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Theorem vitali 21052
Description: If the reals can be well-ordered, then there are non-measurable sets. The proof uses "Vitali sets", named for Giuseppe Vitali (1905). (Contributed by Mario Carneiro, 16-Jun-2014.)
Assertion
Ref Expression
vitali  |-  (  .<  We  RR  ->  dom  vol  C.  ~P RR )

Proof of Theorem vitali
Dummy variables  a 
b  c  f  g  m  n  s  t  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reex 9369 . . . 4  |-  RR  e.  _V
21pwex 4472 . . 3  |-  ~P RR  e.  _V
3 weinxp 4902 . . . . 5  |-  (  .<  We  RR  <->  (  .<  i^i  ( RR  X.  RR ) )  We  RR )
4 unipw 4539 . . . . . 6  |-  U. ~P RR  =  RR
5 weeq2 4705 . . . . . 6  |-  ( U. ~P RR  =  RR  ->  ( (  .<  i^i  ( RR  X.  RR ) )  We  U. ~P RR  <->  ( 
.<  i^i  ( RR  X.  RR ) )  We  RR ) )
64, 5ax-mp 5 . . . . 5  |-  ( ( 
.<  i^i  ( RR  X.  RR ) )  We  U. ~P RR  <->  (  .<  i^i  ( RR  X.  RR ) )  We  RR )
73, 6bitr4i 252 . . . 4  |-  (  .<  We  RR  <->  (  .<  i^i  ( RR  X.  RR ) )  We  U. ~P RR )
81, 1xpex 6507 . . . . . 6  |-  ( RR 
X.  RR )  e. 
_V
98inex2 4431 . . . . 5  |-  (  .<  i^i  ( RR  X.  RR ) )  e.  _V
10 weeq1 4704 . . . . 5  |-  ( x  =  (  .<  i^i  ( RR  X.  RR ) )  ->  ( x  We 
U. ~P RR  <->  (  .<  i^i  ( RR  X.  RR ) )  We  U. ~P RR ) )
119, 10spcev 3061 . . . 4  |-  ( ( 
.<  i^i  ( RR  X.  RR ) )  We  U. ~P RR  ->  E. x  x  We  U. ~P RR )
127, 11sylbi 195 . . 3  |-  (  .<  We  RR  ->  E. x  x  We  U. ~P RR )
13 dfac8c 8199 . . 3  |-  ( ~P RR  e.  _V  ->  ( E. x  x  We 
U. ~P RR  ->  E. f A. z  e. 
~P  RR ( z  =/=  (/)  ->  ( f `  z )  e.  z ) ) )
142, 12, 13mpsyl 63 . 2  |-  (  .<  We  RR  ->  E. f A. z  e.  ~P  RR ( z  =/=  (/)  ->  (
f `  z )  e.  z ) )
15 qex 10961 . . . . . . 7  |-  QQ  e.  _V
1615inex1 4430 . . . . . 6  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  e. 
