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Theorem vitali 22583
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 9635 . . . 4  |-  RR  e.  _V
21pwex 4589 . . 3  |-  ~P RR  e.  _V
3 weinxp 4905 . . . . 5  |-  (  .<  We  RR  <->  (  .<  i^i  ( RR  X.  RR ) )  We  RR )
4 unipw 4653 . . . . . 6  |-  U. ~P RR  =  RR
5 weeq2 4826 . . . . . 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 256 . . . 4  |-  (  .<  We  RR  <->  (  .<  i^i  ( RR  X.  RR ) )  We  U. ~P RR )
81, 1xpex 6600 . . . . . 6  |-  ( RR 
X.  RR )  e. 
_V
98inex2 4548 . . . . 5  |-  (  .<  i^i  ( RR  X.  RR ) )  e.  _V
10 weeq1 4825 . . . . 5  |-  ( x  =  (  .<  i^i  ( RR  X.  RR ) )  ->  ( x  We 
U. ~P RR  <->  (  .<  i^i  ( RR  X.  RR ) )  We  U. ~P RR ) )
119, 10spcev 3143 . . . 4  |-  ( ( 
.<  i^i  ( RR  X.  RR ) )  We  U. ~P RR  ->  E. x  x  We  U. ~P RR )
127, 11sylbi 199 . . 3  |-  (  .<  We  RR  ->  E. x  x  We  U. ~P RR )
13 dfac8c 8469 . . 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 65 . 2  |-  (  .<  We  RR  ->  E. f A. z  e.  ~P  RR ( z  =/=  (/)  ->  (
f `  z )  e.  z ) )
15 qex 11283 . . . . . . 7  |-  QQ  e.  _V
1615inex1 4547 . . . . . 6  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  e. 
_V
17 nnrecq 11294 . . . . . . . 8  |-  ( x  e.  NN  ->  (
1  /  x )  e.  QQ )
18 nnrecre 10653 . . . . . . . . 9  |-  ( x  e.  NN  ->  (
1  /  x )  e.  RR )
19 neg1rr 10721 . . . . . . . . . . 11  |-  -u 1  e.  RR
2019a1i 11 . . . . . . . . . 10  |-  ( x  e.  NN  ->  -u 1  e.  RR )
21 0re 9648 . . . . . . . . . . 11  |-  0  e.  RR
2221a1i 11 . . . . . . . . . 10  |-  ( x  e.  NN  ->  0  e.  RR )
23 neg1lt0 10723 . . . . . . . . . . . 12  |-  -u 1  <  0
2419, 21, 23ltleii 9762 . . . . . . . . . . 11  |-  -u 1  <_  0
2524a1i 11 . . . . . . . . . 10  |-  ( x  e.  NN  ->  -u 1  <_  0 )
26 nnrp 11318 . . . . . . . . . . . 12  |-  ( x  e.  NN  ->  x  e.  RR+ )
2726rpreccld 11358 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  (
1  /  x )  e.  RR+ )
2827rpge0d 11352 . . . . . . . . . 10  |-  ( x  e.  NN  ->  0  <_  ( 1  /  x
) )
2920, 22, 18, 25, 28letrd 9797 . . . . . . . . 9  |-  ( x  e.  NN  ->  -u 1  <_  ( 1  /  x
) )
30 nnge1 10642 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  1  <_  x )
31 nnre 10623 . . . . . . . . . . . 12  |-  ( x  e.  NN  ->  x  e.  RR )
32 nngt0 10645 . . . . . . . . . . . 12  |-  ( x  e.  NN  ->  0  <  x )
33 1re 9647 . . . . . . . . . . . . 13  |-  1  e.  RR
34 0lt1 10143 . . . . . . . . . . . . 13  |-  0  <  1
35 lerec 10496 . . . . . . . . . . . . 13  |-  ( ( ( 1  e.  RR  /\  0  <  1 )  /\  ( x  e.  RR  /\  0  < 
x ) )  -> 
( 1  <_  x  <->  ( 1  /  x )  <_  ( 1  / 
1 ) ) )
3633, 34, 35mpanl12 689 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  0  <  x )  -> 
( 1  <_  x  <->  ( 1  /  x )  <_  ( 1  / 
1 ) ) )
