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Theorem vitalilem2 21048
Description: Lemma for vitali 21052. (Contributed by Mario Carneiro, 16-Jun-2014.)
Hypotheses
Ref Expression
vitali.1  |-  .~  =  { <. x ,  y
>.  |  ( (
x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,] 1 ) )  /\  ( x  -  y )  e.  QQ ) }
vitali.2  |-  S  =  ( ( 0 [,] 1 ) /.  .~  )
vitali.3  |-  ( ph  ->  F  Fn  S )
vitali.4  |-  ( ph  ->  A. z  e.  S  ( z  =/=  (/)  ->  ( F `  z )  e.  z ) )
vitali.5  |-  ( ph  ->  G : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) ) )
vitali.6  |-  T  =  ( n  e.  NN  |->  { s  e.  RR  |  ( s  -  ( G `  n ) )  e.  ran  F } )
vitali.7  |-  ( ph  ->  -.  ran  F  e.  ( ~P RR  \  dom  vol ) )
Assertion
Ref Expression
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 2614 . . . . . . . . 9  |-  ( [ v ]  .~  =  z  ->  ( [ v ]  .~  =/=  (/)  <->  z  =/=  (/) ) )
5 vitali.1 . . . . . . . . . . . . . 14  |-  .~  =  { <. x ,  y
>.  |  ( (
x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,] 1 ) )  /\  ( x  -  y )  e.  QQ ) }
65vitalilem1 21047 . . . . . . . . . . . . 13  |-  .~  Er  ( 0 [,] 1
)
7 erdm 7107 . . . . . . . . . . . . 13  |-  (  .~  Er  ( 0 [,] 1
)  ->  dom  .~  =  ( 0 [,] 1
) )
86, 7ax-mp 5 . . . . . . . . . . . 12  |-  dom  .~  =  ( 0 [,] 1 )
98eleq2i 2505 . . . . . . . . . . 11  |-  ( v  e.  dom  .~  <->  v  e.  ( 0 [,] 1
) )
10 ecdmn0 7139 . . . . . . . . . . 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 7164 . . . . . . . 8  |-  ( z  e.  S  ->  z  =/=  (/) )
1413adantl 463 . . . . . . 7  |-  ( (
ph  /\  z  e.  S )  ->  z  =/=  (/) )
15 sseq1 3374 . . . . . . . . . 10  |-  ( [ w ]  .~  =  z  ->  ( [ w ]  .~  C_  ( 0 [,] 1 )  <->  z  C_  ( 0 [,] 1
) ) )
166a1i 11 . . . . . . . . . . 11  |-  ( w  e.  ( 0 [,] 1 )  ->  .~  Er  ( 0 [,] 1
) )
1716ecss 7138 . . . . . . . . . 10  |-  ( w  e.  ( 0 [,] 1 )  ->  [ w ]  .~  C_  ( 0 [,] 1 ) )
183, 15, 17ectocl 7164 . . . . . . . . 9  |-  ( z  e.  S  ->  z  C_  ( 0 [,] 1
) )
1918adantl 463 . . . . . . . 8  |-  ( (
ph  /\  z  e.  S )  ->  z  C_  ( 0 [,] 1
) )
2019sseld 3352 . . . . . . 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 2792 . . . . 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 5866 . . . 4  |-  ( F : S --> ( 0 [,] 1 )  <->  ( F  Fn  S  /\  A. z  e.  S  ( F `  z )  e.  ( 0 [,] 1 ) ) )
251, 23, 24sylanbrc 659 . . 3  |-  ( ph  ->  F : S --> ( 0 [,] 1 ) )
26 frn 5562 . . 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 462 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  G : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) ) )
30 f1ocnv 5650 . . . . . . . 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 5638 . . . . . . . 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 6115 . . . . . . . . . . . . . . 15  |-  ( 0 [,] 1 )  e. 