_V
17 nnrecq 10972 . . . . . . . 8  |-  ( x  e.  NN  ->  (
1  /  x )  e.  QQ )
18 nnrecre 10354 . . . . . . . . 9  |-  ( x  e.  NN  ->  (
1  /  x )  e.  RR )
19 neg1rr 10422 . . . . . . . . . . 11  |-  -u 1  e.  RR
2019a1i 11 . . . . . . . . . 10  |-  ( x  e.  NN  ->  -u 1  e.  RR )
21 0re 9382 . . . . . . . . . . 11  |-  0  e.  RR
2221a1i 11 . . . . . . . . . 10  |-  ( x  e.  NN  ->  0  e.  RR )
23 neg1lt0 10424 . . . . . . . . . . . 12  |-  -u 1  <  0
2419, 21, 23ltleii 9493 . . . . . . . . . . 11  |-  -u 1  <_  0
2524a1i 11 . . . . . . . . . 10  |-  ( x  e.  NN  ->  -u 1  <_  0 )
26 nnrp 10996 . . . . . . . . . . . 12  |-  ( x  e.  NN  ->  x  e.  RR+ )
2726rpreccld 11033 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  (
1  /  x )  e.  RR+ )
2827rpge0d 11027 . . . . . . . . . 10  |-  ( x  e.  NN  ->  0  <_  ( 1  /  x
) )
2920, 22, 18, 25, 28letrd 9524 . . . . . . . . 9  |-  ( x  e.  NN  ->  -u 1  <_  ( 1  /  x
) )
30 nnge1 10344 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  1  <_  x )
31 nnre 10325 . . . . . . . . . . . 12  |-  ( x  e.  NN  ->  x  e.  RR )
32 nngt0 10347 . . . . . . . . . . . 12  |-  ( x  e.  NN  ->  0  <  x )
33 1re 9381 . . . . . . . . . . . . 13  |-  1  e.  RR
34 0lt1 9858 . . . . . . . . . . . . 13  |-  0  <  1
35 lerec 10210 . . . . . . . . . . . . 13  |-  ( ( ( 1  e.  RR  /\  0  <  1 )  /\  ( x  e.  RR  /\  0  < 
x ) )  -> 
( 1  <_  x  <->  ( 1  /  x )  <_  ( 1  / 
1 ) ) )
3633, 34, 35mpanl12 677 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  0  <  x )  -> 
( 1  <_  x  <->  ( 1  /  x )  <_  ( 1  / 
1 ) ) )
3731, 32, 36syl2anc 656 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  (
1  <_  x  <->  ( 1  /  x )  <_ 
( 1  /  1
) ) )
3830, 37mpbid 210 . . . . . . . . . 10  |-  ( x  e.  NN  ->  (
1  /  x )  <_  ( 1  / 
1 ) )
39 1div1e1 10020 . . . . . . . . . 10  |-  ( 1  /  1 )  =  1
4038, 39syl6breq 4328 . . . . . . . . 9  |-  ( x  e.  NN  ->  (
1  /  x )  <_  1 )
4119, 33elicc2i 11357 . . . . . . . . 9  |-  ( ( 1  /  x )  e.  ( -u 1 [,] 1 )  <->  ( (
1  /  x )  e.  RR  /\  -u 1  <_  ( 1  /  x
)  /\  ( 1  /  x )  <_ 
1 ) )
4218, 29, 40, 41syl3anbrc 1167 . . . . . . . 8  |-  ( x  e.  NN  ->  (
1  /  x )  e.  ( -u 1 [,] 1 ) )
4317, 42elind 3537 . . . . . . 7  |-  ( x  e.  NN  ->  (
1  /  x )  e.  ( QQ  i^i  ( -u 1 [,] 1
) ) )
44 oveq2 6098 . . . . . . . . 9  |-  ( ( 1  /  x )  =  ( 1  / 
y )  ->  (
1  /  ( 1  /  x ) )  =  ( 1  / 
( 1  /  y
) ) )
45 nncn 10326 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  x  e.  CC )
46 nnne0 10350 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  x  =/=  0 )
4745, 46recrecd 10100 . . . . . . . . . 10  |-  ( x  e.  NN  ->  (
1  /  ( 1  /  x ) )  =  x )
48 nncn 10326 . . . . . . . . . . 11  |-  ( y  e.  NN  ->  y  e.  CC )
49 nnne0 10350 . . . . . . . . . . 11  |-  ( y  e.  NN  ->  y  =/=  0 )
5048, 49recrecd 10100 . . . . . . . . . 10  |-  ( y  e.  NN  ->  (
1  /  ( 1  /  y ) )  =  y )
5147, 50eqeqan12d 2456 . . . . . . . . 9  |-  ( ( x  e.  NN  /\  y  e.  NN )  ->  ( ( 1  / 
