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Theorem vitali 19497
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 9073 . . . 4  |-  RR  e.  _V
21pwex 4374 . . 3  |-  ~P RR  e.  _V
3 weinxp 4937 . . . . 5  |-  (  .<  We  RR  <->  (  .<  i^i  ( RR  X.  RR ) )  We  RR )
4 unipw 4406 . . . . . 6  |-  U. ~P RR  =  RR
5 weeq2 4563 . . . . . 6  |-  ( U. ~P RR  =  RR  ->  ( (  .<  i^i  ( RR  X.  RR ) )  We  U. ~P RR  <->  ( 
.<  i^i  ( RR  X.  RR ) )  We  RR ) )
64, 5ax-mp 8 . . . . 5  |-  ( ( 
.<  i^i  ( RR  X.  RR ) )  We  U. ~P RR  <->  (  .<  i^i  ( RR  X.  RR ) )  We  RR )
73, 6bitr4i 244 . . . 4  |-  (  .<  We  RR  <->  (  .<  i^i  ( RR  X.  RR ) )  We  U. ~P RR )
81, 1xpex 4982 . . . . . 6  |-  ( RR 
X.  RR )  e. 
_V
98inex2 4337 . . . . 5  |-  (  .<  i^i  ( RR  X.  RR ) )  e.  _V
10 weeq1 4562 . . . . 5  |-  ( x  =  (  .<  i^i  ( RR  X.  RR ) )  ->  ( x  We 
U. ~P RR  <->  (  .<  i^i  ( RR  X.  RR ) )  We  U. ~P RR ) )
119, 10spcev 3035 . . . 4  |-  ( ( 
.<  i^i  ( RR  X.  RR ) )  We  U. ~P RR  ->  E. x  x  We  U. ~P RR )
127, 11sylbi 188 . . 3  |-  (  .<  We  RR  ->  E. x  x  We  U. ~P RR )
13 dfac8c 7906 . . 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 61 . 2  |-  (  .<  We  RR  ->  E. f A. z  e.  ~P  RR ( z  =/=  (/)  ->  (
f `  z )  e.  z ) )
15 qex 10578 . . . . . . 7  |-  QQ  e.  _V
1615inex1 4336 . . . . . 6  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  e. 
_V
17 nnrecq 10589 . . . . . . . 8  |-  ( x  e.  NN  ->  (
1  /  x )  e.  QQ )
18 nnrecre 10028 . . . . . . . . 9  |-  ( x  e.  NN  ->  (
1  /  x )  e.  RR )
19 1re 9082 . . . . . . . . . . . 12  |-  1  e.  RR
2019renegcli 9354 . . . . . . . . . . 11  |-  -u 1  e.  RR
2120a1i 11 . . . . . . . . . 10  |-  ( x  e.  NN  ->  -u 1  e.  RR )
22 0re 9083 . . . . . . . . . . 11  |-  0  e.  RR
2322a1i 11 . . . . . . . . . 10  |-  ( x  e.  NN  ->  0  e.  RR )
24 0lt1 9542 . . . . . . . . . . . . 13  |-  0  <  1
25 lt0neg2 9527 . . . . . . . . . . . . . 14  |-  ( 1  e.  RR  ->  (
0  <  1  <->  -u 1  <  0 ) )
2619, 25ax-mp 8 . . . . . . . . . . . . 13  |-  ( 0  <  1  <->  -u 1  <  0 )
2724, 26mpbi 200 . . . . . . . . . . . 12  |-  -u 1  <  0
2820, 22, 27ltleii 9188 . . . . . . . . . . 11  |-  -u 1  <_  0
2928a1i 11 . . . . . . . . . 10  |-  ( x  e.  NN  ->  -u 1  <_  0 )
30 nnrp 10613 . . . . . . . . . . . 12  |-  ( x  e.  NN  ->  x  e.  RR+ )
3130rpreccld 10650 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  (
1  /  x )  e.  RR+ )