_V
34 erex 7121 . . . . . . . . . . . . . . 15  |-  (  .~  Er  ( 0 [,] 1
)  ->  ( (
0 [,] 1 )  e.  _V  ->  .~  e.  _V ) )
356, 33, 34mp2 9 . . . . . . . . . . . . . 14  |-  .~  e.  _V
3635ecelqsi 7152 . . . . . . . . . . . . 13  |-  ( v  e.  ( 0 [,] 1 )  ->  [ v ]  .~  e.  ( ( 0 [,] 1
) /.  .~  )
)
3736adantl 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  [ v ]  .~  e.  ( ( 0 [,] 1
) /.  .~  )
)
3837, 3syl6eleqr 2532 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  [ v ]  .~  e.  S
)
392adantr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  A. z  e.  S  ( z  =/=  (/)  ->  ( F `  z )  e.  z ) )
40 simpr 458 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  v  e.  ( 0 [,] 1
) )
4140, 11sylib 196 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  [ v ]  .~  =/=  (/) )
42 neeq1 2614 . . . . . . . . . . . . 13  |-  ( z  =  [ v ]  .~  ->  ( z  =/=  (/)  <->  [ v ]  .~  =/=  (/) ) )
43 fveq2 5688 . . . . . . . . . . . . . 14  |-  ( z  =  [ v ]  .~  ->  ( F `  z )  =  ( F `  [ v ]  .~  ) )
44 id 22 . . . . . . . . . . . . . 14  |-  ( z  =  [ v ]  .~  ->  z  =  [ v ]  .~  )
4543, 44eleq12d 2509 . . . . . . . . . . . . 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 3066 . . . . . . . . . . 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 5698 . . . . . . . . . . . 12  |-  ( F `
 [ v ]  .~  )  e.  _V
50 vex 2973 . . . . . . . . . . . 12  |-  v  e. 
_V
5149, 50elec 7136 . . . . . . . . . . 11  |-  ( ( F `  [ v ]  .~  )  e. 
[ v ]  .~  <->  v  .~  ( F `  [ v ]  .~  ) )
52 oveq12 6099 . . . . . . . . . . . . 13  |-  ( ( x  =  v  /\  y  =  ( F `  [ v ]  .~  ) )  ->  (
x  -  y )  =  ( v  -  ( F `  [ v ]  .~  ) ) )
5352eleq1d 2507 . . . . . . . . . . . 12  |-  ( ( x  =  v  /\  y  =  ( F `  [ v ]  .~  ) )  ->  (
( x  -  y
)  e.  QQ  <->  ( v  -  ( F `  [ v ]  .~  ) )  e.  QQ ) )
5453, 5brab2ga 4908 . . . . . . . . . . 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 460 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( F `
 [ v ]  .~  ) )  e.  QQ )
58 0re 9382 . . . . . . . . . . . . 13  |-  0  e.  RR
59 1re 9381 . . . . . . . . . . . . 13  |-  1  e.  RR
6058, 59elicc2i 11357 . . . . . . . . . . . 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 995 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  v  e.  RR )
6356simpld 456 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  e.  ( 0 [,] 1 )  /\  ( F `  [ v ]  .~  )  e.  ( 0 [,] 1
) ) )
6463simprd 460 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( F `  [ v ]  .~  )  e.  ( 0 [,] 1 ) )
6558, 59elicc2i 11357 . . . . . . . . . . . 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 995 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( F `  [ v ]  .~  )  e.  RR )
6862, 67resubcld 9772 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( F `
 [ v ]  .~  ) )  e.  RR )
6967, 62resubcld 9772 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
( F `  [
v ]  .~  )  -  v )  e.  RR )
70 1red 9397 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  1  e.  RR )
7161simp2d 996 . . . . . . . . . . . . 13  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  0  <_  v )
7267, 62subge02d 9927 . . . . . . . . . . . . 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 997 . . . . . . . . . . . 12  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( F `  [ v ]  .~  )  <_  1
)
7569, 67, 70, 73, 74letrd 9524 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
( F `  [
v ]  .~  )  -  v )  <_ 
1 )
7669, 70lenegd 9914 . . . . . . . . . . 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 9408 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( F `  [ v ]  .~  )  e.  CC )
7962recnd 9408 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  v  e.  CC )
8078, 79negsubdi2d 9731 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  -u (
( F `  [
v ]  .~  )  -  v )  =  ( v  -  ( F `  [ v ]  .~  ) ) )
8177, 80breqtrd 4313 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  -u 1  <_  ( v  -  ( F `  [ v ]  .~  ) ) )
8266simp2d 996 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  0  <_  ( F `  [
v ]  .~  )
)
8362, 67subge02d 9927 . . . . . . . . . . 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 997 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  v  <_  1 )
8668, 62, 70, 84, 85letrd 9524 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( F `
 [ v ]  .~  ) )  <_ 
1 )
87 neg1rr 10422 . . . . . . . . . 10  |-  -u 1  e.  RR
8887, 59elicc2i 11357 . . . . . . . . 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 1167 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( F `
 [ v ]  .~  ) )  e.  ( -u 1 [,] 1 ) )
9057, 89elind 3537 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( F `
