )  -  v )  e.  RR )
7059a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  1  e.  RR )
7161simp2d 971 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  0  <_  v )
7267, 62subge02d 9623 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
0  <_  v  <->  ( ( F `  [ v ]  .~  )  -  v
)  <_  ( F `  [ v ]  .~  ) ) )
7371, 72mpbid 203 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
( F `  [
v ]  .~  )  -  v )  <_ 
( F `  [
v ]  .~  )
)
7466simp3d 972 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( F `  [ v ]  .~  )  <_  1
)
7569, 67, 70, 73, 74letrd 9232 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
( F `  [
v ]  .~  )  -  v )  <_ 
1 )
7669, 70lenegd 9610 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
( ( F `  [ v ]  .~  )  -  v )  <_  1  <->  -u 1  <_  -u (
( F `  [
v ]  .~  )  -  v ) ) )
7775, 76mpbid 203 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  -u 1  <_ 
-u ( ( F `
 [ v ]  .~  )  -  v
) )
7867recnd 9119 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( F `  [ v ]  .~  )  e.  CC )
7962recnd 9119 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  v  e.  CC )
8078, 79negsubdi2d 9432 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  -u (
( F `  [
v ]  .~  )  -  v )  =  ( v  -  ( F `  [ v ]  .~  ) ) )
8177, 80breqtrd 4239 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  -u 1  <_  ( v  -  ( F `  [ v ]  .~  ) ) )
8266simp2d 971 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  0  <_  ( F `  [
v ]  .~  )
)
8362, 67subge02d 9623 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
0  <_  ( F `  [ v ]  .~  ) 
<->  ( v  -  ( F `  [ v ]  .~  ) )  <_ 
v ) )
8482, 83mpbid 203 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( F `
 [ v ]  .~  ) )  <_ 
v )
8561simp3d 972 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  v  <_  1 )
8668, 62, 70, 84, 85letrd 9232 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( F `
 [ v ]  .~  ) )  <_ 
1 )
8759renegcli 9367 . . . . . . . . . 10  |-  -u 1  e.  RR
8887, 59elicc2i 10981 . . . . . . . . 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 1139 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( F `
 [ v ]  .~  ) )  e.  ( -u 1 [,] 1 ) )
90 elin 3532 . . . . . . . 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 647 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( F `
 [ v ]  .~  ) )  e.  ( QQ  i^i  ( -u 1 [,] 1 ) ) )
9232, 91ffvelrnd 5874 . . . . . 6  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) )  e.  NN )
93 f1ocnvfv2 6018 . . . . . . . . . . . 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 644 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) )  =  ( v  -  ( F `  [ v ]  .~  ) ) )
9594oveq2d 6100 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  =  ( v  -  ( v  -  ( F `  [ v ]  .~  ) ) ) )
9679, 78nncand 9421 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( v  -  ( F `  [ v ]  .~  ) ) )  =  ( F `  [
v ]  .~  )
)
9795, 96eqtrd 2470 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  =  ( F `  [ v ]  .~  ) )
981adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  F  Fn  S )
99 fnfvelrn 5870 . . . . . . . . . 10  |-  ( ( F  Fn  S  /\  [ v ]  .~  e.  S )  ->  ( F `  [ v ]  .~  )  e.  ran  F )
10098, 38, 99syl2anc 644 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( F `  [ v ]  .~  )  e.  ran  F )
10197, 100eqeltrd 2512 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F )
102 oveq1 6091 . . . . . . . . . 10  |-  ( s  =  v  ->  (
s  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  =  ( v  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) ) )
103102eleq1d 2504 . . . . . . . . 9  |-  ( s  =  v  ->  (
( s  -  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F  <->  ( v  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F ) )
104103elrab 3094 . . . . . . . 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 647 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  v  e.  { s  e.  RR  |  ( s  -  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F } )
106 fveq2 5731 . . . . . . . . . . . 12  |-  ( n  =  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) )  ->  ( G `  n )  =  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )
107106oveq2d 6100 . . . . . . . . . . 11  |-  ( n  =  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) )  ->  ( s  -  ( G `  n ) )  =  ( s  -  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) ) )
108107eleq1d 2504 . . . . . . . . . 10  |-  ( n  =  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) )  ->  ( ( s  -  ( G `  n ) )  e. 
