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Theorem vitali 22650
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 9648 . . . 4  |-  RR  e.  _V
21pwex 4584 . . 3  |-  ~P RR  e.  _V
3 weinxp 4907 . . . . 5  |-  (  .<  We  RR  <->  (  .<  i^i  ( RR  X.  RR ) )  We  RR )
4 unipw 4650 . . . . . 6  |-  U. ~P RR  =  RR
5 weeq2 4828 . . . . . 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 260 . . . 4  |-  (  .<  We  RR  <->  (  .<  i^i  ( RR  X.  RR ) )  We  U. ~P RR )
81, 1xpex 6614 . . . . . 6  |-  ( RR 
X.  RR )  e. 
_V
98inex2 4538 . . . . 5  |-  (  .<  i^i  ( RR  X.  RR ) )  e.  _V
10 weeq1 4827 . . . . 5  |-  ( x  =  (  .<  i^i  ( RR  X.  RR ) )  ->  ( x  We 
U. ~P RR  <->  (  .<  i^i  ( RR  X.  RR ) )  We  U. ~P RR ) )
119, 10spcev 3127 . . . 4  |-  ( ( 
.<  i^i  ( RR  X.  RR ) )  We  U. ~P RR  ->  E. x  x  We  U. ~P RR )
127, 11sylbi 200 . . 3  |-  (  .<  We  RR  ->  E. x  x  We  U. ~P RR )
13 dfac8c 8482 . . 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 64 . 2  |-  (  .<  We  RR  ->  E. f A. z  e.  ~P  RR ( z  =/=  (/)  ->  (
f `  z )  e.  z ) )
15 qex 11299 . . . . . . 7  |-  QQ  e.  _V
1615inex1 4537 . . . . . 6  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  e. 
_V
17 nnrecq 11310 . . . . . . . 8  |-  ( x  e.  NN  ->  (
1  /  x )  e.  QQ )
18 nnrecre 10668 . . . . . . . . 9  |-  ( x  e.  NN  ->  (
1  /  x )  e.  RR )
19 neg1rr 10736 . . . . . . . . . . 11  |-  -u 1  e.  RR
2019a1i 11 . . . . . . . . . 10  |-  ( x  e.  NN  ->  -u 1  e.  RR )
21 0re 9661 . . . . . . . . . . 11  |-  0  e.  RR
2221a1i 11 . . . . . . . . . 10  |-  ( x  e.  NN  ->  0  e.  RR )
23 neg1lt0 10738 . . . . . . . . . . . 12  |-  -u 1  <  0
2419, 21, 23ltleii 9775 . . . . . . . . . . 11  |-  -u 1  <_  0
2524a1i 11 . . . . . . . . . 10  |-  ( x  e.  NN  ->  -u 1  <_  0 )
26 nnrp 11334 . . . . . . . . . . . 12  |-  ( x  e.  NN  ->  x  e.  RR+ )
2726rpreccld 11374 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  (
1  /  x )  e.  RR+ )
2827rpge0d 11368 . . . . . . . . . 10  |-  ( x  e.  NN  ->  0  <_  ( 1  /  x
) )
2920, 22, 18, 25, 28letrd 9809 . . . . . . . . 9  |-  ( x  e.  NN  ->  -u 1  <_  ( 1  /  x
) )
30 nnge1 10657 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  1  <_  x )
31 nnre 10638 . . . . . . . . . . . 12  |-  ( x  e.  NN  ->  x  e.  RR )
32 nngt0 10660 . . . . . . . . . . . 12  |-  ( x  e.  NN  ->  0  <  x )
33 1re 9660 . . . . . . . . . . . . 13  |-  1  e.  RR
34 0lt1 10157 . . . . . . . . . . . . 13  |-  0  <  1
35 lerec 10511 . . . . . . . . . . . . 13  |-  ( ( ( 1  e.  RR  /\  0  <  1 )  /\  ( x  e.  RR  /\  0  < 
x ) )  -> 
( 1  <_  x  <->  ( 1  /  x )  <_  ( 1  / 
1 ) ) )
3633, 34, 35mpanl12 696 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  0  <  x )  -> 
( 1  <_  x  <->  ( 1  /  x )  <_  ( 1  / 
1 ) ) )