( 1  /  x
) )  =  ( 1  /  ( 1  /  y ) )  <-> 
x  =  y ) )
5244, 51syl5ib 219 . . . . . . . 8  |-  ( ( x  e.  NN  /\  y  e.  NN )  ->  ( ( 1  /  x )  =  ( 1  /  y )  ->  x  =  y ) )
53 oveq2 6098 . . . . . . . 8  |-  ( x  =  y  ->  (
1  /  x )  =  ( 1  / 
y ) )
5452, 53impbid1 203 . . . . . . 7  |-  ( ( x  e.  NN  /\  y  e.  NN )  ->  ( ( 1  /  x )  =  ( 1  /  y )  <-> 
x  =  y ) )
5543, 54dom2 7348 . . . . . 6  |-  ( ( QQ  i^i  ( -u
1 [,] 1 ) )  e.  _V  ->  NN  ~<_  ( QQ  i^i  ( -u 1 [,] 1 ) ) )
5616, 55ax-mp 5 . . . . 5  |-  NN  ~<_  ( QQ 
i^i  ( -u 1 [,] 1 ) )
57 inss1 3567 . . . . . . 7  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  QQ
58 ssdomg 7351 . . . . . . 7  |-  ( QQ  e.  _V  ->  (
( QQ  i^i  ( -u 1 [,] 1 ) )  C_  QQ  ->  ( QQ  i^i  ( -u
1 [,] 1 ) )  ~<_  QQ ) )
5915, 57, 58mp2 9 . . . . . 6  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  ~<_  QQ
60 qnnen 13492 . . . . . 6  |-  QQ  ~~  NN
61 domentr 7364 . . . . . 6  |-  ( ( ( QQ  i^i  ( -u 1 [,] 1 ) )  ~<_  QQ  /\  QQ  ~~  NN )  ->  ( QQ 
i^i  ( -u 1 [,] 1 ) )  ~<_  NN )
6259, 60, 61mp2an 667 . . . . 5  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  ~<_  NN
63 sbth 7427 . . . . 5  |-  ( ( NN  ~<_  ( QQ  i^i  ( -u 1 [,] 1
) )  /\  ( QQ  i^i  ( -u 1 [,] 1 ) )  ~<_  NN )  ->  NN  ~~  ( QQ  i^i  ( -u 1 [,] 1 ) ) )
6456, 62, 63mp2an 667 . . . 4  |-  NN  ~~  ( QQ  i^i  ( -u 1 [,] 1 ) )
65 bren 7315 . . . 4  |-  ( NN 
~~  ( QQ  i^i  ( -u 1 [,] 1
) )  <->  E. g 
g : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) ) )
6664, 65mpbi 208 . . 3  |-  E. g 
g : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) )
67 eleq1 2501 . . . . . . . . . . . . 13  |-  ( a  =  x  ->  (
a  e.  ( 0 [,] 1 )  <->  x  e.  ( 0 [,] 1
) ) )
68 eleq1 2501 . . . . . . . . . . . . 13  |-  ( b  =  y  ->  (
b  e.  ( 0 [,] 1 )  <->  y  e.  ( 0 [,] 1
) ) )
6967, 68bi2anan9 863 . . . . . . . . . . . 12  |-  ( ( a  =  x  /\  b  =  y )  ->  ( ( a  e.  ( 0 [,] 1
)  /\  b  e.  ( 0 [,] 1
) )  <->  ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) ) ) )
70 oveq12 6099 . . . . . . . . . . . . 13  |-  ( ( a  =  x  /\  b  =  y )  ->  ( a  -  b
)  =  ( x  -  y ) )
7170eleq1d 2507 . . . . . . . . . . . 12  |-  ( ( a  =  x  /\  b  =  y )  ->  ( ( a  -  b )  e.  QQ  <->  ( x  -  y )  e.  QQ ) )
7269, 71anbi12d 705 . . . . . . . . . . 11  |-  ( ( a  =  x  /\  b  =  y )  ->  ( ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1
) )  /\  (
a  -  b )  e.  QQ )  <->  ( (
x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,] 1 ) )  /\  ( x  -  y )  e.  QQ ) ) )
7372cbvopabv 4358 . . . . . . . . . 10  |-  { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) }  =  { <. x ,  y
>.  |  ( (
x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,] 1 ) )  /\  ( x  -  y )  e.  QQ ) }
74 eqid 2441 . . . . . . . . . 10  |-  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } )  =  ( ( 0 [,] 1 ) /. { <. a ,  b
>.  |  ( (
a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } )
75 fvex 5698 . . . . . . . . . . . 12  |-  ( f `
 c )  e. 