ran  F  <->  ( s  -  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F
) )
109108rabbidv 2950 . . . . . . . . 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 9086 . . . . . . . . . 10  |-  RR  e.  _V
112111rabex 4357 . . . . . . . . 9  |-  { s  e.  RR  |  ( s  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F }  e.  _V
113109, 110, 112fvmpt 5809 . . . . . . . 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 2515 . . . . . 6  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  v  e.  ( T `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )
116 fveq2 5731 . . . . . . . 8  |-  ( m  =  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) )  ->  ( T `  m )  =  ( T `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )
117116eleq2d 2505 . . . . . . 7  |-  ( m  =  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) )  ->  ( v  e.  ( T `  m
)  <->  v  e.  ( T `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) ) )
118117rspcev 3054 . . . . . 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 644 . . . . 5  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  E. m  e.  NN  v  e.  ( T `  m ) )
120 eliun 4099 . . . . 5  |-  ( v  e.  U_ m  e.  NN  ( T `  m )  <->  E. m  e.  NN  v  e.  ( T `  m ) )
121119, 120sylibr 205 . . . 4  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  v  e.  U_ m  e.  NN  ( T `  m ) )
122121ex 425 . . 3  |-  ( ph  ->  ( v  e.  ( 0 [,] 1 )  ->  v  e.  U_ m  e.  NN  ( T `  m )
) )
123122ssrdv 3356 . 2  |-  ( ph  ->  ( 0 [,] 1
)  C_  U_ m  e.  NN  ( T `  m ) )
124 eliun 4099 . . . 4  |-  ( x  e.  U_ m  e.  NN  ( T `  m )  <->  E. m  e.  NN  x  e.  ( T `  m ) )
125 fveq2 5731 . . . . . . . . . . . . . . . 16  |-  ( n  =  m  ->  ( G `  n )  =  ( G `  m ) )
126125oveq2d 6100 . . . . . . . . . . . . . . 15  |-  ( n  =  m  ->  (
s  -  ( G `
 n ) )  =  ( s  -  ( G `  m ) ) )
127126eleq1d 2504 . . . . . . . . . . . . . 14  |-  ( n  =  m  ->  (
( s  -  ( G `  n )
)  e.  ran  F  <->  ( s  -  ( G `
 m ) )  e.  ran  F ) )
128127rabbidv 2950 . . . . . . . . . . . . 13  |-  ( n  =  m  ->  { s  e.  RR  |  ( s  -  ( G `
 n ) )  e.  ran  F }  =  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } )
129111rabex 4357 . . . . . . . . . . . . 13  |-  { s  e.  RR  |  ( s  -  ( G `
 m ) )  e.  ran  F }  e.  _V
130128, 110, 129fvmpt 5809 . . . . . . . . . . . 12  |-  ( m  e.  NN  ->  ( T `  m )  =  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } )
131130adantl 454 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN )  ->  ( T `
 m )  =  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } )
132131eleq2d 2505 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN )  ->  ( x  e.  ( T `  m )  <->  x  e.  { s  e.  RR  | 
( s  -  ( G `  m )
)  e.  ran  F } ) )
133132biimpa 472 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  x  e.  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } )
134 oveq1 6091 . . . . . . . . . . 11  |-  ( s  =  x  ->  (
s  -  ( G `
 m ) )  =  ( x  -  ( G `  m ) ) )
135134eleq1d 2504 . . . . . . . . . 10  |-  ( s  =  x  ->  (
( s  -  ( G `  m )
)  e.  ran  F  <->  ( x  -  ( G `
 m ) )  e.  ran  F ) )
136135elrab 3094 . . . . . . . . 9  |-  ( x  e.  { s  e.  RR  |  ( s  -  ( G `  m ) )  e. 
ran  F }  <->  ( x  e.  RR  /\  ( x  -  ( G `  m ) )  e. 
