MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ismbl Structured version   Unicode version

Theorem ismbl 21014
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 3551 . . . . . . 7  |-  ( y  =  A  ->  (
x  i^i  y )  =  ( x  i^i 
A ) )
21fveq2d 5700 . . . . . 6  |-  ( y  =  A  ->  ( vol* `  ( x  i^i  y ) )  =  ( vol* `  ( x  i^i  A
) ) )
3 difeq2 3473 . . . . . . 7  |-  ( y  =  A  ->  (
x  \  y )  =  ( x  \  A ) )
43fveq2d 5700 . . . . . 6  |-  ( y  =  A  ->  ( vol* `  ( x 
\  y ) )  =  ( vol* `  ( x  \  A
) ) )
52, 4oveq12d 6114 . . . . 5  |-  ( y  =  A  ->  (
( vol* `  ( x  i^i  y
) )  +  ( vol* `  (
x  \  y )
) )  =  ( ( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) ) )
65eqeq2d 2454 . . . 4  |-  ( y  =  A  ->  (
( vol* `  x )  =  ( ( vol* `  ( x  i^i  y
) )  +  ( vol* `  (
x  \  y )
) )  <->  ( vol* `  x )  =  ( ( vol* `  ( x  i^i  A
) )  +  ( vol* `  (
x  \  A )
) ) ) )
76ralbidv 2740 . . 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 20954 . . . . . 6  |-  vol  =  ( vol*  |`  { y  |  A. x  e.  ( `' vol* " RR ) ( vol* `  x )  =  ( ( vol* `  ( x  i^i  y ) )  +  ( vol* `  ( x  \  y
) ) ) } )
98dmeqi 5046 . . . . 5  |-  dom  vol  =  dom  ( vol*  |` 
{ y  |  A. x  e.  ( `' vol* " RR ) ( vol* `  x )  =  ( ( vol* `  ( x  i^i  y
) )  +  ( vol* `  (
x  \  y )
) ) } )
10 dmres 5136 . . . . 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 20970 . . . . . . 7  |-  vol* : ~P RR --> ( 0 [,] +oo )
1211fdmi 5569 . . . . . 6  |-  dom  vol*  =  ~P RR
1312ineq2i 3554 . . . . 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 2467 . . . 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 3631 . . . 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 2466 . . 3  |-  dom  vol  =  { y  e.  ~P RR  |  A. x  e.  ( `' vol* " RR ) ( vol* `  x )  =  ( ( vol* `  ( x  i^i  y ) )  +  ( vol* `  ( x  \  y
) ) ) }
177, 16elrab2 3124 . 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 9378 . . . 4  |-  RR  e.  _V
1918elpw2 4461 . . 3  |-  ( A  e.  ~P RR  <->  A  C_  RR )
20 ffn 5564 . . . . . . 7  |-  ( vol* : ~P RR --> ( 0 [,] +oo )  ->  vol*  Fn  ~P RR )
21 elpreima 5828 . . . . . . 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 325 . . . . 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 446 . . . . 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 249 . . . 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 2748 . . 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 697 . 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 249 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2429   A.wral 2720   {crab 2724    \ cdif 3330    i^i cin 3332    C_ wss 3333   ~Pcpw 3865   `'ccnv 4844   dom cdm 4845    |` cres 4847   "cima 4848    Fn wfn 5418   -->wf 5419   ` cfv 5423  (class class class)co 6096   RRcr 9286   0cc0 9287    + caddc 9290   +oocpnf 9420   [,]cicc 11308   vol*covol 20951   volcvol 20952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-ico 11311  df-icc 11312  df-fz 11443  df-seq 11812  df-exp 11871  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-ovol 20953  df-vol 20954
This theorem is referenced by:  ismbl2  21015  mblss  21019  mblsplit  21020  cmmbl  21021  shftmbl  21025  voliunlem2  21037
  Copyright terms: Public domain W3C validator