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Theorem ismbl 22062
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 3690 . . . . . . 7  |-  ( y  =  A  ->  (
x  i^i  y )  =  ( x  i^i 
A ) )
21fveq2d 5876 . . . . . 6  |-  ( y  =  A  ->  ( vol* `  ( x  i^i  y ) )  =  ( vol* `  ( x  i^i  A
) ) )
3 difeq2 3612 . . . . . . 7  |-  ( y  =  A  ->  (
x  \  y )  =  ( x  \  A ) )
43fveq2d 5876 . . . . . 6  |-  ( y  =  A  ->  ( vol* `  ( x 
\  y ) )  =  ( vol* `  ( x  \  A
) ) )
52, 4oveq12d 6314 . . . . 5  |-  ( y  =  A  ->  (
( vol* `  ( x  i^i  y
) )  +  ( vol* `  (
x  \  y )
) )  =  ( ( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) ) )
65eqeq2d 2471 . . . 4  |-  ( y  =  A  ->  (
( vol* `  x )  =  ( ( vol* `  ( x  i^i  y
) )  +  ( vol* `  (
x  \  y )
) )  <->  ( vol* `  x )  =  ( ( vol* `  ( x  i^i  A
) )  +  ( vol* `  (
x  \  A )
) ) ) )
76ralbidv 2896 . . 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 22002 . . . . . 6  |-  vol  =  ( vol*  |`  { y  |  A. x  e.  ( `' vol* " RR ) ( vol* `  x )  =  ( ( vol* `  ( x  i^i  y ) )  +  ( vol* `  ( x  \  y
) ) ) } )
98dmeqi 5214 . . . . 5  |-  dom  vol  =  dom  ( vol*  |` 
{ y  |  A. x  e.  ( `' vol* " RR ) ( vol* `  x )  =  ( ( vol* `  ( x  i^i  y
) )  +  ( vol* `  (
x  \  y )
) ) } )
10 dmres 5304 . . . . 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 22018 . . . . . . 7  |-  vol* : ~P RR --> ( 0 [,] +oo )
1211fdmi 5742 . . . . . 6  |-  dom  vol*  =  ~P RR
1312ineq2i 3693 . . . . 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 2490 . . . 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 3781 . . . 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 2489 . . 3  |-  dom  vol  =  { y  e.  ~P RR  |  A. x  e.  ( `' vol* " RR ) ( vol* `  x )  =  ( ( vol* `  ( x  i^i  y ) )  +  ( vol* `  ( x  \  y
) ) ) }
177, 16elrab2 3259 . 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 9600 . . . 4  |-  RR  e.  _V
1918elpw2 4620 . . 3  |-  ( A  e.  ~P RR  <->  A  C_  RR )
20 ffn 5737 . . . . . . 7  |-  ( vol* : ~P RR --> ( 0 [,] +oo )  ->  vol*  Fn  ~P RR )
21 elpreima 6008 . . . . . . 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 2886 . . 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 1395    e. wcel 1819   {cab 2442   A.wral 2807   {crab 2811    \ cdif 3468    i^i cin 3470    C_ wss 3471   ~Pcpw 4015   `'ccnv 5007   dom cdm 5008    |` cres 5010   "cima 5011    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   RRcr 9508   0cc0 9509    + caddc 9512   +oocpnf 9642   [,]cicc 11557   vol*covol 21999   volcvol 22000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-ico 11560  df-icc 11561  df-fz 11698  df-seq 12110  df-exp 12169  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-ovol 22001  df-vol 22002
This theorem is referenced by:  ismbl2  22063  mblss  22067  mblsplit  22068  cmmbl  22070  shftmbl  22074  voliunlem2  22086
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