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Theorem ismbl 19289
Description: The predicate " A is Lebesgue-measurable". A set is measurable if it splits every other set  x in a "nice" way, that is, if the measure of the pieces  x  i^i  A and  x  \  A sum up to the measure of 
x (assuming that the measure of 
x is a real number, so that this addition makes sense). (Contributed by Mario Carneiro, 17-Mar-2014.)
Assertion
Ref Expression
ismbl  |-  ( A  e.  dom  vol  <->  ( A  C_  RR  /\  A. x  e.  ~P  RR ( ( vol * `  x
)  e.  RR  ->  ( vol * `  x
)  =  ( ( vol * `  (
x  i^i  A )
)  +  ( vol
* `  ( x  \  A ) ) ) ) ) )
Distinct variable group:    x, A

Proof of Theorem ismbl
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ineq2 3479 . . . . . . 7  |-  ( y  =  A  ->  (
x  i^i  y )  =  ( x  i^i 
A ) )
21fveq2d 5672 . . . . . 6  |-  ( y  =  A  ->  ( vol * `  ( x  i^i  y ) )  =  ( vol * `  ( x  i^i  A
) ) )
3 difeq2 3402 . . . . . . 7  |-  ( y  =  A  ->  (
x  \  y )  =  ( x  \  A ) )
43fveq2d 5672 . . . . . 6  |-  ( y  =  A  ->  ( vol * `  ( x 
\  y ) )  =  ( vol * `  ( x  \  A
) ) )
52, 4oveq12d 6038 . . . . 5  |-  ( y  =  A  ->  (
( vol * `  ( x  i^i  y
) )  +  ( vol * `  (
x  \  y )
) )  =  ( ( vol * `  ( x  i^i  A ) )  +  ( vol
* `  ( x  \  A ) ) ) )
65eqeq2d 2398 . . . 4  |-  ( y  =  A  ->  (
( vol * `  x )  =  ( ( vol * `  ( x  i^i  y
) )  +  ( vol * `  (
x  \  y )
) )  <->  ( vol * `
 x )  =  ( ( vol * `  ( x  i^i  A
) )  +  ( vol * `  (
x  \  A )
) ) ) )
76ralbidv 2669 . . 3  |-  ( y  =  A  ->  ( A. x  e.  ( `' vol * " RR ) ( vol * `  x )  =  ( ( vol * `  ( x  i^i  y
) )  +  ( vol * `  (
x  \  y )
) )  <->  A. x  e.  ( `' vol * " RR ) ( vol
* `  x )  =  ( ( vol
* `  ( x  i^i  A ) )  +  ( vol * `  ( x  \  A ) ) ) ) )
8 df-vol 19229 . . . . . 6  |-  vol  =  ( vol *  |`  { y  |  A. x  e.  ( `' vol * " RR ) ( vol
* `  x )  =  ( ( vol
* `  ( x  i^i  y ) )  +  ( vol * `  ( x  \  y
) ) ) } )
98dmeqi 5011 . . . . 5  |-  dom  vol  =  dom  ( vol *  |` 
{ y  |  A. x  e.  ( `' vol * " RR ) ( vol * `  x )  =  ( ( vol * `  ( x  i^i  y
) )  +  ( vol * `  (
x  \  y )
) ) } )
10 dmres 5107 . . . . 5  |-  dom  ( vol *  |`  { y  |  A. x  e.  ( `' vol * " RR ) ( vol * `  x )  =  ( ( vol * `  ( x  i^i  y
) )  +  ( vol * `  (
x  \  y )
) ) } )  =  ( { y  |  A. x  e.  ( `' vol * " RR ) ( vol
* `  x )  =  ( ( vol
* `  ( x  i^i  y ) )  +  ( vol * `  ( x  \  y
) ) ) }  i^i  dom  vol * )
11 ovolf 19245 . . . . . . 7  |-  vol * : ~P RR --> ( 0 [,]  +oo )
1211fdmi 5536 . . . . . 6  |-  dom  vol *  =  ~P RR
1312ineq2i 3482 . . . . 5  |-  ( { y  |  A. x  e.  ( `' vol * " RR ) ( vol
* `  x )  =  ( ( vol
* `  ( x  i^i  y ) )  +  ( vol * `  ( x  \  y
) ) ) }  i^i  dom  vol * )  =  ( { y  |  A. x  e.  ( `' vol * " RR ) ( vol
* `  x )  =  ( ( vol
* `  ( x  i^i  y ) )  +  ( vol * `  ( x  \  y
) ) ) }  i^i  ~P RR )
149, 10, 133eqtri 2411 . . . 4  |-  dom  vol  =  ( { y  |  A. x  e.  ( `' vol * " RR ) ( vol
* `  x )  =  ( ( vol
* `  ( x  i^i  y ) )  +  ( vol * `  ( x  \  y
) ) ) }  i^i  ~P RR )
15 dfrab2 3559 . . . 4  |-  { y  e.  ~P RR  |  A. x  e.  ( `' vol * " RR ) ( vol * `  x )  =  ( ( vol * `  ( x  i^i  y
) )  +  ( vol * `  (
x  \  y )
) ) }  =  ( { y  |  A. x  e.  ( `' vol * " RR ) ( vol * `  x )  =  ( ( vol * `  ( x  i^i  y