3231rpge0d 10644 . . . . . . . . . 10  |-  ( x  e.  NN  ->  0  <_  ( 1  /  x
) )
3321, 23, 18, 29, 32letrd 9219 . . . . . . . . 9  |-  ( x  e.  NN  ->  -u 1  <_  ( 1  /  x
) )
34 nnge1 10018 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  1  <_  x )
35 nnre 9999 . . . . . . . . . . . 12  |-  ( x  e.  NN  ->  x  e.  RR )
36 nngt0 10021 . . . . . . . . . . . 12  |-  ( x  e.  NN  ->  0  <  x )
37 lerec 9884 . . . . . . . . . . . . 13  |-  ( ( ( 1  e.  RR  /\  0  <  1 )  /\  ( x  e.  RR  /\  0  < 
x ) )  -> 
( 1  <_  x  <->  ( 1  /  x )  <_  ( 1  / 
1 ) ) )
3819, 24, 37mpanl12 664 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  0  <  x )  -> 
( 1  <_  x  <->  ( 1  /  x )  <_  ( 1  / 
1 ) ) )
3935, 36, 38syl2anc 643 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  (
1  <_  x  <->  ( 1  /  x )  <_ 
( 1  /  1
) ) )
4034, 39mpbid 202 . . . . . . . . . 10  |-  ( x  e.  NN  ->  (
1  /  x )  <_  ( 1  / 
1 ) )
41 ax-1cn 9040 . . . . . . . . . . 11  |-  1  e.  CC
4241div1i 9734 . . . . . . . . . 10  |-  ( 1  /  1 )  =  1
4340, 42syl6breq 4243 . . . . . . . . 9  |-  ( x  e.  NN  ->  (
1  /  x )  <_  1 )
4420, 19elicc2i 10968 . . . . . . . . 9  |-  ( ( 1  /  x )  e.  ( -u 1 [,] 1 )  <->  ( (
1  /  x )  e.  RR  /\  -u 1  <_  ( 1  /  x
)  /\  ( 1  /  x )  <_ 
1 ) )
4518, 33, 43, 44syl3anbrc 1138 . . . . . . . 8  |-  ( x  e.  NN  ->  (
1  /  x )  e.  ( -u 1 [,] 1 ) )
46 elin 3522 . . . . . . . 8  |-  ( ( 1  /  x )  e.  ( QQ  i^i  ( -u 1 [,] 1
) )  <->  ( (
1  /  x )  e.  QQ  /\  (
1  /  x )  e.  ( -u 1 [,] 1 ) ) )
4717, 45, 46sylanbrc 646 . . . . . . 7  |-  ( x  e.  NN  ->  (
1  /  x )  e.  ( QQ  i^i  ( -u 1 [,] 1
) ) )
48 oveq2 6081 . . . . . . . . 9  |-  ( ( 1  /  x )  =  ( 1  / 
y )  ->  (
1  /  ( 1  /  x ) )  =  ( 1  / 
( 1  /  y
) ) )
49 nncn 10000 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  x  e.  CC )
50 nnne0 10024 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  x  =/=  0 )
5149, 50recrecd 9779 . . . . . . . . . 10  |-  ( x  e.  NN  ->  (
1  /  ( 1  /  x ) )  =  x )
52 nncn 10000 . . . . . . . . . . 11  |-  ( y  e.  NN  ->  y  e.  CC )
53 nnne0 10024 . . . . . . . . . . 11  |-  ( y  e.  NN  ->  y  =/=  0 )
5452, 53recrecd 9779 . . . . . . . . . 10  |-  ( y  e.  NN  ->  (
1  /  ( 1  /  y ) )  =  y )
5551, 54eqeqan12d 2450 . . . . . . . . 9  |-  ( ( x  e.  NN  /\  y  e.  NN )  ->  ( ( 1  / 
( 1  /  x
) )  =  ( 1  /  ( 1  /  y ) )  <-> 
x  =  y ) )
5648, 55syl5ib 211 . . . . . . . 8  |-  ( ( x  e.  NN  /\  y  e.  NN )  ->  ( ( 1  /  x )  =  ( 1  /  y )  ->  x  =  y ) )