_V
76 eqid 2441 . . . . . . . . . . . 12  |-  ( c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b
>.  |  ( (
a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } )  |->  ( f `
 c ) )  =  ( c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1
) )  /\  (
a  -  b )  e.  QQ ) } )  |->  ( f `  c ) )
7775, 76fnmpti 5536 . . . . . . . . . . 11  |-  ( c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b
>.  |  ( (
a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } )  |->  ( f `
 c ) )  Fn  ( ( 0 [,] 1 ) /. { <. a ,  b
>.  |  ( (
a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } )
7877a1i 11 . . . . . . . . . 10  |-  ( ( (  .<  We  RR  /\ 
A. z  e.  ~P  RR ( z  =/=  (/)  ->  (
f `  z )  e.  z ) )  /\  ( g : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) )  /\  -.  ran  ( c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1
) )  /\  (
a  -  b )  e.  QQ ) } )  |->  ( f `  c ) )  e.  ( ~P RR  \  dom  vol ) ) )  ->  ( c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1
) )  /\  (
a  -  b )  e.  QQ ) } )  |->  ( f `  c ) )  Fn  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1
) )  /\  (
a  -  b )  e.  QQ ) } ) )
79 neeq1 2614 . . . . . . . . . . . . . . 15  |-  ( z  =  w  ->  (
z  =/=  (/)  <->  w  =/=  (/) ) )
80 fveq2 5688 . . . . . . . . . . . . . . . 16  |-  ( z  =  w  ->  (
f `  z )  =  ( f `  w ) )
81 id 22 . . . . . . . . . . . . . . . 16  |-  ( z  =  w  ->  z  =  w )
8280, 81eleq12d 2509 . . . . . . . . . . . . . . 15  |-  ( z  =  w  ->  (
( f `  z
)  e.  z  <->  ( f `  w )  e.  w
) )
8379, 82imbi12d 320 . . . . . . . . . . . . . 14  |-  ( z  =  w  ->  (
( z  =/=  (/)  ->  (
f `  z )  e.  z )  <->  ( w  =/=  (/)  ->  ( f `  w )  e.  w
) ) )
8483cbvralv 2945 . . . . . . . . . . . . 13  |-  ( A. z  e.  ~P  RR ( z  =/=  (/)  ->  (
f `  z )  e.  z )  <->  A. w  e.  ~P  RR ( w  =/=  (/)  ->  ( f `  w )  e.  w
) )
8573vitalilem1 21047 . . . . . . . . . . . . . . . . . 18  |-  { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) }  Er  ( 0 [,] 1
)
8685a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( T. 
->  { <. a ,  b
>.  |  ( (
a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) }  Er  (
0 [,] 1 ) )
8786qsss 7157 . . . . . . . . . . . . . . . 16  |-  ( T. 
