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Theorem cmmbl 19289
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 difssd 3411 . 2  |-  ( A  e.  dom  vol  ->  ( RR  \  A ) 
C_  RR )
2 elpwi 3743 . . . 4  |-  ( x  e.  ~P RR  ->  x 
C_  RR )
3 inss1 3497 . . . . . . . . . 10  |-  ( x  i^i  A )  C_  x
4 ovolsscl 19242 . . . . . . . . . 10  |-  ( ( ( x  i^i  A
)  C_  x  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  i^i  A ) )  e.  RR )
53, 4mp3an1 1266 . . . . . . . . 9  |-  ( ( x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  i^i  A ) )  e.  RR )
653adant1 975 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  i^i  A ) )  e.  RR )
76recnd 9040 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  i^i  A ) )  e.  CC )
8 difss 3410 . . . . . . . . . 10  |-  ( x 
\  A )  C_  x
9 ovolsscl 19242 . . . . . . . . . 10  |-  ( ( ( x  \  A
)  C_  x  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  \  A ) )  e.  RR )
108, 9mp3an1 1266 . . . . . . . . 9  |-  ( ( x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  \  A ) )  e.  RR )
11103adant1 975 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  \  A ) )  e.  RR )
1211recnd 9040 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  \  A ) )  e.  CC )
137, 12addcomd 9193 . . . . . 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 ) ) ) )
14 mblsplit 19288 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  x )  =  ( ( vol * `  ( x  i^i  A ) )  +  ( vol
* `  ( x  \  A ) ) ) )
15 indifcom 3522 . . . . . . . . 9  |-  ( RR 
i^i  ( x  \  A ) )  =  ( x  i^i  ( RR  \  A ) )
16 simp2 958 . . . . . . . . . . 11  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  ->  x  C_  RR )
1716ssdifssd 3421 . . . . . . . . . 10  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( x  \  A
)  C_  RR )
18 sseqin2 3496 . . . . . . . . . 10  |-  ( ( x  \  A ) 
C_  RR  <->  ( RR  i^i  ( x  \  A ) )  =  ( x 
\  A ) )
1917, 18sylib 189 . . . . . . . . 9  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( RR  i^i  (
x  \  A )
)  =  ( x 
\  A ) )
2015, 19syl5eqr 2426 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( x  i^i  ( RR  \  A ) )  =  ( x  \  A ) )
2120fveq2d 5665 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  i^i  ( RR  \  A ) ) )  =  ( vol
* `  ( x  \  A ) ) )
22 difin 3514 . . . . . . . . . 10  |-  ( x 
\  ( x  i^i  ( RR  \  A
) ) )  =  ( x  \  ( RR  \  A ) )
2320difeq2d 3401 . . . . . . . . . 10  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( x  \  (
x  i^i  ( RR  \  A ) ) )  =  ( x  \ 
( x  \  A
) ) )
2422, 23syl5eqr 2426 . . . . . . . . 9  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( x  \  ( RR  \  A ) )  =  ( x  \ 
( x  \  A
) ) )
25 dfin4 3517 . . . . . . . . 9  |-  ( x  i^i  A )  =  ( x  \  (
x  \  A )
)
2624, 25syl6eqr 2430 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( x  \  ( RR  \  A ) )  =  ( x  i^i 
A ) )
2726fveq2d 5665 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  \  ( RR  \  A ) ) )  =  ( vol
* `  ( x  i^i  A ) ) )
2821, 27oveq12d 6031 . . . . . 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 ) ) ) )
2913, 14, 283eqtr4d 2422 . . . . 5  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  x )  =  ( ( vol * `  ( x  i^i  ( RR  \  A ) ) )  +  ( vol
* `  ( x  \  ( RR  \  A
) ) ) ) )
30293expia 1155 . . . 4  |-  ( ( A  e.  dom  vol  /\  x  C_  RR )  ->  ( ( vol * `  x )  e.  RR  ->  ( vol * `  x )  =  ( ( vol * `  ( x  i^i  ( RR  \  A ) ) )  +  ( vol
* `  ( x  \  ( RR  \  A
) ) ) ) ) )
312, 30sylan2 461 . . 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
) ) ) ) ) )
3231ralrimiva 2725 . 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
) ) ) ) ) )
33 ismbl 19282 . 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 ) ) ) ) ) ) )
341, 32, 33sylanbrc 646 1  |-  ( A  e.  dom  vol  ->  ( RR  \  A )  e.  dom  vol )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2642    \ cdif 3253    i^i cin 3255    C_ wss 3256   ~Pcpw 3735   dom cdm 4811   ` cfv 5387  (class class class)co 6013   RRcr 8915    + caddc 8919   vol *covol 19219   volcvol 19220
This theorem is referenced by:  rembl  19295  inmbl  19296  difmbl  19297  iccmbl  19320  itg2uba  19495  itg2monolem1  19502  itg2cnlem1  19513  itg2cnlem2  19514  dmvlsiga  24301
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-sup 7374  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-rp 10538  df-ico 10847  df-icc 10848  df-fz 10969  df-seq 11244  df-exp 11303  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-ovol 19221  df-vol 19222
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