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Theorem vitalilem2 21089
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
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 2616 . . . . . . . . 9  |-  ( [ v ]  .~  =  z  ->  ( [ v ]  .~  =/=  (/)  <->  z  =/=  (/) ) )
5 vitali.1 . . . . . . . . . . . . . 14  |-  .~  =  { <. x ,  y
>.  |  ( (
x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,] 1 ) )  /\  ( x  -  y )  e.  QQ ) }
65vitalilem1 21088 . . . . . . . . . . . . 13  |-  .~  Er  ( 0 [,] 1
)
7 erdm 7111 . . . . . . . . . . . . 13  |-  (  .~  Er  ( 0 [,] 1
)  ->  dom  .~  =  ( 0 [,] 1
) )
86, 7ax-mp 5 . . . . . . . . . . . 12  |-  dom  .~  =  ( 0 [,] 1 )
98eleq2i 2507 . . . . . . . . . . 11  |-  ( v  e.  dom  .~  <->  v  e.  ( 0 [,] 1
) )
10 ecdmn0 7143 . . . . . . . . . . 11  |-  ( v  e.  dom  .~  <->  [ v ]  .~  =/=  (/) )
119, 10bitr3i 251 . . . . . . . . . 10  |-  ( v  e.  ( 0 [,] 1 )  <->  [ v ]  .~  =/=  (/) )
1211biimpi 194 . . . . . . . . 9  |-  ( v  e.  ( 0 [,] 1 )  ->  [ v ]  .~  =/=  (/) )
133, 4, 12ectocl 7168 . . . . . . . 8  |-  ( z  e.  S  ->  z  =/=  (/) )
1413adantl 466 . . . . . . 7  |-  ( (
ph  /\  z  e.  S )  ->  z  =/=  (/) )
15 sseq1 3377 . . . . . . . . . 10  |-  ( [ w ]  .~  =  z  ->  ( [ w ]  .~  C_  ( 0 [,] 1 )  <->  z  C_  ( 0 [,] 1
) ) )
166a1i 11 . . . . . . . . . . 11  |-  ( w  e.  ( 0 [,] 1 )  ->  .~  Er  ( 0 [,] 1
) )
1716ecss 7142 . . . . . . . . . 10  |-  ( w  e.  ( 0 [,] 1 )  ->  [ w ]  .~  C_  ( 0 [,] 1 ) )
183, 15, 17ectocl 7168 . . . . . . . . 9  |-  ( z  e.  S  ->  z  C_  ( 0 [,] 1
) )
1918adantl 466 . . . . . . . 8  |-  ( (
ph  /\  z  e.  S )  ->  z  C_  ( 0 [,] 1
) )
2019sseld 3355 . . . . . . 7  |-  ( (
ph  /\  z  e.  S )  ->  (
( F `  z
)  e.  z  -> 
( F `  z
)  e.  ( 0 [,] 1 ) ) )
2114, 20embantd 54 . . . . . 6  |-  ( (
ph  /\  z  e.  S )  ->  (
( z  =/=  (/)  ->  ( F `  z )  e.  z )  ->  ( F `  z )  e.  ( 0 [,] 1
) ) )
2221ralimdva 2794 . . . . 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 5869 . . . 4  |-  ( F : S --> ( 0 [,] 1 )  <->  ( F  Fn  S  /\  A. z  e.  S  ( F `  z )  e.  ( 0 [,] 1 ) ) )
251, 23, 24sylanbrc 664 . . 3  |-  ( ph  ->  F : S --> ( 0 [,] 1 ) )
26 frn 5565 . . 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 465 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  G : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) ) )
30 f1ocnv 5653 . . . . . . . 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 5641 . . . . . . . 8  |-  ( `' G : ( QQ 
i^i  ( -u 1 [,] 1 ) ) -1-1-onto-> NN  ->  `' G : ( QQ 
i^i  ( -u 1 [,] 1 ) ) --> NN )
3229, 30, 313syl 20 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  `' G : ( QQ  i^i  ( -u 1 [,] 1
) ) --> NN )
33 ovex 6116 . . . . . . . . . . . . . . 15  |-  ( 0 [,] 1 )  e. 