->  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1
) )  /\  (
a  -  b )  e.  QQ ) } )  C_  ~P (
0 [,] 1 ) )
8887trud 1373 . . . . . . . . . . . . . . 15  |-  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } ) 
C_  ~P ( 0 [,] 1 )
89 unitssre 11428 . . . . . . . . . . . . . . . 16  |-  ( 0 [,] 1 )  C_  RR
90 sspwb 4538 . . . . . . . . . . . . . . . 16  |-  ( ( 0 [,] 1 ) 
C_  RR  <->  ~P ( 0 [,] 1 )  C_  ~P RR )
9189, 90mpbi 208 . . . . . . . . . . . . . . 15  |-  ~P (
0 [,] 1 ) 
C_  ~P RR
9288, 91sstri 3362 . . . . . . . . . . . . . 14  |-  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } ) 
C_  ~P RR
93 ssralv 3413 . . . . . . . . . . . . . 14  |-  ( ( ( 0 [,] 1
) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } ) 
C_  ~P RR  ->  ( A. w  e.  ~P  RR ( w  =/=  (/)  ->  (
f `  w )  e.  w )  ->  A. w  e.  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1
) )  /\  (
a  -  b )  e.  QQ ) } ) ( w  =/=  (/)  ->  ( f `  w )  e.  w
) ) )
9492, 93ax-mp 5 . . . . . . . . . . . . 13  |-  ( A. w  e.  ~P  RR ( w  =/=  (/)  ->  (
f `  w )  e.  w )  ->  A. w  e.  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1
) )  /\  (
a  -  b )  e.  QQ ) } ) ( w  =/=  (/)  ->  ( f `  w )  e.  w
) )
9584, 94sylbi 195 . . . . . . . . . . . 12  |-  ( A. z  e.  ~P  RR ( z  =/=  (/)  ->  (
f `  z )  e.  z )  ->  A. w  e.  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1
) )  /\  (
a  -  b )  e.  QQ ) } ) ( w  =/=  (/)  ->  ( f `  w )  e.  w
) )
96 fveq2 5688 . . . . . . . . . . . . . . . 16  |-  ( c  =  w  ->  (
f `  c )  =  ( f `  w ) )
97 fvex 5698 . . . . . . . . . . . . . . . 16  |-  ( f `
 w )  e. 
_V
9896, 76, 97fvmpt 5771 . . . . . . . . . . . . . . 15  |-  ( w  e.  ( ( 0 [,] 1 ) /. { <. a ,  b
>.  |  ( (
a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } )  ->  (
( c  e.  ( ( 0 [,] 1
) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } ) 
|->  ( f `  c
) ) `  w
)  =  ( f `
 w ) )
9998eleq1d 2507 . . . . . . . . . . . . . 14  |-  ( w  e.  ( ( 0 [,] 1 ) /. { <. a ,  b
>.  |  ( (
a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } )  ->  (
( ( c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1
) )  /\  (
a  -  b )  e.  QQ ) } )  |->  ( f `  c ) ) `  w )  e.  w  <->  ( f `  w )  e.  w ) )
10099imbi2d 316 . . . . . . . . . . . . 13  |-  ( w  e.  ( ( 0 [,] 1 ) /. { <. a ,  b
>.  |  ( (
a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } )  ->  (
( w  =/=  (/)  ->  (
( c  e.  ( ( 0 [,] 1
) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } ) 
|->  ( f `  c
) ) `  w
)  e.  w )  <-> 
( w  =/=  (/)  ->  (
f `  w )  e.  w ) ) )
101100ralbiia 2745 . . . . . . . . . . . 12  |-  ( A. w  e.  ( (
0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } ) ( w  =/=  (/)  ->  (
( c  e.  ( ( 0 [,] 1
) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } ) 
|->  ( f `  c
