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Theorem cmmbl 18908
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 3316 . . 3  |-  ( RR 
\  A )  C_  RR
21a1i 10 . 2  |-  ( A  e.  dom  vol  ->  ( RR  \  A ) 
C_  RR )
3 elpwi 3646 . . . 4  |-  ( x  e.  ~P RR  ->  x 
C_  RR )
4 inss1 3402 . . . . . . . . . 10  |-  ( x  i^i  A )  C_  x
5 ovolsscl 18861 . . . . . . . . . 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 8877 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  i^i  A ) )  e.  CC )
9 difss 3316 . . . . . . . . . 10  |-  ( x 
\  A )  C_  x
10 ovolsscl 18861 . . . . . . . . . 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 8877 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  \  A ) )  e.  CC )
148, 13addcomd 9030 . . . . . 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 18907 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  x )  =  ( ( vol * `  ( x  i^i  A ) )  +  ( vol
* `  ( x  \  A ) ) ) )
16 indifcom 3427 . . . . . . . . 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 3203 . . . . . . . . . 10  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( x  \  A
)  C_  RR )
19 sseqin2 3401 . . . . . . . . . 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 2342 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( x  i^i  ( RR  \  A ) )  =  ( x  \  A ) )
2221fveq2d 5545 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  i^i  ( RR  \  A ) ) )  =  ( vol
* `  ( x  \  A ) ) )
23 difin 3419 . . . . . . . . . 10  |-  ( x 
\  ( x  i^i  ( RR  \  A
) ) )  =  ( x  \  ( RR  \  A ) )
2421difeq2d 3307 . . . . . . . . . 10  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( x  \  (
x  i^i  ( RR  \  A ) ) )  =  ( x  \ 
( x  \  A
) ) )
2523, 24syl5eqr 2342 . . . . . . . . 9  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( x  \  ( RR  \  A ) )  =  ( x  \ 
( x  \  A
) ) )
26 dfin4 3422 . . . . . . . . 9  |-  ( x  i^i  A )  =  ( x  \  (
x  \  A )
)
2725, 26syl6eqr 2346 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( x  \  ( RR  \  A ) )  =  ( x  i^i 
A ) )
2827fveq2d 5545 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  \  ( RR  \  A ) ) )  =  ( vol
* `  ( x  i^i  A ) ) )
2922, 28oveq12d 5892 . . . . . 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 2338 . . . . 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 2639 . 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 18901 . 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 1632    e. wcel 1696   A.wral 2556    \ cdif 3162    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   dom cdm 4705   ` cfv 5271  (class class class)co 5874   RRcr 8752    + caddc 8756   vol *covol 18838   volcvol 18839
This theorem is referenced by:  rembl  18914  inmbl  18915  difmbl  18916  iccmbl  18939  itg2uba  19114  itg2monolem1  19121  itg2cnlem1  19132  itg2cnlem2  19133
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-ico 10678  df-icc 10679  df-fz 10799  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-ovol 18840  df-vol 18841
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