_V
34 erex 7125 . . . . . . . . . . . . . . 15  |-  (  .~  Er  ( 0 [,] 1
)  ->  ( (
0 [,] 1 )  e.  _V  ->  .~  e.  _V ) )
356, 33, 34mp2 9 . . . . . . . . . . . . . 14  |-  .~  e.  _V
3635ecelqsi 7156 . . . . . . . . . . . . 13  |-  ( v  e.  ( 0 [,] 1 )  ->  [ v ]  .~  e.  ( ( 0 [,] 1
) /.  .~  )
)
3736adantl 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  [ v ]  .~  e.  ( ( 0 [,] 1
) /.  .~  )
)
3837, 3syl6eleqr 2534 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  [ v ]  .~  e.  S
)
392adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  A. z  e.  S  ( z  =/=  (/)  ->  ( F `  z )  e.  z ) )
40 simpr 461 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  v  e.  ( 0 [,] 1
) )
4140, 11sylib 196 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  [ v ]  .~  =/=  (/) )
42 neeq1 2616 . . . . . . . . . . . . 13  |-  ( z  =  [ v ]  .~  ->  ( z  =/=  (/)  <->  [ v ]  .~  =/=  (/) ) )
43 fveq2 5691 . . . . . . . . . . . . . 14  |-  ( z  =  [ v ]  .~  ->  ( F `  z )  =  ( F `  [ v ]  .~  ) )
44 id 22 . . . . . . . . . . . . . 14  |-  ( z  =  [ v ]  .~  ->  z  =  [ v ]  .~  )
4543, 44eleq12d 2511 . . . . . . . . . . . . 13  |-  ( z  =  [ v ]  .~  ->  ( ( F `  z )  e.  z  <->  ( F `  [ v ]  .~  )  e.  [ v ]  .~  ) )
4642, 45imbi12d 320 . . . . . . . . . . . 12  |-  ( z  =  [ v ]  .~  ->  ( (
z  =/=  (/)  ->  ( F `  z )  e.  z )  <->  ( [
v ]  .~  =/=  (/) 
->  ( F `  [
v ]  .~  )  e.  [ v ]  .~  ) ) )
4746rspcv 3069 . . . . . . . . . . 11  |-  ( [ v ]  .~  e.  S  ->  ( A. z  e.  S  ( z  =/=  (/)  ->  ( F `  z )  e.  z )  ->  ( [
v ]  .~  =/=  (/) 
->  ( F `  [
v ]  .~  )  e.  [ v ]  .~  ) ) )
4838, 39, 41, 47syl3c 61 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( F `  [ v ]  .~  )  e.  [
v ]  .~  )
49 fvex 5701 . . . . . . . . . . . 12  |-  ( F `
 [ v ]  .~  )  e.  _V
50 vex 2975 . . . . . . . . . . . 12  |-  v  e. 
_V
5149, 50elec 7140 . . . . . . . . . . 11  |-  ( ( F `  [ v ]  .~  )  e. 
[ v ]  .~  <->  v  .~  ( F `  [ v ]  .~  ) )
52 oveq12 6100 . . . . . . . . . . . . 13  |-  ( ( x  =  v  /\  y  =  ( F `  [ v ]  .~  ) )  ->  (
x  -  y )  =  ( v  -  ( F `  [ v ]  .~  ) ) )
5352eleq1d 2509 . . . . . . . . . . . 12  |-  ( ( x  =  v  /\  y  =  ( F `  [ v ]  .~  ) )  ->  (
( x  -  y
)  e.  QQ  <->  ( v  -  ( F `  [ v ]  .~  ) )  e.  QQ ) )
5453, 5brab2ga 4912 . . . . . . . . . . 11  |-  ( v  .~  ( F `  [ v ]  .~  ) 
<->  ( ( v  e.  ( 0 [,] 1
)  /\  ( F `  [ v ]  .~  )  e.  ( 0 [,] 1 ) )  /\  ( v  -  ( F `  [ v ]  .~  ) )  e.  QQ ) )
5551, 54bitri 249 . . . . . . . . . 10  |-  ( ( F `  [ v ]  .~  )  e. 