 [ v ]  .~  ) )  e.  ( QQ  i^i  ( -u 1 [,] 1 ) ) )
9132, 90ffvelrnd 5841 . . . . . 6  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) )  e.  NN )
92 f1ocnvfv2 5981 . . . . . . . . . . . 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 656 . . . . . . . . . . 11  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) )  =  ( v  -  ( F `  [ v ]  .~  ) ) )
9493oveq2d 6106 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  =  ( v  -  ( v  -  ( F `  [ v ]  .~  ) ) ) )
9579, 78nncand 9720 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( v  -  ( F `  [ v ]  .~  ) ) )  =  ( F `  [
v ]  .~  )
)
9694, 95eqtrd 2473 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  =  ( F `  [ v ]  .~  ) )
971adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  F  Fn  S )
98 fnfvelrn 5837 . . . . . . . . . 10  |-  ( ( F  Fn  S  /\  [ v ]  .~  e.  S )  ->  ( F `  [ v ]  .~  )  e.  ran  F )
9997, 38, 98syl2anc 656 . . . . . . . . 9  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  ( F `  [ v ]  .~  )  e.  ran  F )
10096, 99eqeltrd 2515 . . . . . . . 8  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  (
v  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F )
101 oveq1 6097 . . . . . . . . . 10  |-  ( s  =  v  ->  (
s  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  =  ( v  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) ) )
102101eleq1d 2507 . . . . . . . . 9  |-  ( s  =  v  ->  (
( s  -  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F  <->  ( v  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F ) )
103102elrab 3114 . . . . . . . 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 659 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  v  e.  { s  e.  RR  |  ( s  -  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F } )
105 fveq2 5688 . . . . . . . . . . . 12  |-  ( n  =  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) )  ->  ( G `  n )  =  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )
106105oveq2d 6106 . . . . . . . . . . 11  |-  ( n  =  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) )  ->  ( s  -  ( G `  n ) )  =  ( s  -  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) ) )
107106eleq1d 2507 . . . . . . . . . 10  |-  ( n  =  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) )  ->  ( ( s  -  ( G `  n ) )  e. 
ran  F  <->  ( s  -  ( G `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F
) )
108107rabbidv 2962 . . . . . . . . 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 9369 . . . . . . . . . 10  |-  RR  e.  _V
111110rabex 4440 . . . . . . . . 9  |-  { s  e.  RR  |  ( s  -  ( G `
 ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )  e.  ran  F }  e.  _V
112108, 109, 111fvmpt 5771 . . . . . . . 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 2518 . . . . . 6  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  v  e.  ( T `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )
115 fveq2 5688 . . . . . . . 8  |-  ( m  =  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) )  ->  ( T `  m )  =  ( T `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) )
116115eleq2d 2508 . . . . . . 7  |-  ( m  =  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) )  ->  ( v  e.  ( T `  m
)  <->  v  e.  ( T `  ( `' G `  ( v  -  ( F `  [ v ]  .~  ) ) ) ) ) )
117116rspcev 3070 . . . . . 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 656 . . . . 5  |-  ( (
ph  /\  v  e.  ( 0 [,] 1
) )  ->  E. m  e.  NN  v  e.  ( T `  m ) )
119 eliun 4172 . . . . 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 3359 . 2  |-  ( ph  ->  ( 0 [,] 1
)  C_  U_ m  e.  NN  ( T `  m ) )
123 eliun 4172 . . . 4  |-  ( x  e.  U_ m  e.  NN  ( T `  m )  <->  E. m  e.  NN  x  e.  ( T `  m ) )
124 fveq2 5688 . . . . . . . . . . . . . . . 16  |-  ( n  =  m  ->  ( G `  n )  =  ( G `  m ) )
125124oveq2d 6106 . . . . . . . . . . . . . . 15  |-  ( n  =  m  ->  (
s  -  ( G `
 n ) )  =  ( s  -  ( G `  m ) ) )
126125eleq1d 2507 . . . . . . . . . . . . . 14  |-  ( n  =  m  ->  (
( s  -  ( G `  n )
)  e.  ran  F  <->  ( s  -  ( G `
 m ) )  e.  ran  F ) )
127126rabbidv 2962 . . . . . . . . . . . . 13  |-  ( n  =  m  ->  { s  e.  RR  |  ( s  -  ( G `
 n ) )  e.  ran  F }  =  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } )
128110rabex 4440 . . . . . . . . . . . . 13  |-  { s  e.  RR  |  ( s  -  ( G `
 m ) )  e.  ran  F }  e.  _V
129127, 109, 128fvmpt 5771 . . . . . . . . . . . 12  |-  ( m  e.  NN  ->  ( T `  m )  =  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } )
130129adantl 463 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN )  ->  ( T `
 m )  =  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } )
131130eleq2d 2508 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN )  ->  ( x  e.  ( T `  m )  <->  x  e.  { s  e.  RR  | 
( s  -  ( G `  m )
)  e.  ran  F } ) )
132131biimpa 481 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  x  e.  { s  e.  RR  |  ( s  -  ( G `  m ) )  e.  ran  F } )
133 oveq1 6097 . . . . . . . . . . 11  |-  ( s  =  x  ->  (
s  -  ( G `
 m ) )  =  ( x  -  ( G `  m ) ) )
134133eleq1d 2507 . . . . . . . . . 10  |-  ( s  =  x  ->  (
( s  -  ( G `  m )
)  e.  ran  F  <->  ( x  -  ( G `
 m ) )  e.  ran  F ) )
135134elrab 3114 . . . . . . . . 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 456 . . . . . . 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 11373 . . . . . . . . . . 11  |-  ( (
-u 1  e.  RR  /\  1  e.  RR )  ->  ( -u 1 [,] 1 )  C_  RR )
14087, 59, 139mp2an 667 . . . . . . . . . 10  |-  ( -u
1 [,] 1 ) 
C_  RR
141 inss2 3568 . . . . . . . . . . 11  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  ( -u 1 [,] 1
)
142 f1of 5638 . . . . . . . . . . . . 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 5840 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  ( QQ  i^i  ( -u 1 [,] 1 ) ) )
145141, 144sseldi 3351 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  ( -u 1 [,] 1 ) )
146140, 145sseldi 3351 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  RR )
147146adantr 462 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  ( G `  m )  e.  RR )
148145adantr 462 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  ( G `  m )  e.  ( -u 1 [,] 1 ) )
14987, 59elicc2i 11357 . . . . . . . . . 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 996 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  -u 1  <_  ( G `  m
) )
15227ad2antrr 720 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  ran  F 
C_  ( 0 [,] 1 ) )
153136simprd 460 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
x  -  ( G `
 m ) )  e.  ran  F )
154152, 153sseldd 3354 . . . . . . . . . . 11  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
x  -  ( G `
 m ) )  e.  ( 0 [,] 1 ) )
15558, 59elicc2i 11357 . . . . . . . . . . 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 996 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  0  <_  ( x  -  ( G `  m )
) )
158137, 147subge0d 9925 . . . . . . . . 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 9524 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  -u 1  <_  x )
161 peano2re 9538 . . . . . . . . 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 10387 . . . . . . . . 9  |-  2  e.  RR
164163a1i 11 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  2  e.  RR )
165156simp3d 997 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
x  -  ( G `
 m ) )  <_  1 )
166 1red 9397 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  1  e.  RR )
167137, 147, 166lesubadd2d 9934 . . . . . . . . 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 997 . . . . . . . . . 10  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  ( G `  m )  <_  1 )
170147, 166, 166, 169leadd1dd 9949 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
( G `  m
)  +  1 )  <_  ( 1  +  1 ) )
171 df-2 10376 . . . . . . . . 9  |-  2  =  ( 1  +  1 )
172170, 171syl6breqr 4329 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  (
( G `  m
)  +  1 )  <_  2 )
173137, 162, 164, 168, 172letrd 9524 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  NN )  /\  x  e.  ( T `  m
) )  ->  x  <_  2 )
17487, 163elicc2i 11357 . . . . . . 7  |-  ( x  e.  ( -u 1 [,] 2 )  <->  ( x  e.  RR  /\  -u 1  <_  x  /\  x  <_ 
2 ) )
175137, 160, 173, 174syl3anbrc 1167 . . . . . 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 2839 . . . 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 3359 . 2  |-  ( ph  ->  U_ m  e.  NN  ( T `  m ) 
C_  ( -u 1 [,] 2 ) )
18027, 122, 1793jca 1163 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 960    = wceq 1364    e. wcel 1761    =/= wne 2604   A.wral 2713   E.wrex 2714   {crab 2717   _Vcvv 2970    \ cdif 3322    i^i cin 3324    C_ wss 3325   (/)c0 3634   ~Pcpw 3857   U_ciun 4168   class class class wbr 4289   {copab 4346    e. cmpt 4347   `'ccnv 4835   dom cdm 4836   ran crn 4837    Fn wfn 5410   -->wf 5411   -1-1-onto->wf1o 5414   ` cfv 5415  (class class class)co 6090    Er wer 7094   [cec 7095   /.cqs 7096   RRcr 9277   0cc0 9278   1c1 9279    + caddc 9281    <_ cle 9415    - cmin 9591   -ucneg 9592   NNcn 10318   2c2 10367   QQcq 10949   [,]cicc 11299   volcvol 20906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-ec 7099  df-qs 7103  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-n0 10576  df-z 10643  df-q 10950  df-icc 11303
This theorem is referenced by:  vitalilem3  21049  vitalilem4  21050  vitalilem5  21051
  Copyright terms: Public domain W3C validator