3731, 32, 36syl2anc 667 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  (
1  <_  x  <->  ( 1  /  x )  <_ 
( 1  /  1
) ) )
3830, 37mpbid 214 . . . . . . . . . 10  |-  ( x  e.  NN  ->  (
1  /  x )  <_  ( 1  / 
1 ) )
39 1div1e1 10307 . . . . . . . . . 10  |-  ( 1  /  1 )  =  1
4038, 39syl6breq 4445 . . . . . . . . 9  |-  ( x  e.  NN  ->  (
1  /  x )  <_  1 )
4119, 33elicc2i 11707 . . . . . . . . 9  |-  ( ( 1  /  x )  e.  ( -u 1 [,] 1 )  <->  ( (
1  /  x )  e.  RR  /\  -u 1  <_  ( 1  /  x
)  /\  ( 1  /  x )  <_ 
1 ) )
4218, 29, 40, 41syl3anbrc 1193 . . . . . . . 8  |-  ( x  e.  NN  ->  (
1  /  x )  e.  ( -u 1 [,] 1 ) )
4317, 42elind 3620 . . . . . . 7  |-  ( x  e.  NN  ->  (
1  /  x )  e.  ( QQ  i^i  ( -u 1 [,] 1
) ) )
44 oveq2 6303 . . . . . . . . 9  |-  ( ( 1  /  x )  =  ( 1  / 
y )  ->  (
1  /  ( 1  /  x ) )  =  ( 1  / 
( 1  /  y
) ) )
45 nncn 10624 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  x  e.  CC )
46 nnne0 10649 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  x  =/=  0 )
4745, 46recrecd 10387 . . . . . . . . . 10  |-  ( x  e.  NN  ->  (
1  /  ( 1  /  x ) )  =  x )
48 nncn 10624 . . . . . . . . . . 11  |-  ( y  e.  NN  ->  y  e.  CC )
49 nnne0 10649 . . . . . . . . . . 11  |-  ( y  e.  NN  ->  y  =/=  0 )
5048, 49recrecd 10387 . . . . . . . . . 10  |-  ( y  e.  NN  ->  (
1  /  ( 1  /  y ) )  =  y )
5147, 50eqeqan12d 2469 . . . . . . . . 9  |-  ( ( x  e.  NN  /\  y  e.  NN )  ->  ( ( 1  / 
( 1  /  x
) )  =  ( 1  /  ( 1  /  y ) )  <-> 
x  =  y ) )
5244, 51syl5ib 223 . . . . . . . 8  |-  ( ( x  e.  NN  /\  y  e.  NN )  ->  ( ( 1  /  x )  =  ( 1  /  y )  ->  x  =  y ) )
53 oveq2 6303 . . . . . . . 8  |-  ( x  =  y  ->  (
1  /  x )  =  ( 1  / 
y ) )
5452, 53impbid1 207 . . . . . . 7  |-  ( ( x  e.  NN  /\  y  e.  NN )  ->  ( ( 1  /  x )  =  ( 1  /  y )  <-> 
x  =  y ) )
5543, 54dom2 7617 . . . . . 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 3654 . . . . . . 7  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  QQ
58 ssdomg 7620 . . . . . . 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 14278 . . . . . 6  |-  QQ  ~~  NN
61 domentr 7633 . . . . . 6  |-  ( ( ( QQ  i^i  ( -u 1 [,] 1 ) )  ~<_  QQ  /\  QQ  ~~  NN )  ->  ( QQ 
i^i  ( -u 1 [,] 1 ) )  ~<_  NN )
6259, 60, 61mp2an 679 . . . . 5  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  ~<_  NN
63 sbth 7697 . . . . 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 679 . . . 4  |-  NN  ~~  ( QQ  i^i  ( -u 1 [,] 1 ) )
65 bren 7583 . . . 4  |-  ( NN 
~~  ( QQ  i^i  ( -u 1 [,] 1
) )  <->  E. g 
g : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) ) )
6664, 65mpbi 212 . . 3  |-  E. g 
g : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) )
67 eleq1 2519 . . . . . . . . . . . . 13  |-  ( a  =  x  ->  (
a  e.  ( 0 [,] 1 )  <->  x  e.  ( 0 [,] 1
) ) )