[ v ]  .~  <->  ( ( v  e.  ( 0 [,] 1 )  /\  ( F `  [ v ]  .~  )  e.  ( 0 [,] 1 ) )  /\  ( v  -  ( F `  [ v ]  .~  ) )  e.  QQ ) )
5648, 55sylib 196 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
( v  e.  ( 0 [,] 1 )  /\  ( F `  [ v ]  .~  )  e.  ( 0 [,] 1 ) )  /\  ( v  -  ( F `  [ v ]  .~  ) )  e.  QQ ) )
5756simprd 463 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( F `
 [ v ]  .~  ) )  e.  QQ )
58 0re 9386 . . . . . . . . . . . . 13  |-  0  e.  RR
59 1re 9385 . . . . . . . . . . . . 13  |-  1  e.  RR
6058, 59elicc2i 11361 . . . . . . . . . . . 12  |-  ( v  e.  ( 0 [,] 1 )  <->  ( v  e.  RR  /\  0  <_ 
v  /\  v  <_  1 ) )
6140, 60sylib 196 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  e.  RR  /\  0  <_  v  /\  v  <_  1 ) )
6261simp1d 1000 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  v  e.  RR )
6356simpld 459 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  e.  ( 0 [,] 1 )  /\  ( F `  [ v ]  .~  )  e.  ( 0 [,] 1
) ) )
6463simprd 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( F `  [ v ]  .~  )  e.  ( 0 [,] 1 ) )
6558, 59elicc2i 11361 . . . . . . . . . . . 12  |-  ( ( F `  [ v ]  .~  )  e.  ( 0 [,] 1
)  <->  ( ( F `
 [ v ]  .~  )  e.  RR  /\  0  <_  ( F `  [ v ]  .~  )  /\  ( F `  [ v ]  .~  )  <_  1 ) )
6664, 65sylib 196 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
( F `  [
v ]  .~  )  e.  RR  /\  0  <_ 
( F `  [
v ]  .~  )  /\  ( F `  [
v ]  .~  )  <_  1 ) )
6766simp1d 1000 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( F `  [ v ]  .~  )  e.  RR )
6862, 67resubcld 9776 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( F `
 [ v ]  .~  ) )  e.  RR )
6967, 62resubcld 9776 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
( F `  [
v ]  .~  )  -  v )  e.  RR )
70 1red 9401 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  1  e.  RR )
7161simp2d 1001 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  0  <_  v )
7267, 62subge02d 9931 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
0  <_  v  <->  ( ( F `  [ v ]  .~  )  -  v
)  <_  ( F `  [ v ]  .~  ) ) )
7371, 72mpbid 210 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
( F `  [
v ]  .~  )  -  v )  <_ 
( F `  [
v ]  .~  )
)
7466simp3d 1002 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( F `  [ v ]  .~  )  <_  1
)
7569, 67, 70, 73, 74letrd 9528 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
( F `  [
v ]  .~  )  -  v )  <_ 
1 )
7669, 70lenegd 9918 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
( ( F `  [ v ]  .~  )  -  v )  <_  1  <->  -u 1  <_  -u (
( F `  [
v ]  .~  )  -  v ) ) )
7775, 76mpbid 210 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  -u 1  <_ 
-u ( ( F `
 [ v ]  .~  )  -  v
) )
7867recnd 9412 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( F `  [ v ]  .~  )  e.  CC )
7962recnd 9412 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  v  e.  CC )
8078, 79negsubdi2d 9735 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  -u (
( F `  [
v ]  .~  )  -  v )  =  ( v  -  ( F `  [ v ]  .~  ) ) )
8177, 80breqtrd 4316 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  -u 1  <_  ( v  -  ( F `  [ v ]  .~  ) ) )
8266simp2d 1001 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  0  <_  ( F `  [
v ]  .~  )
)
8362, 67subge02d 9931 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
0  <_  ( F `  [ v ]  .~  ) 
<->  ( v  -  ( F `  [ v ]  .~  ) )  <_ 
v ) )
8482, 83mpbid 210 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( F `
 [ v ]  .~  ) )  <_ 
v )
8561simp3d 1002 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  v  <_  1 )
8668, 62, 70, 84, 85letrd 9528 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( F `
 [ v ]  .~  ) )  <_ 
1 )
87 neg1rr 10426 . . . . . . . . . 10  |-  -u 1  e.  RR
8887, 59elicc2i 11361 . . . . . . . . 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 1172 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( F `
 [ v ]  .~  ) )  e.  ( -u 1 [,] 1 ) )
9057, 89elind 3540 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( F `
 [ v ]  .~  ) )  e.  ( QQ  i^i  ( -u 1 [,] 1 ) ) )
