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Theorem cmmbl 18892
Description: The complement of a measurable set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
Assertion
Ref Expression
cmmbl  |-  ( A  e.  dom  vol  ->  ( RR  \  A )  e.  dom  vol )

Proof of Theorem cmmbl
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 difss 3303 . . 3  |-  ( RR 
\  A )  C_  RR
21a1i 10 . 2  |-  ( A  e.  dom  vol  ->  ( RR  \  A ) 
C_  RR )
3 elpwi 3633 . . . 4  |-  ( x  e.  ~P RR  ->  x 
C_  RR )
4 inss1 3389 . . . . . . . . . 10  |-  ( x  i^i  A )  C_  x
5 ovolsscl 18845 . . . . . . . . . 10  |-  ( ( ( x  i^i  A
)  C_  x  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  i^i  A ) )  e.  RR )
64, 5mp3an1 1264 . . . . . . . . 9  |-  ( ( x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  i^i  A ) )  e.  RR )
763adant1 973 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  i^i  A ) )  e.  RR )
87recnd 8861 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  i^i  A ) )  e.  CC )
9 difss 3303 . . . . . . . . . 10  |-  ( x 
\  A )  C_  x
10 ovolsscl 18845 . . . . . . . . . 10  |-  ( ( ( x  \  A
)  C_  x  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  \  A ) )  e.  RR )
119, 10mp3an1 1264 . . . . . . . . 9  |-  ( ( x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  \  A ) )  e.  RR )
12113adant1 973 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  \  A ) )  e.  RR )
1312recnd 8861 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  \  A ) )  e.  CC )
148, 13addcomd 9014 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( ( vol * `  ( x  i^i  A
) )  +  ( vol * `  (
x  \  A )
) )  =  ( ( vol * `  ( x  \  A ) )  +  ( vol
* `  ( x  i^i  A ) ) ) )
15 mblsplit 18891 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  x )  =  ( ( vol * `  ( x  i^i  A ) )  +  ( vol
* `  ( x  \  A ) ) ) )
16 indifcom 3414 . . . . . . . . 9  |-  ( RR 
i^i  ( x  \  A ) )  =  ( x  i^i  ( RR  \  A ) )
17 simp2 956 . . . . . . . . . . 11  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  ->  x  C_  RR )
189, 17syl5ss 3190 . . . . . . . . . 10  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( x  \  A
)  C_  RR )
19 sseqin2 3388 . . . . . . . . . 10  |-  ( ( x  \  A ) 
C_  RR  <->  ( RR  i^i  ( x  \  A ) )  =  ( x 
\  A ) )
2018, 19sylib 188 . . . . . . . . 9  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( RR  i^i  (
x  \  A )
)  =  ( x 
\  A ) )
2116, 20syl5eqr 2329 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( x  i^i  ( RR  \  A ) )  =  ( x  \  A ) )
2221fveq2d 5529 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  i^i  ( RR  \  A ) ) )  =  ( vol
* `  ( x  \  A ) ) )
23 difin 3406 . . . . . . . . . 10  |-  ( x 
\  ( x  i^i  ( RR  \  A
) ) )  =  ( x  \  ( RR  \  A ) )
2421difeq2d 3294 . . . . . . . . . 10  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( x  \  (
x  i^i  ( RR  \  A ) ) )  =  ( x  \ 
( x  \  A
) ) )
2523, 24syl5eqr 2329 . . . . . . . . 9  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( x  \  ( RR  \  A ) )  =  ( x  \ 
( x  \  A
) ) )
26 dfin4 3409 . . . . . . . . 9  |-  ( x  i^i  A )  =  ( x  \  (
x  \  A )
)
2725, 26syl6eqr 2333 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( x  \  ( RR  \  A ) )  =  ( x  i^i 
A ) )
2827fveq2d 5529 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  \  ( RR  \  A ) ) )  =  ( vol
* `  ( x  i^i  A ) ) )
2922, 28oveq12d 5876 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( ( vol * `  ( x  i^i  ( RR  \  A ) ) )  +  ( vol
* `  ( x  \  ( RR  \  A
) ) ) )  =  ( ( vol
* `  ( x  \  A ) )  +  ( vol * `  ( x  i^i  A ) ) ) )
3014, 15, 293eqtr4d 2325 . . . . 5  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  x )  =  ( ( vol * `  ( x  i^i  ( RR  \  A ) ) )  +  ( vol
* `  ( x  \  ( RR  \  A
) ) ) ) )
31303expia 1153 . . . 4  |-  ( ( A  e.  dom  vol  /\  x  C_  RR )  ->  ( ( vol * `  x )  e.  RR  ->  ( vol * `  x )  =  ( ( vol * `  ( x  i^i  ( RR  \  A ) ) )  +  ( vol
* `  ( x  \  ( RR  \  A
) ) ) ) ) )
323, 31sylan2 460 . . 3  |-  ( ( A  e.  dom  vol  /\  x  e.  ~P RR )  ->  ( ( vol
* `  x )  e.  RR  ->  ( vol * `
 x )  =  ( ( vol * `  ( x  i^i  ( RR  \  A ) ) )  +  ( vol
* `  ( x  \  ( RR  \  A
) ) ) ) ) )
3332ralrimiva 2626 . 2  |-  ( A  e.  dom  vol  ->  A. x  e.  ~P  RR ( ( vol * `  x )  e.  RR  ->  ( vol * `  x )  =  ( ( vol * `  ( x  i^i  ( RR  \  A ) ) )  +  ( vol
* `  ( x  \  ( RR  \  A
) ) ) ) ) )
34 ismbl 18885 . 2  |-  ( ( RR  \  A )  e.  dom  vol  <->  ( ( RR  \  A )  C_  RR  /\  A. x  e. 
~P  RR ( ( vol * `  x
)  e.  RR  ->  ( vol * `  x
)  =  ( ( vol * `  (
x  i^i  ( RR  \  A ) ) )  +  ( vol * `  ( x  \  ( RR  \  A ) ) ) ) ) ) )
352, 33, 34sylanbrc 645 1  |-  ( A  e.  dom  vol  ->  ( RR  \  A )  e.  dom  vol )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    \ cdif 3149    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   dom cdm 4689   ` cfv 5255  (class class class)co 5858   RRcr 8736    + caddc 8740   vol *covol 18822   volcvol 18823
This theorem is referenced by:  rembl  18898  inmbl  18899  difmbl  18900  iccmbl  18923  itg2uba  19098  itg2monolem1  19105  itg2cnlem1  19116  itg2cnlem2  19117
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ico 10662  df-icc 10663  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-ovol 18824  df-vol 18825
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