) ) `  w
)  e.  w )  <->  A. w  e.  (
( 0 [,] 1
) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } ) ( w  =/=  (/)  ->  (
f `  w )  e.  w ) )
10295, 101sylibr 212 . . . . . . . . . . 11  |-  ( A. z  e.  ~P  RR ( z  =/=  (/)  ->  (
f `  z )  e.  z )  ->  A. w  e.  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1
) )  /\  (
a  -  b )  e.  QQ ) } ) ( w  =/=  (/)  ->  ( ( c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b
>.  |  ( (
a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } )  |->  ( f `
 c ) ) `
 w )  e.  w ) )
103102ad2antlr 721 . . . . . . . . . 10  |-  ( ( (  .<  We  RR  /\ 
A. z  e.  ~P  RR ( z  =/=  (/)  ->  (
f `  z )  e.  z ) )  /\  ( g : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) )  /\  -.  ran  ( c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1
) )  /\  (
a  -  b )  e.  QQ ) } )  |->  ( f `  c ) )  e.  ( ~P RR  \  dom  vol ) ) )  ->  A. w  e.  ( ( 0 [,] 1
) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } ) ( w  =/=  (/)  ->  (
( c  e.  ( ( 0 [,] 1
) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } ) 
|->  ( f `  c
) ) `  w
)  e.  w ) )
104 simprl 750 . . . . . . . . . 10  |-  ( ( (  .<  We  RR  /\ 
A. z  e.  ~P  RR ( z  =/=  (/)  ->  (
f `  z )  e.  z ) )  /\  ( g : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) )  /\  -.  ran  ( c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1
) )  /\  (
a  -  b )  e.  QQ ) } )  |->  ( f `  c ) )  e.  ( ~P RR  \  dom  vol ) ) )  ->  g : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) ) )
105 oveq1 6097 . . . . . . . . . . . . . 14  |-  ( t  =  s  ->  (
t  -  ( g `
 m ) )  =  ( s  -  ( g `  m
) ) )
106105eleq1d 2507 . . . . . . . . . . . . 13  |-  ( t  =  s  ->  (
( t  -  (
g `  m )
)  e.  ran  (
c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } ) 
|->  ( f `  c
) )  <->  ( s  -  ( g `  m ) )  e. 
ran  ( c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1
) )  /\  (
a  -  b )  e.  QQ ) } )  |->  ( f `  c ) ) ) )
107106cbvrabv 2969 . . . . . . . . . . . 12  |-  { t  e.  RR  |  ( t  -  ( g `
 m ) )  e.  ran  ( c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b
>.  |  ( (
a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } )  |->  ( f `
 c ) ) }  =  { s  e.  RR  |  ( s  -  ( g `
 m ) )  e.  ran  ( c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b
>.  |  ( (
a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } )  |->  ( f `
 c ) ) }
108 fveq2 5688 . . . . . . . . . . . . . . 15  |-  ( m  =  n  ->  (
g `  m )  =  ( g `  n ) )
109108oveq2d 6106 . . . . . . . . . . . . . 14  |-  ( m  =  n  ->  (
s  -  ( g `
 m ) )  =  ( s  -  ( g `  n
) ) )
110109eleq1d 2507 . . . . . . . . . . . . 13  |-  ( m  =  n  ->  (
( s  -  (
g `  m )
)  e.  ran  (
c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } ) 
|->  ( f `  c
) )  <->  ( s  -  ( g `  n ) )  e. 
