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Theorem ismbl 22229
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 3635 . . . . . . 7  |-  ( y  =  A  ->  (
x  i^i  y )  =  ( x  i^i 
A ) )
21fveq2d 5853 . . . . . 6  |-  ( y  =  A  ->  ( vol* `  ( x  i^i  y ) )  =  ( vol* `  ( x  i^i  A
) ) )
3 difeq2 3555 . . . . . . 7  |-  ( y  =  A  ->  (
x  \  y )  =  ( x  \  A ) )
43fveq2d 5853 . . . . . 6  |-  ( y  =  A  ->  ( vol* `  ( x 
\  y ) )  =  ( vol* `  ( x  \  A
) ) )
52, 4oveq12d 6296 . . . . 5  |-  ( y  =  A  ->  (
( vol* `  ( x  i^i  y
) )  +  ( vol* `  (
x  \  y )
) )  =  ( ( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) ) )
65eqeq2d 2416 . . . 4  |-  ( y  =  A  ->  (
( vol* `  x )  =  ( ( vol* `  ( x  i^i  y
) )  +  ( vol* `  (
x  \  y )
) )  <->  ( vol* `  x )  =  ( ( vol* `  ( x  i^i  A
) )  +  ( vol* `  (
x  \  A )
) ) ) )
76ralbidv 2843 . . 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 22169 . . . . . 6  |-  vol  =  ( vol*  |`  { y  |  A. x  e.  ( `' vol* " RR ) ( vol* `  x )  =  ( ( vol* `  ( x  i^i  y ) )  +  ( vol* `  ( x  \  y
) ) ) } )
98dmeqi 5025 . . . . 5  |-  dom  vol  =  dom  ( vol*  |` 
{ y  |  A. x  e.  ( `' vol* " RR ) ( vol* `  x )  =  ( ( vol* `  ( x  i^i  y
) )  +  ( vol* `  (
x  \  y )
) ) } )
10 dmres 5114 . . . . 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 22185 . . . . . . 7  |-  vol* : ~P RR --> ( 0 [,] +oo )
1211fdmi 5719 . . . . . 6  |-  dom  vol*  =  ~P RR
1312ineq2i 3638 . . . . 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 2435 . . . 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 3726 . . . 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 2434 . . 3  |-  dom  vol  =  { y  e.  ~P RR  |  A. x  e.  ( `' vol* " RR ) ( vol* `  x )  =  ( ( vol* `  ( x  i^i  y ) )  +  ( vol* `  ( x  \  y
) ) ) }
177, 16elrab2 3209 . 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 9613 . . . 4  |-  RR  e.  _V
1918elpw2 4558 . . 3  |-  ( A  e.  ~P RR  <->  A  C_  RR )
20 ffn 5714 . . . . . . 7  |-  ( vol* : ~P RR --> ( 0 [,] +oo )  ->  vol*  Fn  ~P RR )
21 elpreima 5985 . . . . . . 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 323 . . . . 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 444 . . . . 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 2833 . . 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 695 . 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 367    = wceq 1405    e. wcel 1842   {cab 2387   A.wral 2754   {crab 2758    \ cdif 3411    i^i cin 3413    C_ wss 3414   ~Pcpw 3955   `'ccnv 4822   dom cdm 4823    |` cres 4825   "cima 4826    Fn wfn 5564   -->wf 5565   ` cfv 5569  (class class class)co 6278   RRcr 9521   0cc0 9522    + caddc 9525   +oocpnf 9655   [,]cicc 11585   vol*covol 22166   volcvol 22167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-sup 7935  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-ico 11588  df-icc 11589  df-fz 11727  df-seq 12152  df-exp 12211  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-ovol 22168  df-vol 22169
This theorem is referenced by:  ismbl2  22230  mblss  22234  mblsplit  22235  cmmbl  22237  shftmbl  22241  voliunlem2  22253
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