57 oveq2 6081 . . . . . . . 8  |-  ( x  =  y  ->  (
1  /  x )  =  ( 1  / 
y ) )
5856, 57impbid1 195 . . . . . . 7  |-  ( ( x  e.  NN  /\  y  e.  NN )  ->  ( ( 1  /  x )  =  ( 1  /  y )  <-> 
x  =  y ) )
5947, 58dom2 7142 . . . . . 6  |-  ( ( QQ  i^i  ( -u
1 [,] 1 ) )  e.  _V  ->  NN  ~<_  ( QQ  i^i  ( -u 1 [,] 1 ) ) )
6016, 59ax-mp 8 . . . . 5  |-  NN  ~<_  ( QQ 
i^i  ( -u 1 [,] 1 ) )
61 inss1 3553 . . . . . . 7  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  C_  QQ
62 ssdomg 7145 . . . . . . 7  |-  ( QQ  e.  _V  ->  (
( QQ  i^i  ( -u 1 [,] 1 ) )  C_  QQ  ->  ( QQ  i^i  ( -u
1 [,] 1 ) )  ~<_  QQ ) )
6315, 61, 62mp2 9 . . . . . 6  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  ~<_  QQ
64 qnnen 12805 . . . . . 6  |-  QQ  ~~  NN
65 domentr 7158 . . . . . 6  |-  ( ( ( QQ  i^i  ( -u 1 [,] 1 ) )  ~<_  QQ  /\  QQ  ~~  NN )  ->  ( QQ 
i^i  ( -u 1 [,] 1 ) )  ~<_  NN )
6663, 64, 65mp2an 654 . . . . 5  |-  ( QQ 
i^i  ( -u 1 [,] 1 ) )  ~<_  NN
67 sbth 7219 . . . . 5  |-  ( ( NN  ~<_  ( QQ  i^i  ( -u 1 [,] 1
) )  /\  ( QQ  i^i  ( -u 1 [,] 1 ) )  ~<_  NN )  ->  NN  ~~  ( QQ  i^i  ( -u 1 [,] 1 ) ) )
6860, 66, 67mp2an 654 . . . 4  |-  NN  ~~  ( QQ  i^i  ( -u 1 [,] 1 ) )
69 bren 7109 . . . 4  |-  ( NN 
~~  ( QQ  i^i  ( -u 1 [,] 1
) )  <->  E. g 
g : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) ) )
7068, 69mpbi 200 . . 3  |-  E. g 
g : NN -1-1-onto-> ( QQ  i^i  ( -u 1 [,] 1 ) )
71 eleq1 2495 . . . . . . . . . . . . 13  |-  ( a  =  x  ->  (
a  e.  ( 0 [,] 1 )  <->  x  e.  ( 0 [,] 1
) ) )
72 eleq1 2495 . . . . . . . . . . . . 13  |-  ( b  =  y  ->  (
b  e.  ( 0 [,] 1 )  <->  y  e.  ( 0 [,] 1
) ) )
7371, 72bi2anan9 844 . . . . . . . . . . . 12  |-  ( ( a  =  x  /\  b  =  y )  ->  ( ( a  e.  ( 0 [,] 1
)  /\  b  e.  ( 0 [,] 1
) )  <->  ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) ) ) )
74 oveq12 6082 . . . . . . . . . . . . 13  |-  ( ( a  =  x  /\  b  =  y )  ->  ( a  -  b
)  =  ( x  -  y ) )
7574eleq1d 2501 . . . . . . . . . . . 12  |-  ( ( a  =  x  /\  b  =  y )  ->  ( ( a  -  b )  e.  QQ  <->  ( x  -  y )  e.  QQ ) )
7673, 75anbi12d 692 . . . . . . . . . . 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 ) ) )
7776cbvopabv 4269 . . . . . . . . . 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 ) }
78 eqid 2435 . . . . . . . . . 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 ) } )
79 fvex 5734 . . . . . . . . . . . 12  |-  ( f `
 c )  e. 