ran  ( c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1
) )  /\  (
a  -  b )  e.  QQ ) } )  |->  ( f `  c ) ) ) )
111110rabbidv 2962 . . . . . . . . . . . 12  |-  ( m  =  n  ->  { s  e.  RR  |  ( s  -  ( g `
 m ) )  e.  ran  ( c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b
>.  |  ( (
a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } )  |->  ( f `
 c ) ) }  =  { s  e.  RR  |  ( s  -  ( g `
 n ) )  e.  ran  ( c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b
>.  |  ( (
a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } )  |->  ( f `
 c ) ) } )
112107, 111syl5eq 2485 . . . . . . . . . . 11  |-  ( m  =  n  ->  { t  e.  RR  |  ( t  -  ( g `
 m ) )  e.  ran  ( c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b
>.  |  ( (
a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } )  |->  ( f `
 c ) ) }  =  { s  e.  RR  |  ( s  -  ( g `
 n ) )  e.  ran  ( c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b
>.  |  ( (
a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } )  |->  ( f `
 c ) ) } )
113112cbvmptv 4380 . . . . . . . . . 10  |-  ( m  e.  NN  |->  { t  e.  RR  |  ( t  -  ( g `
 m ) )  e.  ran  ( c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b
>.  |  ( (
a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } )  |->  ( f `
 c ) ) } )  =  ( n  e.  NN  |->  { s  e.  RR  | 
( s  -  (
g `  n )
)  e.  ran  (
c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } ) 
|->  ( f `  c
) ) } )
114 simprr 751 . . . . . . . . . 10  |-  ( ( (  .<  We  RR  /\ 
A. z  e.  ~P  RR ( z  =/=  (/)  ->  (
f `  z )  e.  z ) )  /\  ( g : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) )  /\  -.  ran  ( c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1
) )  /\  (
a  -  b )  e.  QQ ) } )  |->  ( f `  c ) )  e.  ( ~P RR  \  dom  vol ) ) )  ->  -.  ran  ( c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b
>.  |  ( (
a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } )  |->  ( f `
 c ) )  e.  ( ~P RR  \  dom  vol ) )
11573, 74, 78, 103, 104, 113, 114vitalilem5 21051 . . . . . . . . 9  |-  -.  (
(  .<  We  RR  /\  A. z  e.  ~P  RR ( z  =/=  (/)  ->  (
f `  z )  e.  z ) )  /\  ( g : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) )  /\  -.  ran  ( c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1
) )  /\  (
a  -  b )  e.  QQ ) } )  |->  ( f `  c ) )  e.  ( ~P RR  \  dom  vol ) ) )
116115pm2.21i 131 . . . . . . . 8  |-  ( ( (  .<  We  RR  /\ 
A. z  e.  ~P  RR ( z  =/=  (/)  ->  (
f `  z )  e.  z ) )  /\  ( g : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) )  /\  -.  ran  ( c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1
) )  /\  (
a  -  b )  e.  QQ ) } )  |->  ( f `  c ) )  e.  ( ~P RR  \  dom  vol ) ) )  ->  ran  ( c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1
) )  /\  (
a  -  b )  e.  QQ ) } )  |->  ( f `  c ) )  e.  ( ~P RR  \  dom  vol ) )
117116expr 612 . . . . . . 7  |-  ( ( (  .<  We  RR  /\ 
A. z  e.  ~P  RR ( z  =/=  (/)  ->  (
f `  z )  e.  z ) )  /\  g : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) ) )  ->  ( -.  ran  ( c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1
) )  /\  (
a  -  b )  e.  QQ ) } )  |->  ( f `  c ) )  e.  ( ~P RR  \  dom  vol )  ->  ran  ( c  e.  ( ( 0 [,] 1
) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } ) 
|->  ( f `  c
) )  e.  ( ~P RR  \  dom  vol ) ) )
118117pm2.18d 111 . . . . . 6  |-  ( ( (  .<  We  RR  /\ 
A. z  e.  ~P  RR ( z  =/=  (/)  ->  (
f `  z )  e.  z ) )  /\  g : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) ) )  ->  ran  ( c  e.  ( ( 0 [,] 1
) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } ) 
|->  ( f `  c
) )  e.  ( ~P RR  \  dom  vol ) )
119 eldif 3335 . . . . . . 7  |-  ( ran  ( c  e.  ( ( 0 [,] 1
) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } ) 
|->  ( f `  c
) )  e.  ( ~P RR  \  dom  vol )  <->  ( ran  (
c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } ) 
|->  ( f `  c
) )  e.  ~P RR  /\  -.  ran  (
c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } ) 
|->  ( f `  c
) )  e.  dom  vol ) )
120 mblss 20973 . . . . . . . . . 10  |-  ( x  e.  dom  vol  ->  x 
C_  RR )
121 selpw 3864 . . . . . . . . . 10  |-  ( x  e.  ~P RR  <->  x  C_  RR )
122120, 121sylibr 212 . . . . . . . . 9  |-  ( x  e.  dom  vol  ->  x  e.  ~P RR )
123122ssriv 3357 . . . . . . . 8  |-  dom  vol  C_ 
~P RR
124 ssnelpss 3739 . . . . . . . 8  |-  ( dom 
vol  C_  ~P RR  ->  ( ( ran  ( c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b
>.  |  ( (
a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } )  |->  ( f `
 c ) )  e.  ~P RR  /\  -.  ran  ( c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1
) )  /\  (
a  -  b )  e.  QQ ) } )  |->  ( f `  c ) )  e. 