ran  F ) )
137133, 136sylib 190 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
x  e.  RR  /\  ( x  -  ( G `  m )
)  e.  ran  F
) )
138137simpld 447 . . . . . . 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 10997 . . . . . . . . . . 11  |-  ( (
-u 1  e.  RR  /\  1  e.  RR )  ->  ( -u 1 [,] 1 )  C_  RR )
14187, 59, 140mp2an 655 . . . . . . . . . 10  |-  ( -u
1 [,] 1 ) 
C_  RR
142 inss2 3564 . . . . . . . . . . 11  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  ( -u 1 [,] 1
)
143 f1of 5677 . . . . . . . . . . . . 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 5873 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  ( QQ  i^i  ( -u 1 [,] 1 ) ) )
146142, 145sseldi 3348 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  ( -u 1 [,] 1 ) )
147141, 146sseldi 3348 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  RR )
148147adantr 453 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  ( G `  m )  e.  RR )
149146adantr 453 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  ( G `  m )  e.  ( -u 1 [,] 1 ) )
15087, 59elicc2i 10981 . . . . . . . . . 10  |-  ( ( G `  m )  e.  ( -u 1 [,] 1 )  <->  ( ( G `  m )  e.  RR  /\  -u 1  <_  ( G `  m
)  /\  ( G `  m )  <_  1
) )
151149, 150sylib 190 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
( G `  m
)  e.  RR  /\  -u 1  <_  ( G `  m )  /\  ( G `  m )  <_  1 ) )
152151simp2d 971 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  -u 1  <_  ( G `  m
) )
15327ad2antrr 708 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  ran  F 
C_  ( 0 [,] 1 ) )
154137simprd 451 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
x  -  ( G `
 m ) )  e.  ran  F )
155153, 154sseldd 3351 . . . . . . . . . . 11  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
x  -  ( G `
 m ) )  e.  ( 0 [,] 1 ) )
15658, 59elicc2i 10981 . . . . . . . . . . 11  |-  ( ( x  -  ( G `
 m ) )  e.  ( 0 [,] 1 )  <->  ( (
x  -  ( G `
 m ) )  e.  RR  /\  0  <_  ( x  -  ( G `  m )
)  /\  ( x  -  ( G `  m ) )  <_ 
1 ) )
157155, 156sylib 190 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
( x  -  ( G `  m )
)  e.  RR  /\  0  <_  ( x  -  ( G `  m ) )  /\  ( x  -  ( G `  m ) )  <_ 
1 ) )
158157simp2d 971 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  0  <_  ( x  -  ( G `  m )
) )
159138, 148subge0d 9621 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
0  <_  ( x  -  ( G `  m ) )  <->  ( G `  m )  <_  x
) )
160158, 159mpbid 203 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  ( G `  m )  <_  x )
161139, 148, 138, 152, 160letrd 9232 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  -u 1  <_  x )
162 peano2re 9244 . . . . . . . . 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 10074 . . . . . . . . 9  |-  2  e.  RR
165164a1i 11 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  2  e.  RR )
166157simp3d 972 . . . . . . . . 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 9630 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
( x  -  ( G `  m )
)  <_  1  <->  x  <_  ( ( G `  m
)  +  1 ) ) )
169166, 168mpbid 203 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  x  <_  ( ( G `  m )  +  1 ) )
170151simp3d 972 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  ( G `  m )  <_  1 )
171148, 167, 167, 170leadd1dd 9645 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
( G `  m
)  +  1 )  <_  ( 1  +  1 ) )
172 df-2 10063 . . . . . . . . 9  |-  2  =  ( 1  +  1 )
173171, 172syl6breqr 4255 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
( G `  m
)  +  1 )  <_  2 )
174138, 163, 165, 169, 173letrd 9232 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  x  <_  2 )
17587, 164elicc2i 10981 . . . . . . 7  |-  ( x  e.  ( -u 1 [,] 2 )  <->  ( x  e.  RR  /\  -u 1  <_  x  /\  x  <_ 
2 ) )
176138, 161, 174, 175syl3anbrc 1139 . . . . . 6  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  x  e.  ( -u 1 [,] 2 ) )
177176ex 425 . . . . 5  |-  ( (
ph  /\  m  e.  NN )  ->  ( x  e.  ( T `  m )  ->  x  e.  ( -u 1 [,] 2 ) ) )
178177rexlimdva 2832 . . . 4  |-  ( ph  ->  ( E. m  e.  NN  x  e.  ( T `  m )  ->  x  e.  (
-u 1 [,] 2
) ) )
179124, 178syl5bi 210 . . 3  |-  ( ph  ->  ( x  e.  U_ m  e.  NN  ( T `  m )  ->  x  e.  ( -u
1 [,] 2 ) ) )
180179ssrdv 3356 . 2  |-  ( ph  ->  U_ m  e.  NN  ( T `  m ) 
C_  ( -u 1 [,] 2 ) )
18127, 123, 1803jca 1135 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   {crab 2711   _Vcvv 2958    \ cdif 3319    i^i cin 3321    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   U_ciun 4095   class class class wbr 4215   {copab 4268    e. cmpt 4269   `'ccnv 4880   dom cdm 4881   ran crn 4882    Fn wfn 5452   -->wf 5453   -1-1-onto->wf1o 5456   ` cfv 5457  (class class class)co 6084    Er wer 6905   [cec 6906   /.cqs 6907   RRcr 8994   0cc0 8995   1c1 8996    + caddc 8998    <_ cle 9126    - cmin 9296   -ucneg 9297   NNcn 10005   2c2 10054   QQcq 10579   [,]cicc 10924   volcvol 19365
This theorem is referenced by:  vitalilem3  19507  vitalilem4  19508  vitalilem5  19509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-ec 6910  df-qs 6914  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-n0 10227  df-z 10288  df-q 10580  df-icc 10928
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