_V
80 eqid 2435 . . . . . . . . . . . 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 ) )
8179, 80fnmpti 5565 . . . . . . . . . . 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 ) } )
8281a1i 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 ) } ) )
83 neeq1 2606 . . . . . . . . . . . . . . 15  |-  ( z  =  w  ->  (
z  =/=  (/)  <->  w  =/=  (/) ) )
84 fveq2 5720 . . . . . . . . . . . . . . . 16  |-  ( z  =  w  ->  (
f `  z )  =  ( f `  w ) )
85 id 20 . . . . . . . . . . . . . . . 16  |-  ( z  =  w  ->  z  =  w )
8684, 85eleq12d 2503 . . . . . . . . . . . . . . 15  |-  ( z  =  w  ->  (
( f `  z
)  e.  z  <->  ( f `  w )  e.  w
) )
8783, 86imbi12d 312 . . . . . . . . . . . . . 14  |-  ( z  =  w  ->  (
( z  =/=  (/)  ->  (
f `  z )  e.  z )  <->  ( w  =/=  (/)  ->  ( f `  w )  e.  w
) ) )
8887cbvralv 2924 . . . . . . . . . . . . 13  |-  ( A. z  e.  ~P  RR ( z  =/=  (/)  ->  (
f `  z )  e.  z )  <->  A. w  e.  ~P  RR ( w  =/=  (/)  ->  ( f `  w )  e.  w
) )
8977vitalilem1 19492 . . . . . . . . . . . . . . . . . 18  |-  { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) }  Er  ( 0 [,] 1
)
9089a1i 11 . . . . . . . . . . . . . . . . 17  |-  (  T. 
->  { <. a ,  b
>.  |  ( (
a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) }  Er  (
0 [,] 1 ) )
9190qsss 6957 . . . . . . . . . . . . . . . 16  |-  (  T. 
->  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1
) )  /\  (
a  -  b )  e.  QQ ) } )  C_  ~P (
0 [,] 1 ) )
9291trud 1332 . . . . . . . . . . . . . . 15  |-  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } ) 
C_  ~P ( 0 [,] 1 )
93 unitssre 11034 . . . . . . . . . . . . . . . 16  |-  ( 0 [,] 1 )  C_  RR
94 sspwb 4405 . . . . . . . . . . . . . . . 16  |-  ( ( 0 [,] 1 ) 
C_  RR  <->  ~P ( 0 [,] 1 )  C_  ~P RR )
9593, 94mpbi 200 . . . . . . . . . . . . . . 15  |-  ~P (
0 [,] 1 ) 
C_  ~P RR
9692, 95sstri 3349 . . . . . . . . . . . . . 14  |-  ( ( 0 [,] 1 ) /. { <. a ,  b >.  |  ( ( a  e.  ( 0 [,] 1 )  /\  b  e.  ( 0 [,] 1 ) )  /\  ( a  -  b )  e.  QQ ) } ) 
C_  ~P RR
97 ssralv 3399 . . . . . . . . . . . . . 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
) ) )
9896, 97ax-mp 8 . . . . . . . . . . . . 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
) )
9988, 98sylbi 188 . . . . . . . . . . . 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
) )
100 fveq2 5720 . . . . . . . . . . . . . . . 16  |-  ( c  =  w  ->  (
f `  c )  =  ( f `  w ) )
101 fvex 5734 . . . . . . . . . . . . . . . 16  |-  ( f `
 w )  e. 
