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

Theorem ismbl 22417
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 3596 . . . . . . 7  |-  ( y  =  A  ->  (
x  i^i  y )  =  ( x  i^i 
A ) )
21fveq2d 5824 . . . . . 6  |-  ( y  =  A  ->  ( vol* `  ( x  i^i  y ) )  =  ( vol* `  ( x  i^i  A
) ) )
3 difeq2 3515 . . . . . . 7  |-  ( y  =  A  ->  (
x  \  y )  =  ( x  \  A ) )
43fveq2d 5824 . . . . . 6  |-  ( y  =  A  ->  ( vol* `  ( x 
\  y ) )  =  ( vol* `  ( x  \  A
) ) )
52, 4oveq12d 6262 . . . . 5  |-  ( y  =  A  ->  (
( vol* `  ( x  i^i  y
) )  +  ( vol* `  (
x  \  y )
) )  =  ( ( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) ) )
65eqeq2d 2433 . . . 4  |-  ( y  =  A  ->  (
( vol* `  x )  =  ( ( vol* `  ( x  i^i  y
) )  +  ( vol* `  (
x  \  y )
) )  <->  ( vol* `  x )  =  ( ( vol* `  ( x  i^i  A
) )  +  ( vol* `  (
x  \  A )
) ) ) )
76ralbidv 2799 . . 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 22355 . . . . . 6  |-  vol  =  ( vol*  |`  { y  |  A. x  e.  ( `' vol* " RR ) ( vol* `  x )  =  ( ( vol* `  ( x  i^i  y ) )  +  ( vol* `  ( x  \  y
) ) ) } )
98dmeqi 4993 . . . . 5  |-  dom  vol  =  dom  ( vol*  |` 
{ y  |  A. x  e.  ( `' vol* " RR ) ( vol* `  x )  =  ( ( vol* `  ( x  i^i  y
) )  +  ( vol* `  (
x  \  y )
) ) } )
10 dmres 5082 . . . . 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 22372 . . . . . . 7  |-  vol* : ~P RR --> ( 0 [,] +oo )
1211fdmi 5689 . . . . . 6  |-  dom  vol*  =  ~P RR
1312ineq2i 3599 . . . . 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 2449 . . . 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 3687 . . . 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 2448 . . 3  |-  dom  vol  =  { y  e.  ~P RR  |  A. x  e.  ( `' vol* " RR ) ( vol* `  x )  =  ( ( vol* `  ( x  i^i  y ) )  +  ( vol* `  ( x  \  y
) ) ) }
177, 16elrab2 3168 . 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 9576 . . . 4  |-  RR  e.  _V
1918elpw2 4526 . . 3  |-  ( A  e.  ~P RR  <->  A  C_  RR )
20 ffn 5684 . . . . . . 7  |-  ( vol* : ~P RR --> ( 0 [,] +oo )  ->  vol*  Fn  ~P RR )
21 elpreima 5956 . . . . . . 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 326 . . . . 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 447 . . . . 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 252 . . . 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 2789 . . 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 701 . 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 252 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 187    /\ wa 370    = wceq 1437    e. wcel 1872   {cab 2409   A.wral 2709   {crab 2713    \ cdif 3371    i^i cin 3373    C_ wss 3374   ~Pcpw 3919   `'ccnv 4790   dom cdm 4791    |` cres 4793   "cima 4794    Fn wfn 5534   -->wf 5535   ` cfv 5539  (class class class)co 6244   RRcr 9484   0cc0 9485    + caddc 9488   +oocpnf 9618   [,]cicc 11584   vol*covol 22350   volcvol 22352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536  ax-cnex 9541  ax-resscn 9542  ax-1cn 9543  ax-icn 9544  ax-addcl 9545  ax-addrcl 9546  ax-mulcl 9547  ax-mulrcl 9548  ax-mulcom 9549  ax-addass 9550  ax-mulass 9551  ax-distr 9552  ax-i2m1 9553  ax-1ne0 9554  ax-1rid 9555  ax-rnegex 9556  ax-rrecex 9557  ax-cnre 9558  ax-pre-lttri 9559  ax-pre-lttrn 9560  ax-pre-ltadd 9561  ax-pre-mulgt0 9562  ax-pre-sup 9563
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-nel 2597  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-pss 3390  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4158  df-iun 4239  df-br 4362  df-opab 4421  df-mpt 4422  df-tr 4457  df-eprel 4702  df-id 4706  df-po 4712  df-so 4713  df-fr 4750  df-we 4752  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-pred 5337  df-ord 5383  df-on 5384  df-lim 5385  df-suc 5386  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-riota 6206  df-ov 6247  df-oprab 6248  df-mpt2 6249  df-om 6646  df-1st 6746  df-2nd 6747  df-wrecs 6978  df-recs 7040  df-rdg 7078  df-er 7313  df-map 7424  df-en 7520  df-dom 7521  df-sdom 7522  df-sup 7904  df-inf 7905  df-pnf 9623  df-mnf 9624  df-xr 9625  df-ltxr 9626  df-le 9627  df-sub 9808  df-neg 9809  df-div 10216  df-nn 10556  df-2 10614  df-3 10615  df-n0 10816  df-z 10884  df-uz 11106  df-rp 11249  df-ico 11587  df-icc 11588  df-fz 11731  df-seq 12159  df-exp 12218  df-cj 13101  df-re 13102  df-im 13103  df-sqrt 13237  df-abs 13238  df-ovol 22353  df-vol 22355
This theorem is referenced by:  ismbl2  22418  mblss  22422  mblsplit  22423  cmmbl  22425  shftmbl  22429  voliunlem2  22441
  Copyright terms: Public domain W3C validator