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Theorem cmmbl 21790
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 3637 . 2  |-  ( A  e.  dom  vol  ->  ( RR  \  A ) 
C_  RR )
2 elpwi 4024 . . . 4  |-  ( x  e.  ~P RR  ->  x 
C_  RR )
3 inss1 3723 . . . . . . . . . 10  |-  ( x  i^i  A )  C_  x
4 ovolsscl 21742 . . . . . . . . . 10  |-  ( ( ( x  i^i  A
)  C_  x  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  A ) )  e.  RR )
53, 4mp3an1 1311 . . . . . . . . 9  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  A ) )  e.  RR )
653adant1 1014 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  A ) )  e.  RR )
76recnd 9632 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  A ) )  e.  CC )
8 difss 3636 . . . . . . . . . 10  |-  ( x 
\  A )  C_  x
9 ovolsscl 21742 . . . . . . . . . 10  |-  ( ( ( x  \  A
)  C_  x  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  \  A ) )  e.  RR )
108, 9mp3an1 1311 . . . . . . . . 9  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  \  A ) )  e.  RR )
11103adant1 1014 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  \  A ) )  e.  RR )
1211recnd 9632 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  \  A ) )  e.  CC )
137, 12addcomd 9791 . . . . . 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 21788 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  x )  =  ( ( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) ) )
15 indifcom 3748 . . . . . . . . 9  |-  ( RR 
i^i  ( x  \  A ) )  =  ( x  i^i  ( RR  \  A ) )
16 simp2 997 . . . . . . . . . . 11  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  x  C_  RR )
1716ssdifssd 3647 . . . . . . . . . 10  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( x  \  A
)  C_  RR )
18 sseqin2 3722 . . . . . . . . . 10  |-  ( ( x  \  A ) 
C_  RR  <->  ( RR  i^i  ( x  \  A ) )  =  ( x 
\  A ) )
1917, 18sylib 196 . . . . . . . . 9  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( RR  i^i  (
x  \  A )
)  =  ( x 
\  A ) )
2015, 19syl5eqr 2522 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( x  i^i  ( RR  \  A ) )  =  ( x  \  A ) )
2120fveq2d 5875 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  ( RR  \  A ) ) )  =  ( vol* `  ( x  \  A ) ) )
22 difin 3740 . . . . . . . . . 10  |-  ( x 
\  ( x  i^i  ( RR  \  A
) ) )  =  ( x  \  ( RR  \  A ) )
2320difeq2d 3627 . . . . . . . . . 10  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( x  \  (
x  i^i  ( RR  \  A ) ) )  =  ( x  \ 
( x  \  A
) ) )
2422, 23syl5eqr 2522 . . . . . . . . 9  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( x  \  ( RR  \  A ) )  =  ( x  \ 
( x  \  A
) ) )
25 dfin4 3743 . . . . . . . . 9  |-  ( x  i^i  A )  =  ( x  \  (
x  \  A )
)
2624, 25syl6eqr 2526 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( x  \  ( RR  \  A ) )  =  ( x  i^i 
A ) )
2726fveq2d 5875 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  \  ( RR  \  A ) ) )  =  ( vol* `  ( x  i^i  A ) ) )
2821, 27oveq12d 6312 . . . . . 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 2518 . . . . 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 1198 . . . 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 474 . . 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 2881 . 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 21782 . 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 664 1  |-  ( A  e.  dom  vol  ->  ( RR  \  A )  e.  dom  vol )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817    \ cdif 3478    i^i cin 3480    C_ wss 3481   ~Pcpw 4015   dom cdm 5004   ` cfv 5593  (class class class)co 6294   RRcr 9501    + caddc 9505   vol*covol 21719   volcvol 21720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579  ax-pre-sup 9580
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-recs 7052  df-rdg 7086  df-er 7321  df-map 7432  df-en 7527  df-dom 7528  df-sdom 7529  df-sup 7911  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-div 10217  df-nn 10547  df-2 10604  df-3 10605  df-n0 10806  df-z 10875  df-uz 11093  df-rp 11231  df-ico 11545  df-icc 11546  df-fz 11683  df-seq 12086  df-exp 12145  df-cj 12907  df-re 12908  df-im 12909  df-sqrt 13043  df-abs 13044  df-ovol 21721  df-vol 21722
This theorem is referenced by:  rembl  21796  inmbl  21797  difmbl  21798  iccmbl  21821  itg2uba  21995  itg2monolem1  22002  itg2cnlem1  22013  itg2cnlem2  22014  dmvlsiga  27922  ftc1anclem5  29989
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