9132, 90ffvelrnd 5844 . . . . . 6  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) )  e.  NN )
92 f1ocnvfv2 5984 . . . . . . . . . . . 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 ]  .~  ) ) )
9329, 90, 92syl2anc 661 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) )  =  ( v  -  ( F `  [ v ]  .~  ) ) )
9493oveq2d 6107 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  =  ( v  -  ( v  -  ( F `  [ v ]  .~  ) ) ) )
9579, 78nncand 9724 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( v  -  ( F `  [ v ]  .~  ) ) )  =  ( F `  [
v ]  .~  )
)
9694, 95eqtrd 2475 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  =  ( F `  [ v ]  .~  ) )
971adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  F  Fn  S )
98 fnfvelrn 5840 . . . . . . . . . 10  |-  ( ( F  Fn  S  /\  [ v ]  .~  e.  S )  ->  ( F `  [ v ]  .~  )  e.  ran  F )
9997, 38, 98syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( F `  [ v ]  .~  )  e.  ran  F )
10096, 99eqeltrd 2517 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F )
101 oveq1 6098 . . . . . . . . . 10  |-  ( s  =  v  ->  (
s  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  =  ( v  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) ) )
102101eleq1d 2509 . . . . . . . . 9  |-  ( s  =  v  ->  (
( s  -  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F  <->  ( v  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F ) )
103102elrab 3117 . . . . . . . 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
) )
10462, 100, 103sylanbrc 664 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  v  e.  { s  e.  RR  |  ( s  -  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F } )
105 fveq2 5691 . . . . . . . . . . . 12  |-  ( n  =  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) )  ->  ( G `  n )  =  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )
106105oveq2d 6107 . . . . . . . . . . 11  |-  ( n  =  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) )  ->  ( s  -  ( G `  n ) )  =  ( s  -  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) ) )
107106eleq1d 2509 . . . . . . . . . 10  |-  ( n  =  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) )  ->  ( ( s  -  ( G `  n ) )  e. 
ran  F  <->  ( s  -  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F
) )
108107rabbidv 2964 . . . . . . . . 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 } )
109 vitali.6 . . . . . . . . 9  |-  T  =  ( n  e.  NN  |->  { s  e.  RR  |  ( s  -  ( G `  n ) )  e.  ran  F } )
110 reex 9373 . . . . . . . . . 10  |-  RR  e.  _V
111110rabex 4443 . . . . . . . . 9  |-  { s  e.  RR  |  ( s  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F }  e.  _V
112108, 109, 111fvmpt 5774 . . . . . . . 8  |-  ( ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) )  e.  NN  ->  ( T `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) )  =  { s  e.  RR  |  ( s  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F } )
11391, 112syl 16 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( T `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) )  =  { s  e.  RR  |  ( s  -  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F } )
114104, 113eleqtrrd 2520 . . . . . 6  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  v  e.  ( T `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )
115 fveq2 5691 . . . . . . . 8  |-  ( m  =  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) )  ->  ( T `  m )  =  ( T `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )
116115eleq2d 2510 . . . . . . 7  |-  ( m  =  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) )  ->  ( v  e.  ( T `  m
)  <->  v  e.  ( T `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) ) )
117116rspcev 3073 . . . . . 6  |-  ( ( ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) )  e.  NN  /\  v  e.  ( T `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  ->  E. m  e.  NN  v  e.  ( T `  m ) )
11891, 114, 117syl2anc 661 . . . . 5  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  E. m  e.  NN  v  e.  ( T `  m ) )
119 eliun 4175 . . . . 5  |-  ( v  e.  U_ m  e.  NN  ( T `  m )  <->  E. m  e.  NN  v  e.  ( T `  m ) )