dom  vol )  ->  dom  vol  C.  ~P RR ) )
125123, 124ax-mp 5 . . . . . . 7  |-  ( ( ran  ( c  e.  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1
) )  /\  (
a  -  b )  e.  QQ ) } )  |->  ( f `  c ) )  e. 
~P RR  /\  -.  ran  ( c  e.  ( ( 0 [,] 1
) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } ) 
|->  ( f `  c
) )  e.  dom  vol )  ->  dom  vol  C.  ~P RR )
126119, 125sylbi 195 . . . . . 6  |-  ( ran  ( c  e.  ( ( 0 [,] 1
) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } ) 
|->  ( f `  c
) )  e.  ( ~P RR  \  dom  vol )  ->  dom  vol  C.  ~P RR )
127118, 126syl 16 . . . . 5  |-  ( ( (  .<  We  RR  /\ 
A. z  e.  ~P  RR ( z  =/=  (/)  ->  (
f `  z )  e.  z ) )  /\  g : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) ) )  ->  dom  vol  C.  ~P RR )
128127ex 434 . . . 4  |-  ( ( 
.<  We  RR  /\  A. z  e.  ~P  RR ( z  =/=  (/)  ->  (
f `  z )  e.  z ) )  -> 
( g : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) )  ->  dom  vol  C.  ~P RR ) )
129128exlimdv 1695 . . 3  |-  ( ( 
.<  We  RR  /\  A. z  e.  ~P  RR ( z  =/=  (/)  ->  (
f `  z )  e.  z ) )  -> 
( E. g  g : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) )  ->  dom  vol  C.  ~P RR ) )
13066, 129mpi 17 . 2  |-  ( ( 
.<  We  RR  /\  A. z  e.  ~P  RR ( z  =/=  (/)  ->  (
f `  z )  e.  z ) )  ->  dom  vol  C.  ~P RR )
13114, 130exlimddv 1697 1  |-  (  .<  We  RR  ->  dom  vol  C.  ~P RR )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364   T. wtru 1365   E.wex 1591    e. wcel 1761    =/= wne 2604   A.wral 2713   {crab 2717   _Vcvv 2970    \ cdif 3322    i^i cin 3324    C_ wss 3325    C. wpss 3326   (/)c0 3634   ~Pcpw 3857   U.cuni 4088   class class class wbr 4289   {copab 4346    e. cmpt 4347    We wwe 4674    X. cxp 4834   dom cdm 4836   ran crn 4837    Fn wfn 5410   -1-1-onto->wf1o 5414   ` cfv 5415  (class class class)co 6090    Er wer 7094   /.cqs 7096    ~~ cen 7303    ~<_ cdom 7304   RRcr 9277   0cc0 9278   1c1 9279    < clt 9414    <_ cle 9415    - cmin 9591   -ucneg 9592    / cdiv 9989   NNcn 10318   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-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cc 8600  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  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  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-int 4126  df-iun 4170  df-disj 4260  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-se 4676  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-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-omul 6921  df-er 7097  df-ec 7099  df-qs 7103  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-acn 8108  df-cda 8333  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-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-rlim 12963  df-sum 13160  df-rest 14357  df-topgen 14378  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-top 18462  df-bases 18464  df-topon 18465  df-cmp 18949  df-ovol 20907  df-vol 20908
This theorem is referenced by: (None)
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