68 eleq1 2519 . . . . . . . . . . . . 13  |-  ( b  =  y  ->  (
b  e.  ( 0 [,] 1 )  <->  y  e.  ( 0 [,] 1
) ) )
6967, 68bi2anan9 885 . . . . . . . . . . . 12  |-  ( ( a  =  x  /\  b  =  y )  ->  ( ( a  e.  ( 0 [,] 1
)  /\  b  e.  ( 0 [,] 1
) )  <->  ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) ) ) )
70 oveq12 6304 . . . . . . . . . . . . 13  |-  ( ( a  =  x  /\  b  =  y )  ->  ( a  -  b
)  =  ( x  -  y ) )
7170eleq1d 2515 . . . . . . . . . . . 12  |-  ( ( a  =  x  /\  b  =  y )  ->  ( ( a  -  b )  e.  QQ  <->  ( x  -  y )  e.  QQ ) )
7269, 71anbi12d 718 . . . . . . . . . . 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 4475 . . . . . . . . . 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 2453 . . . . . . . . . 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 5880 . . . . . . . . . . . 12  |-  ( f `
 c )  e. 
_V
76 eqid 2453 . . . . . . . . . . . 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 5711 . . . . . . . . . . 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 2688 . . . . . . . . . . . . . . 15  |-  ( z  =  w  ->  (
z  =/=  (/)  <->  w  =/=  (/) ) )
80 fveq2 5870 . . . . . . . . . . . . . . . 16  |-  ( z  =  w  ->  (
f `  z )  =  ( f `  w ) )
81 id 22 . . . . . . . . . . . . . . . 16  |-  ( z  =  w  ->  z  =  w )
8280, 81eleq12d 2525 . . . . . . . . . . . . . . 15  |-  ( z  =  w  ->  (
( f `  z
)  e.  z  <->  ( f `  w )  e.  w
) )
8379, 82imbi12d 322 . . . . . . . . . . . . . 14  |-  ( z  =  w  ->  (
( z  =/=  (/)  ->  (
f `  z )  e.  z )  <->  ( w  =/=  (/)  ->  ( f `  w )  e.  w
) ) )
8483cbvralv 3021 . . . . . . . . . . . . 13  |-  ( A. z  e.  ~P  RR ( z  =/=  (/)  ->  (
f `  z )  e.  z )  <->  A. w  e.  ~P  RR ( w  =/=  (/)  ->  ( f `  w )  e.  w
) )
8573vitalilem1 22578 . . . . . . . . . . . . . . . . . 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 7429 . . . . . . . . . . . . . . . 16  |-  ( T. 
->  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1
) )  /\  (
a  -  b )  e.  QQ ) } )  C_  ~P (
0 [,] 1 ) )
8887trud 1455 . . . . . . . . . . . . . . 15  |-  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } ) 
C_  ~P ( 0 [,] 1 )
89 unitssre 11786 . . . . . . . . . . . . . . . 16  |-  ( 0 [,] 1 )  C_  RR
90 sspwb 4652 . . . . . . . . . . . . . . . 16  |-  ( ( 0 [,] 1 ) 
C_  RR  <->  ~P ( 0 [,] 1 )  C_  ~P RR )
9189, 90mpbi 212 . . . . . . . . . . . . . . 15  |-  ~P (
0 [,] 1 ) 
C_  ~P RR
9288, 91sstri 3443 . . . . . . . . . . . . . 14  |-  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } ) 
C_  ~P RR
93 ssralv 3495 . . . . . . . . . . . . . 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 199 . . . . . . . . . . . 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 5870 . . . . . . . . . . . . . . . 16  |-  ( c  =  w  ->  (
f `  c )  =  ( f `  w ) )
97 fvex 5880 . . . . . . . . . . . . . . . 16  |-  ( f `
 w )  e. 