3731, 32, 36syl2anc 673 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  (
1  <_  x  <->  ( 1  /  x )  <_ 
( 1  /  1
) ) )
3830, 37mpbid 215 . . . . . . . . . 10  |-  ( x  e.  NN  ->  (
1  /  x )  <_  ( 1  / 
1 ) )
39 1div1e1 10322 . . . . . . . . . 10  |-  ( 1  /  1 )  =  1
4038, 39syl6breq 4435 . . . . . . . . 9  |-  ( x  e.  NN  ->  (
1  /  x )  <_  1 )
4119, 33elicc2i 11725 . . . . . . . . 9  |-  ( ( 1  /  x )  e.  ( -u 1 [,] 1 )  <->  ( (
1  /  x )  e.  RR  /\  -u 1  <_  ( 1  /  x
)  /\  ( 1  /  x )  <_ 
1 ) )
4218, 29, 40, 41syl3anbrc 1214 . . . . . . . 8  |-  ( x  e.  NN  ->  (
1  /  x )  e.  ( -u 1 [,] 1 ) )
4317, 42elind 3609 . . . . . . 7  |-  ( x  e.  NN  ->  (
1  /  x )  e.  ( QQ  i^i  ( -u 1 [,] 1
) ) )
44 oveq2 6316 . . . . . . . . 9  |-  ( ( 1  /  x )  =  ( 1  / 
y )  ->  (
1  /  ( 1  /  x ) )  =  ( 1  / 
( 1  /  y
) ) )
45 nncn 10639 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  x  e.  CC )
46 nnne0 10664 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  x  =/=  0 )
4745, 46recrecd 10402 . . . . . . . . . 10  |-  ( x  e.  NN  ->  (
1  /  ( 1  /  x ) )  =  x )
48 nncn 10639 . . . . . . . . . . 11  |-  ( y  e.  NN  ->  y  e.  CC )
49 nnne0 10664 . . . . . . . . . . 11  |-  ( y  e.  NN  ->  y  =/=  0 )
5048, 49recrecd 10402 . . . . . . . . . 10  |-  ( y  e.  NN  ->  (
1  /  ( 1  /  y ) )  =  y )
5147, 50eqeqan12d 2487 . . . . . . . . 9  |-  ( ( x  e.  NN  /\  y  e.  NN )  ->  ( ( 1  / 
( 1  /  x
) )  =  ( 1  /  ( 1  /  y ) )  <-> 
x  =  y ) )
5244, 51syl5ib 227 . . . . . . . 8  |-  ( ( x  e.  NN  /\  y  e.  NN )  ->  ( ( 1  /  x )  =  ( 1  /  y )  ->  x  =  y ) )
53 oveq2 6316 . . . . . . . 8  |-  ( x  =  y  ->  (
1  /  x )  =  ( 1  / 
y ) )
5452, 53impbid1 208 . . . . . . 7  |-  ( ( x  e.  NN  /\  y  e.  NN )  ->  ( ( 1  /  x )  =  ( 1  /  y )  <-> 
x  =  y ) )
5543, 54dom2 7630 . . . . . 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 3643 . . . . . . 7  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  QQ
58 ssdomg 7633 . . . . . . 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 14343 . . . . . 6  |-  QQ  ~~  NN
61 domentr 7646 . . . . . 6  |-  ( ( ( QQ  i^i  ( -u 1 [,] 1 ) )  ~<_  QQ  /\  QQ  ~~  NN )  ->  ( QQ 
i^i  ( -u 1 [,] 1 ) )  ~<_  NN )
6259, 60, 61mp2an 686 . . . . 5  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  ~<_  NN
63 sbth 7710 . . . . 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 686 . . . 4  |-  NN  ~~  ( QQ  i^i  ( -u 1 [,] 1 ) )
65 bren 7596 . . . 4  |-  ( NN 
~~  ( QQ  i^i  ( -u 1 [,] 1
) )  <->  E. g 
g : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) ) )
6664, 65mpbi 213 . . 3  |-  E. g 
g : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) )
67 eleq1 2537 . . . . . . . . . . . . 13  |-  ( a  =  x  ->  (
a  e.  ( 0 [,] 1 )  <->  x  e.  ( 0 [,] 1
) ) )
