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Theorem ismbl 19414
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 3528 . . . . . . 7  |-  ( y  =  A  ->  (
x  i^i  y )  =  ( x  i^i 
A ) )
21fveq2d 5724 . . . . . 6  |-  ( y  =  A  ->  ( vol * `  ( x  i^i  y ) )  =  ( vol * `  ( x  i^i  A
) ) )
3 difeq2 3451 . . . . . . 7  |-  ( y  =  A  ->  (
x  \  y )  =  ( x  \  A ) )
43fveq2d 5724 . . . . . 6  |-  ( y  =  A  ->  ( vol * `  ( x 
\  y ) )  =  ( vol * `  ( x  \  A
) ) )
52, 4oveq12d 6091 . . . . 5  |-  ( y  =  A  ->  (
( vol * `  ( x  i^i  y
) )  +  ( vol * `  (
x  \  y )
) )  =  ( ( vol * `  ( x  i^i  A ) )  +  ( vol
* `  ( x  \  A ) ) ) )
65eqeq2d 2446 . . . 4  |-  ( y  =  A  ->  (
( vol * `  x )  =  ( ( vol * `  ( x  i^i  y
) )  +  ( vol * `  (
x  \  y )
) )  <->  ( vol * `
 x )  =  ( ( vol * `  ( x  i^i  A
) )  +  ( vol * `  (
x  \  A )
) ) ) )
76ralbidv 2717 . . 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 19354 . . . . . 6  |-  vol  =  ( vol *  |`  { y  |  A. x  e.  ( `' vol * " RR ) ( vol
* `  x )  =  ( ( vol
* `  ( x  i^i  y ) )  +  ( vol * `  ( x  \  y
) ) ) } )
98dmeqi 5063 . . . . 5  |-  dom  vol  =  dom  ( vol *  |` 
{ y  |  A. x  e.  ( `' vol * " RR ) ( vol * `  x )  =  ( ( vol * `  ( x  i^i  y
) )  +  ( vol * `  (
x  \  y )
) ) } )
10 dmres 5159 . . . . 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 19370 . . . . . . 7  |-  vol * : ~P RR --> ( 0 [,]  +oo )
1211fdmi 5588 . . . . . 6  |-  dom  vol *  =  ~P RR
1312ineq2i 3531 . . . . 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 2459 . . . 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 3608 . . . 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 2458 . . 3  |-  dom  vol  =  { y  e.  ~P RR  |  A. x  e.  ( `' vol * " RR ) ( vol
* `  x )  =  ( ( vol
* `  ( x  i^i  y ) )  +  ( vol * `  ( x  \  y
) ) ) }
177, 16elrab2 3086 . 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 9073 . . . 4  |-  RR  e.  _V
1918elpw2 4356 . . 3  |-  ( A  e.  ~P RR  <->  A  C_  RR )
20 ffn 5583 . . . . . . 7  |-  ( vol
* : ~P RR --> ( 0 [,]  +oo )  ->  vol *  Fn  ~P RR )
21 elpreima 5842 . . . . . . 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 2725 . . 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 1652    e. wcel 1725   {cab 2421   A.wral 2697   {crab 2701    \ cdif 3309    i^i cin 3311    C_ wss 3312   ~Pcpw 3791   `'ccnv 4869   dom cdm 4870    |` cres 4872   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   RRcr 8981   0cc0 8982    + caddc 8985    +oocpnf 9109   [,]cicc 10911   vol
*covol 19351   volcvol 19352
This theorem is referenced by:  ismbl2  19415  mblss  19419  mblsplit  19420  cmmbl  19421  shftmbl  19425  voliunlem2  19437
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-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  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-iun 4087  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-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-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  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-rp 10605  df-ico 10914  df-icc 10915  df-fz 11036  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-ovol 19353  df-vol 19354
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