_V
9896, 76, 97fvmpt 5953 . . . . . . . . . . . . . . 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 2515 . . . . . . . . . . . . . 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 318 . . . . . . . . . . . . 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 2820 . . . . . . . . . . . 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 216 . . . . . . . . . . 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 734 . . . . . . . . . 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 765 . . . . . . . . . 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 6302 . . . . . . . . . . . . . 14  |-  ( t  =  s  ->  (
t  -  ( g `
 m ) )  =  ( s  -  ( g `  m
) ) )
106105eleq1d 2515 . . . . . . . . . . . . 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 3046 . . . . . . . . . . . 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 5870 . . . . . . . . . . . . . . 15  |-  ( m  =  n  ->  (
g `  m )  =  ( g `  n ) )
109108oveq2d 6311 . . . . . . . . . . . . . 14  |-  ( m  =  n  ->  (
s  -  ( g `
 m ) )  =  ( s  -  ( g `  n
) ) )
110109eleq1d 2515 . . . . . . . . . . . . 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 3038 . . . . . . . . . . . 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 2499 . . . . . . . . . . 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 4498 . . . . . . . . . 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 767 . . . . . . . . . 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 22582 . . . . . . . . 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 135 . . . . . . . 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 620 . . . . . . 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 3416 . . . . . . 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 22497 . . . . . . . . . 10  |-  ( x  e.  dom  vol  ->  x 
C_  RR )
121 selpw 3960 . . . . . . . . . 10  |-  ( x  e.  ~P RR  <->  x  C_  RR )
122120, 121sylibr 216 . . . . . . . . 9  |-  ( x  e.  dom  vol  ->  x  e.  ~P RR )
123122ssriv 3438 . . . . . . . 8  |-  dom  vol  C_ 
~P RR
124 ssnelpss 3831 . . . . . . . 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 199 . . . . . 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 436 . . . 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 1781 . . 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 1783 1  |-  (  .<  We  RR  ->  dom  vol  C.  ~P RR )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1446   T. wtru 1447   E.wex 1665    e. wcel 1889    =/= wne 2624   A.wral 2739   {crab 2743   _Vcvv 3047    \ cdif 3403    i^i cin 3405    C_ wss 3406    C. wpss 3407   (/)c0 3733   ~Pcpw 3953   U.cuni 4201   class class class wbr 4405   {copab 4463    |-> cmpt 4464    We wwe 4795    X. cxp 4835   dom cdm 4837   ran crn 4838    Fn wfn 5580   -1-1-onto->wf1o 5584   ` cfv 5585  (class class class)co 6295    Er wer 7365   /.cqs 7367    ~~ cen 7571    ~<_ cdom 7572   RRcr 9543   0cc0 9544   1c1 9545    < clt 9680    <_ cle 9681    - cmin 9865   -ucneg 9866    / cdiv 10276   NNcn 10616   QQcq 11271   [,]cicc 11645   volcvol 22427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cc 8870  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-disj 4377  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-omul 7192  df-er 7368  df-ec 7370  df-qs 7374  df-map 7479  df-pm 7480  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fi 7930  df-sup 7961  df-inf 7962  df-oi 8030  df-card 8378  df-acn 8381  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-q 11272  df-rp 11310  df-xneg 11416  df-xadd 11417  df-xmul 11418  df-ioo 11646  df-ico 11648  df-icc 11649  df-fz 11792  df-fzo 11923  df-fl 12035  df-seq 12221  df-exp 12280  df-hash 12523  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-clim 13564  df-rlim 13565  df-sum 13765  df-rest 15333  df-topgen 15354  df-psmet 18974  df-xmet 18975  df-met 18976  df-bl 18977  df-mopn 18978  df-top 19933  df-bases 19934  df-topon 19935  df-cmp 20414  df-ovol 22428  df-vol 22430
This theorem is referenced by: (None)
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