68 eleq1 2537 . . . . . . . . . . . . 13  |-  ( b  =  y  ->  (
b  e.  ( 0 [,] 1 )  <->  y  e.  ( 0 [,] 1
) ) )
6967, 68bi2anan9 890 . . . . . . . . . . . 12  |-  ( ( a  =  x  /\  b  =  y )  ->  ( ( a  e.  ( 0 [,] 1
)  /\  b  e.  ( 0 [,] 1
) )  <->  ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) ) ) )
70 oveq12 6317 . . . . . . . . . . . . 13  |-  ( ( a  =  x  /\  b  =  y )  ->  ( a  -  b
)  =  ( x  -  y ) )
7170eleq1d 2533 . . . . . . . . . . . 12  |-  ( ( a  =  x  /\  b  =  y )  ->  ( ( a  -  b )  e.  QQ  <->  ( x  -  y )  e.  QQ ) )
7269, 71anbi12d 725 . . . . . . . . . . 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 4465 . . . . . . . . . 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 2471 . . . . . . . . . 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 5889 . . . . . . . . . . . 12  |-  ( f `
 c )  e. 
_V
76 eqid 2471 . . . . . . . . . . . 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 5716 . . . . . . . . . . 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 2705 . . . . . . . . . . . . . . 15  |-  ( z  =  w  ->  (
z  =/=  (/)  <->  w  =/=  (/) ) )
80 fveq2 5879 . . . . . . . . . . . . . . . 16  |-  ( z  =  w  ->  (
f `  z )  =  ( f `  w ) )
81 id 22 . . . . . . . . . . . . . . . 16  |-  ( z  =  w  ->  z  =  w )
8280, 81eleq12d 2543 . . . . . . . . . . . . . . 15  |-  ( z  =  w  ->  (
( f `  z
)  e.  z  <->  ( f `  w )  e.  w
) )
8379, 82imbi12d 327 . . . . . . . . . . . . . 14  |-  ( z  =  w  ->  (
( z  =/=  (/)  ->  (
f `  z )  e.  z )  <->  ( w  =/=  (/)  ->  ( f `  w )  e.  w
) ) )
8483cbvralv 3005 . . . . . . . . . . . . 13  |-  ( A. z  e.  ~P  RR ( z  =/=  (/)  ->  (
f `  z )  e.  z )  <->  A. w  e.  ~P  RR ( w  =/=  (/)  ->  ( f `  w )  e.  w
) )
8573vitalilem1 22645 . . . . . . . . . . . . . . . . . 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 7442 . . . . . . . . . . . . . . . 16  |-  ( T. 
->  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1
) )  /\  (
a  -  b )  e.  QQ ) } )  C_  ~P (
0 [,] 1 ) )
8887trud 1461 . . . . . . . . . . . . . . 15  |-  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } ) 
C_  ~P ( 0 [,] 1 )
89 unitssre 11805 . . . . . . . . . . . . . . . 16  |-  ( 0 [,] 1 )  C_  RR
90 sspwb 4649 . . . . . . . . . . . . . . . 16  |-  ( ( 0 [,] 1 ) 
C_  RR  <->  ~P ( 0 [,] 1 )  C_  ~P RR )
9189, 90mpbi 213 . . . . . . . . . . . . . . 15  |-  ~P (
0 [,] 1 ) 
C_  ~P RR
9288, 91sstri 3427 . . . . . . . . . . . . . 14  |-  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } ) 
C_  ~P RR
93 ssralv 3479 . . . . . . . . . . . . . 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 200 . . . . . . . . . . . 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 5879 . . . . . . . . . . . . . . . 16  |-  ( c  =  w  ->  (
f `  c )  =  ( f `  w ) )
97 fvex 5889 . . . . . . . . . . . . . . . 16  |-  ( f `
 w )  e. 
