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Theorem ismbl 19375
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 3496 . . . . . . 7  |-  ( y  =  A  ->  (
x  i^i  y )  =  ( x  i^i 
A ) )
21fveq2d 5691 . . . . . 6  |-  ( y  =  A  ->  ( vol * `  ( x  i^i  y ) )  =  ( vol * `  ( x  i^i  A
) ) )
3 difeq2 3419 . . . . . . 7  |-  ( y  =  A  ->  (
x  \  y )  =  ( x  \  A ) )
43fveq2d 5691 . . . . . 6  |-  ( y  =  A  ->  ( vol * `  ( x 
\  y ) )  =  ( vol * `  ( x  \  A
) ) )
52, 4oveq12d 6058 . . . . 5  |-  ( y  =  A  ->  (
( vol * `  ( x  i^i  y
) )  +  ( vol * `  (
x  \  y )
) )  =  ( ( vol * `  ( x  i^i  A ) )  +  ( vol
* `  ( x  \  A ) ) ) )
65eqeq2d 2415 . . . 4  |-  ( y  =  A  ->  (
( vol * `  x )  =  ( ( vol * `  ( x  i^i  y
) )  +  ( vol * `  (
x  \  y )
) )  <->  ( vol * `
 x )  =  ( ( vol * `  ( x  i^i  A
) )  +  ( vol * `  (
x  \  A )
) ) ) )
76ralbidv 2686 . . 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 19315 . . . . . 6  |-  vol  =  ( vol *  |`  { y  |  A. x  e.  ( `' vol * " RR ) ( vol
* `  x )  =  ( ( vol
* `  ( x  i^i  y ) )  +  ( vol * `  ( x  \  y
) ) ) } )
98dmeqi 5030 . . . . 5  |-  dom  vol  =  dom  ( vol *  |` 
{ y  |  A. x  e.  ( `' vol * " RR ) ( vol * `  x )  =  ( ( vol * `  ( x  i^i  y
) )  +  ( vol * `  (
x  \  y )
) ) } )
10 dmres 5126 . . . . 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 19331 . . . . . . 7  |-  vol * : ~P RR --> ( 0 [,]  +oo )
1211fdmi 5555 . . . . . 6  |-  dom  vol *  =  ~P RR
1312ineq2i 3499 . . . . 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 2428 . . . 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 3576 . . . 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 2427 . . 3  |-  dom  vol  =  { y  e.  ~P RR  |  A. x  e.  ( `' vol * " RR ) ( vol
* `  x )  =  ( ( vol
* `  ( x  i^i  y ) )  +  ( vol * `  ( x  \  y
) ) ) }
177, 16elrab2 3054 . 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 9037 . . . 4  |-  RR  e.  _V
1918elpw2 4324 . . 3  |-  ( A  e.  ~P RR  <->  A  C_  RR )
20 ffn 5550 . . . . . . 7  |-  ( vol
* : ~P RR --> ( 0 [,]  +oo )  ->  vol *  Fn  ~P RR )
21 elpreima 5809 . . . . . . 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 2694 . . 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 1721   {cab 2390   A.wral 2666   {crab 2670    \ cdif 3277    i^i cin 3279    C_ wss 3280   ~Pcpw 3759   `'ccnv 4836   dom cdm 4837    |` cres 4839   "cima 4840    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   RRcr 8945   0cc0 8946    + caddc 8949    +oocpnf 9073   [,]cicc 10875   vol
*covol 19312   volcvol 19313
This theorem is referenced by:  ismbl2  19376  mblss  19380  mblsplit  19381  cmmbl  19382  shftmbl  19386  voliunlem2  19398
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  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-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-ico 10878  df-icc 10879  df-fz 11000  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-ovol 19314  df-vol 19315
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