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Theorem cmmbl 21016
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 3484 . 2  |-  ( A  e.  dom  vol  ->  ( RR  \  A ) 
C_  RR )
2 elpwi 3869 . . . 4  |-  ( x  e.  ~P RR  ->  x 
C_  RR )
3 inss1 3570 . . . . . . . . . 10  |-  ( x  i^i  A )  C_  x
4 ovolsscl 20969 . . . . . . . . . 10  |-  ( ( ( x  i^i  A
)  C_  x  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  A ) )  e.  RR )
53, 4mp3an1 1301 . . . . . . . . 9  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  A ) )  e.  RR )
653adant1 1006 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  A ) )  e.  RR )
76recnd 9412 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  A ) )  e.  CC )
8 difss 3483 . . . . . . . . . 10  |-  ( x 
\  A )  C_  x
9 ovolsscl 20969 . . . . . . . . . 10  |-  ( ( ( x  \  A
)  C_  x  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  \  A ) )  e.  RR )
108, 9mp3an1 1301 . . . . . . . . 9  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  \  A ) )  e.  RR )
11103adant1 1006 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  \  A ) )  e.  RR )
1211recnd 9412 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  \  A ) )  e.  CC )
137, 12addcomd 9571 . . . . . 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 21015 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  x )  =  ( ( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) ) )
15 indifcom 3595 . . . . . . . . 9  |-  ( RR 
i^i  ( x  \  A ) )  =  ( x  i^i  ( RR  \  A ) )
16 simp2 989 . . . . . . . . . . 11  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  ->  x  C_  RR )
1716ssdifssd 3494 . . . . . . . . . 10  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( x  \  A
)  C_  RR )
18 sseqin2 3569 . . . . . . . . . 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 2489 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( x  i^i  ( RR  \  A ) )  =  ( x  \  A ) )
2120fveq2d 5695 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  ( RR  \  A ) ) )  =  ( vol* `  ( x  \  A ) ) )
22 difin 3587 . . . . . . . . . 10  |-  ( x 
\  ( x  i^i  ( RR  \  A
) ) )  =  ( x  \  ( RR  \  A ) )
2320difeq2d 3474 . . . . . . . . . 10  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( x  \  (
x  i^i  ( RR  \  A ) ) )  =  ( x  \ 
( x  \  A
) ) )
2422, 23syl5eqr 2489 . . . . . . . . 9  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( x  \  ( RR  \  A ) )  =  ( x  \ 
( x  \  A
) ) )
25 dfin4 3590 . . . . . . . . 9  |-  ( x  i^i  A )  =  ( x  \  (
x  \  A )
)
2624, 25syl6eqr 2493 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( x  \  ( RR  \  A ) )  =  ( x  i^i 
A ) )
2726fveq2d 5695 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  \  ( RR  \  A ) ) )  =  ( vol* `  ( x  i^i  A ) ) )
2821, 27oveq12d 6109 . . . . . 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 2485 . . . . 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 1189 . . . 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 2799 . 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 21009 . 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 965    = wceq 1369    e. wcel 1756   A.wral 2715    \ cdif 3325    i^i cin 3327    C_ wss 3328   ~Pcpw 3860   dom cdm 4840   ` cfv 5418  (class class class)co 6091   RRcr 9281    + caddc 9285   vol*covol 20946   volcvol 20947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-ico 11306  df-icc 11307  df-fz 11438  df-seq 11807  df-exp 11866  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-ovol 20948  df-vol 20949
This theorem is referenced by:  rembl  21022  inmbl  21023  difmbl  21024  iccmbl  21047  itg2uba  21221  itg2monolem1  21228  itg2cnlem1  21239  itg2cnlem2  21240  dmvlsiga  26572  ftc1anclem5  28471
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