) )  +  ( vol * `  (
x  \  y )
) ) }  i^i  ~P RR )
1614, 15eqtr4i 2410 . . 3  |-  dom  vol  =  { y  e.  ~P RR  |  A. x  e.  ( `' vol * " RR ) ( vol
* `  x )  =  ( ( vol
* `  ( x  i^i  y ) )  +  ( vol * `  ( x  \  y
) ) ) }
177, 16elrab2 3037 . 2  |-  ( A  e.  dom  vol  <->  ( A  e.  ~P RR  /\  A. x  e.  ( `' vol * " RR ) ( vol * `  x )  =  ( ( vol * `  ( x  i^i  A ) )  +  ( vol
* `  ( x  \  A ) ) ) ) )
18 reex 9014 . . . 4  |-  RR  e.  _V
1918elpw2 4305 . . 3  |-  ( A  e.  ~P RR  <->  A  C_  RR )
20 ffn 5531 . . . . . . 7  |-  ( vol
* : ~P RR --> ( 0 [,]  +oo )  ->  vol *  Fn  ~P RR )
21 elpreima 5789 . . . . . . 7  |-  ( vol
*  Fn  ~P RR  ->  ( x  e.  ( `' vol * " RR ) 
<->  ( x  e.  ~P RR  /\  ( vol * `  x )  e.  RR ) ) )
2211, 20, 21mp2b 10 . . . . . 6  |-  ( x  e.  ( `' vol *
" RR )  <->  ( x  e.  ~P RR  /\  ( vol * `  x )  e.  RR ) )
2322imbi1i 316 . . . . 5  |-  ( ( x  e.  ( `' vol * " RR )  ->  ( vol * `  x )  =  ( ( vol * `  ( x  i^i  A ) )  +  ( vol
* `  ( x  \  A ) ) ) )  <->  ( ( x  e.  ~P RR  /\  ( vol * `  x
)  e.  RR )  ->  ( vol * `  x )  =  ( ( vol * `  ( x  i^i  A ) )  +  ( vol
* `  ( x  \  A ) ) ) ) )
24 impexp 434 . . . . 5  |-  ( ( ( x  e.  ~P RR  /\  ( vol * `  x )  e.  RR )  ->  ( vol * `  x )  =  ( ( vol * `  ( x  i^i  A ) )  +  ( vol
* `  ( x  \  A ) ) ) )  <->  ( x  e. 
~P RR  ->  (
( vol * `  x )  e.  RR  ->  ( vol * `  x )  =  ( ( vol * `  ( x  i^i  A ) )  +  ( vol
* `  ( x  \  A ) ) ) ) ) )
2523, 24bitri 241 . . . 4  |-  ( ( x  e.  ( `' vol * " RR )  ->  ( vol * `  x )  =  ( ( vol * `  ( x  i^i  A ) )  +  ( vol
* `  ( x  \  A ) ) ) )  <->  ( x  e. 
~P RR  ->  (
( vol * `  x )  e.  RR  ->  ( vol * `  x )  =  ( ( vol * `  ( x  i^i  A ) )  +  ( vol
* `  ( x  \  A ) ) ) ) ) )
2625ralbii2 2677 . . 3  |-  ( A. x  e.  ( `' vol * " RR ) ( vol * `  x )  =  ( ( vol * `  ( x  i^i  A ) )  +  ( vol
* `  ( x  \  A ) ) )  <->  A. x  e.  ~P  RR ( ( vol * `  x )  e.  RR  ->  ( vol * `  x )  =  ( ( vol * `  ( x  i^i  A ) )  +  ( vol
* `  ( x  \  A ) ) ) ) )
2719, 26anbi12i 679 . 2  |-  ( ( A  e.  ~P RR  /\ 
A. x  e.  ( `' vol * " RR ) ( vol * `  x )  =  ( ( vol * `  ( x  i^i  A ) )  +  ( vol
* `  ( x  \  A ) ) ) )  <->  ( A  C_  RR  /\  A. x  e. 
~P  RR ( ( vol * `  x
)  e.  RR  ->  ( vol * `  x
)  =  ( ( vol * `  (
x  i^i  A )
)  +  ( vol
* `  ( x  \  A ) ) ) ) ) )
2817, 27bitri 241 1  |-  ( A  e.  dom  vol  <->  ( A  C_  RR  /\  A. x  e.  ~P  RR ( ( vol * `  x
)  e.  RR  ->  ( vol * `  x
)  =  ( ( vol * `  (
x  i^i  A )
)  +  ( vol
* `  ( x  \  A ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   {cab 2373   A.wral 2649   {crab 2653    \ cdif 3260    i^i cin 3262    C_ wss 3263   ~Pcpw 3742   `'ccnv 4817   dom cdm 4818    |` cres 4820   "cima 4821    Fn wfn 5389   -->wf 5390   ` cfv 5394  (class class class)co 6020   RRcr 8922   0cc0 8923    + caddc 8926    +oocpnf 9050   [,]cicc 10851   vol
*covol 19226   volcvol 19227
This theorem is referenced by:  ismbl2  19290  mblss  19294  mblsplit  19295  cmmbl  19296  shftmbl  19300  voliunlem2  19312
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-ico 10854  df-icc 10855  df-fz 10976  df-seq 11251  df-exp 11310  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-ovol 19228  df-vol 19229
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