_V
102100, 80, 101fvmpt 5798 . . . . . . . . . . . . . . 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 ) )
103102eleq1d 2501 . . . . . . . . . . . . . 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 ) )
104103imbi2d 308 . . . . . . . . . . . . 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 ) ) )
105104ralbiia 2729 . . . . . . . . . . . 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 ) )
10699, 105sylibr 204 . . . . . . . . . . 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 ) )
107106ad2antlr 708 . . . . . . . . . 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 ) )
108 simprl 733 . . . . . . . . . 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 ) ) )
109 oveq1 6080 . . . . . . . . . . . . . 14  |-  ( t  =  s  ->  (
t  -  ( g `
 m ) )  =  ( s  -  ( g `  m
) ) )
110109eleq1d 2501 . . . . . . . . . . . . 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 ) ) ) )
111110cbvrabv 2947 . . . . . . . . . . . 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 ) ) }
112 fveq2 5720 . . . . . . . . . . . . . . 15  |-  ( m  =  n  ->  (
g `  m )  =  ( g `  n ) )
113112oveq2d 6089 . . . . . . . . . . . . . 14  |-  ( m  =  n  ->  (
s  -  ( g `
 m ) )  =  ( s  -  ( g `  n
) ) )
114113eleq1d 2501 . . . . . . . . . . . . 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 ) ) ) )
115114rabbidv 2940 . . . . . . . . . . . 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 ) ) } )
116111, 115syl5eq 2479 . . . . . . . . . . 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 ) ) } )
117116cbvmptv 4292 . . . . . . . . . 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
) ) } )
118 simprr 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 ) ) )  ->  -.  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 ) )
11977, 78, 82, 107, 108, 117, 118vitalilem5 19496 . . . . . . . . 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 ) ) )
120119pm2.21i 125 . . . . . . . 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 ) )
121120expr 599 . . . . . . 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 ) ) )
122121pm2.18d 105 . . . . . 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 ) )
123 eldif 3322 . . . . . . 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 ) )
124 mblss 19419 . . . . . . . . . 10  |-  ( x  e.  dom  vol  ->  x 
C_  RR )
125 vex 2951 . . . . . . . . . . 11  |-  x  e. 
_V
126125elpw 3797 . . . . . . . . . 10  |-  ( x  e.  ~P RR  <->  x  C_  RR )
127124, 126sylibr 204 . . . . . . . . 9  |-  ( x  e.  dom  vol  ->  x  e.  ~P RR )
128127ssriv 3344 . . . . . . . 8  |-  dom  vol  C_ 
~P RR
129 ssnelpss 3683 . . . . . . . 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 ) )
130128, 129ax-mp 8 . . . . . . 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 )
131123, 130sylbi 188 . . . . . 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 )
132122, 131syl 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 )
133132ex 424 . . . 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 ) )
134133exlimdv 1646 . . 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 ) )
13570, 134mpi 17 . 2  |-  ( ( 
.<  We  RR  /\  A. z  e.  ~P  RR ( z  =/=  (/)  ->  (
f `  z )  e.  z ) )  ->  dom  vol  C.  ~P RR )
13614, 135exlimddv 1648 1  |-  (  .<  We  RR  ->  dom  vol  C.  ~P RR )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    T. wtru 1325   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   {crab 2701   _Vcvv 2948    \ cdif 3309    i^i cin 3311    C_ wss 3312    C. wpss 3313   (/)c0 3620   ~Pcpw 3791   U.cuni 4007   class class class wbr 4204   {copab 4257    e. cmpt 4258    We wwe 4532    X. cxp 4868   dom cdm 4870   ran crn 4871    Fn wfn 5441   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073    Er wer 6894   /.cqs 6896    ~~ cen 7098    ~<_ cdom 7099   RRcr 8981   0cc0 8982   1c1 8983    < clt 9112    <_ cle 9113    - cmin 9283   -ucneg 9284    / cdiv 9669   NNcn 9992   QQcq 10566   [,]cicc 10911   volcvol 19352
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cc 8307  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-disj 4175  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-omul 6721  df-er 6897  df-ec 6899  df-qs 6903  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-acn 7821  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ioo 10912  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-rlim 12275  df-sum 12472  df-rest 13642  df-topgen 13659  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-top 16955  df-bases 16957  df-topon 16958  df-cmp 17442  df-ovol 19353  df-vol 19354
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