120118, 119sylibr 212 . . . 4  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  v  e.  U_ m  e.  NN  ( T `  m ) )
121120ex 434 . . 3  |-  ( ph  ->  ( v  e.  ( 0 [,] 1 )  ->  v  e.  U_ m  e.  NN  ( T `  m )
) )
122121ssrdv 3362 . 2  |-  ( ph  ->  ( 0 [,] 1
)  C_  U_ m  e.  NN  ( T `  m ) )
123 eliun 4175 . . . 4  |-  ( x  e.  U_ m  e.  NN  ( T `  m )  <->  E. m  e.  NN  x  e.  ( T `  m ) )
124 fveq2 5691 . . . . . . . . . . . . . . . 16  |-  ( n  =  m  ->  ( G `  n )  =  ( G `  m ) )
125124oveq2d 6107 . . . . . . . . . . . . . . 15  |-  ( n  =  m  ->  (
s  -  ( G `
 n ) )  =  ( s  -  ( G `  m ) ) )
126125eleq1d 2509 . . . . . . . . . . . . . 14  |-  ( n  =  m  ->  (
( s  -  ( G `  n )
)  e.  ran  F  <->  ( s  -  ( G `
 m ) )  e.  ran  F ) )
127126rabbidv 2964 . . . . . . . . . . . . 13  |-  ( n  =  m  ->  { s  e.  RR  |  ( s  -  ( G `
 n ) )  e.  ran  F }  =  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } )
128110rabex 4443 . . . . . . . . . . . . 13  |-  { s  e.  RR  |  ( s  -  ( G `
 m ) )  e.  ran  F }  e.  _V
129127, 109, 128fvmpt 5774 . . . . . . . . . . . 12  |-  ( m  e.  NN  ->  ( T `  m )  =  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } )
130129adantl 466 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN )  ->  ( T `
 m )  =  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } )
131130eleq2d 2510 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN )  ->  ( x  e.  ( T `  m )  <->  x  e.  { s  e.  RR  | 
( s  -  ( G `  m )
)  e.  ran  F } ) )
132131biimpa 484 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  x  e.  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } )
133 oveq1 6098 . . . . . . . . . . 11  |-  ( s  =  x  ->  (
s  -  ( G `
 m ) )  =  ( x  -  ( G `  m ) ) )
134133eleq1d 2509 . . . . . . . . . 10  |-  ( s  =  x  ->  (
( s  -  ( G `  m )
)  e.  ran  F  <->  ( x  -  ( G `
 m ) )  e.  ran  F ) )
135134elrab 3117 . . . . . . . . 9  |-  ( x  e.  { s  e.  RR  |  ( s  -  ( G `  m ) )  e. 
ran  F }  <->  ( x  e.  RR  /\  ( x  -  ( G `  m ) )  e. 
ran  F ) )
136132, 135sylib 196 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
x  e.  RR  /\  ( x  -  ( G `  m )
)  e.  ran  F
) )
137136simpld 459 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  x  e.  RR )
13887a1i 11 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  -u 1  e.  RR )
139 iccssre 11377 . . . . . . . . . . 11  |-  ( (
-u 1  e.  RR  /\  1  e.  RR )  ->  ( -u 1 [,] 1 )  C_  RR )
14087, 59, 139mp2an 672 . . . . . . . . . 10  |-  ( -u
1 [,] 1 ) 
C_  RR
141 inss2 3571 . . . . . . . . . . 11  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  ( -u 1 [,] 1
)
142 f1of 5641 . . . . . . . . . . . . 13  |-  ( G : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) )  ->  G : NN
--> ( QQ  i^i  ( -u 1 [,] 1 ) ) )
14328, 142syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  G : NN --> ( QQ 
i^i  ( -u 1 [,] 1 ) ) )
144143ffvelrnda 5843 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  ( QQ  i^i  ( -u 1 [,] 1 ) ) )
145141, 144sseldi 3354 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  ( -u 1 [,] 1 ) )
146140, 145sseldi 3354 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  RR )
147146adantr 465 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  ( G `  m )  e.  RR )
148145adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  ( G `  m )  e.  ( -u 1 [,] 1 ) )
14987, 59elicc2i 11361 . . . . . . . . . 10  |-  ( ( G `  m )  e.  ( -u 1 [,] 1 )  <->  ( ( G `  m )  e.  RR  /\  -u 1  <_  ( G `  m
)  /\  ( G `  m )  <_  1
) )
150148, 149sylib 196 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
( G `  m
)  e.  RR  /\  -u 1  <_  ( G `  m )  /\  ( G `  m )  <_  1 ) )
151150simp2d 1001 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  -u 1  <_  ( G `  m
) )
15227ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  ran  F 
C_  ( 0 [,] 1 ) )
153136simprd 463 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
x  -  ( G `
 m ) )  e.  ran  F )
154152, 153sseldd 3357 . . . . . . . . . . 11  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