_V
9896, 76, 97fvmpt 5963 . . . . . . . . . . . . . . 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 2533 . . . . . . . . . . . . . 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 323 . . . . . . . . . . . . 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 2822 . . . . . . . . . . . 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 217 . . . . . . . . . . 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 741 . . . . . . . . . 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 772 . . . . . . . . . 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 6315 . . . . . . . . . . . . . 14  |-  ( t  =  s  ->  (
t  -  ( g `
 m ) )  =  ( s  -  ( g `  m
) ) )
106105eleq1d 2533 . . . . . . . . . . . . 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 3030 . . . . . . . . . . . 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 5879 . . . . . . . . . . . . . . 15  |-  ( m  =  n  ->  (
g `  m )  =  ( g `  n ) )
109108oveq2d 6324 . . . . . . . . . . . . . 14  |-  ( m  =  n  ->  (
s  -  ( g `
 m ) )  =  ( s  -  ( g `  n
) ) )
110109eleq1d 2533 . . . . . . . . . . . . 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 3022 . . . . . . . . . . . 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 2517 . . . . . . . . . . 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 4488 . . . . . . . . . 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 774 . . . . . . . . . 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 22649 . . . . . . . . 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 136 . . . . . . . 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 626 . . . . . . 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 115 . . . . . 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 3400 . . . . . . 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 22563 . . . . . . . . . 10  |-  ( x  e.  dom  vol  ->  x 
C_  RR )
121 selpw 3949 . . . . . . . . . 10  |-  ( x  e.  ~P RR  <->  x  C_  RR )
122120, 121sylibr 217 . . . . . . . . 9  |-  ( x  e.  dom  vol  ->  x  e.  ~P RR )
123122ssriv 3422 . . . . . . . 8  |-  dom  vol  C_ 
~P RR
124 ssnelpss 3530 . . . . . . . 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 200 . . . . . 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 17 . . . . 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 441 . . . 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 1787 . . 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 20 . 2  |-  ( ( 
.<  We  RR  /\  A. z  e.  ~P  RR ( z  =/=  (/)  ->  (
f `  z )  e.  z ) )  ->  dom  vol  C.  ~P RR )
13114, 130exlimddv 1789 1  |-  (  .<  We  RR  ->  dom  vol  C.  ~P RR )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452   T. wtru 1453   E.wex 1671    e. wcel 1904    =/= wne 2641   A.wral 2756   {crab 2760   _Vcvv 3031    \ cdif 3387    i^i cin 3389    C_ wss 3390    C. wpss 3391   (/)c0 3722   ~Pcpw 3942   U.cuni 4190   class class class wbr 4395   {copab 4453    |-> cmpt 4454    We wwe 4797    X. cxp 4837   dom cdm 4839   ran crn 4840    Fn wfn 5584   -1-1-onto->wf1o 5588   ` cfv 5589  (class class class)co 6308    Er wer 7378   /.cqs 7380    ~~ cen 7584    ~<_ cdom 7585   RRcr 9556   0cc0 9557   1c1 9558    < clt 9693    <_ cle 9694    - cmin 9880   -ucneg 9881    / cdiv 10291   NNcn 10631   QQcq 11287   [,]cicc 11663   volcvol 22493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cc 8883  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-disj 4367  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-omul 7205  df-er 7381  df-ec 7383  df-qs 7387  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-acn 8394  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-rlim 13630  df-sum 13830  df-rest 15399  df-topgen 15420  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-top 19998  df-bases 19999  df-topon 20000  df-cmp 20479  df-ovol 22494  df-vol 22496
This theorem is referenced by: (None)
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