x  -  ( G `
 m ) )  e.  ( 0 [,] 1 ) )
15558, 59elicc2i 11361 . . . . . . . . . . 11  |-  ( ( x  -  ( G `
 m ) )  e.  ( 0 [,] 1 )  <->  ( (
x  -  ( G `
 m ) )  e.  RR  /\  0  <_  ( x  -  ( G `  m )
)  /\  ( x  -  ( G `  m ) )  <_ 
1 ) )
156154, 155sylib 196 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
( x  -  ( G `  m )
)  e.  RR  /\  0  <_  ( x  -  ( G `  m ) )  /\  ( x  -  ( G `  m ) )  <_ 
1 ) )
157156simp2d 1001 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  0  <_  ( x  -  ( G `  m )
) )
158137, 147subge0d 9929 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
0  <_  ( x  -  ( G `  m ) )  <->  ( G `  m )  <_  x
) )
159157, 158mpbid 210 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  ( G `  m )  <_  x )
160138, 147, 137, 151, 159letrd 9528 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  -u 1  <_  x )
161 peano2re 9542 . . . . . . . . 9  |-  ( ( G `  m )  e.  RR  ->  (
( G `  m
)  +  1 )  e.  RR )
162147, 161syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
( G `  m
)  +  1 )  e.  RR )
163 2re 10391 . . . . . . . . 9  |-  2  e.  RR
164163a1i 11 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  2  e.  RR )
165156simp3d 1002 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
x  -  ( G `
 m ) )  <_  1 )
166 1red 9401 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  1  e.  RR )
167137, 147, 166lesubadd2d 9938 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
( x  -  ( G `  m )
)  <_  1  <->  x  <_  ( ( G `  m
)  +  1 ) ) )
168165, 167mpbid 210 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  x  <_  ( ( G `  m )  +  1 ) )
169150simp3d 1002 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  ( G `  m )  <_  1 )
170147, 166, 166, 169leadd1dd 9953 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
( G `  m
)  +  1 )  <_  ( 1  +  1 ) )
171 df-2 10380 . . . . . . . . 9  |-  2  =  ( 1  +  1 )
172170, 171syl6breqr 4332 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
( G `  m
)  +  1 )  <_  2 )
173137, 162, 164, 168, 172letrd 9528 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  x  <_  2 )
17487, 163elicc2i 11361 . . . . . . 7  |-  ( x  e.  ( -u 1 [,] 2 )  <->  ( x  e.  RR  /\  -u 1  <_  x  /\  x  <_ 
2 ) )
175137, 160, 173, 174syl3anbrc 1172 . . . . . 6  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  x  e.  ( -u 1 [,] 2 ) )
176175ex 434 . . . . 5  |-  ( (
ph  /\  m  e.  NN )  ->  ( x  e.  ( T `  m )  ->  x  e.  ( -u 1 [,] 2 ) ) )
177176rexlimdva 2841 . . . 4  |-  ( ph  ->  ( E. m  e.  NN  x  e.  ( T `  m )  ->  x  e.  (
-u 1 [,] 2
) ) )
178123, 177syl5bi 217 . . 3  |-  ( ph  ->  ( x  e.  U_ m  e.  NN  ( T `  m )  ->  x  e.  ( -u
1 [,] 2 ) ) )
179178ssrdv 3362 . 2  |-  ( ph  ->  U_ m  e.  NN  ( T `  m ) 
C_  ( -u 1 [,] 2 ) )
18027, 122, 1793jca 1168 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   E.wrex 2716   {crab 2719   _Vcvv 2972    \ cdif 3325    i^i cin 3327    C_ wss 3328   (/)c0 3637   ~Pcpw 3860   U_ciun 4171   class class class wbr 4292   {copab 4349    e. cmpt 4350   `'ccnv 4839   dom cdm 4840   ran crn 4841    Fn wfn 5413   -->wf 5414   -1-1-onto->wf1o 5417   ` cfv 5418  (class class class)co 6091    Er wer 7098   [cec 7099   /.cqs 7100   RRcr 9281   0cc0 9282   1c1 9283    + caddc 9285    <_ cle 9419    - cmin 9595   -ucneg 9596   NNcn 10322   2c2 10371   QQcq 10953   [,]cicc 11303   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-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  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
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  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-iun 4173  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-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-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-ec 7103  df-qs 7107  df-en 7311  df-dom 7312  df-sdom 7313  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-n0 10580  df-z 10647  df-q 10954  df-icc 11307
This theorem is referenced by:  vitalilem3  21090  vitalilem4  21091  vitalilem5  21092
  Copyright terms: Public domain W3C validator