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Theorem cmmbl 19382
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 3435 . 2  |-  ( A  e.  dom  vol  ->  ( RR  \  A ) 
C_  RR )
2 elpwi 3767 . . . 4  |-  ( x  e.  ~P RR  ->  x 
C_  RR )
3 inss1 3521 . . . . . . . . . 10  |-  ( x  i^i  A )  C_  x
4 ovolsscl 19335 . . . . . . . . . 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 9070 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  i^i  A ) )  e.  CC )
8 difss 3434 . . . . . . . . . 10  |-  ( x 
\  A )  C_  x
9 ovolsscl 19335 . . . . . . . . . 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 9070 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  \  A ) )  e.  CC )
137, 12addcomd 9224 . . . . . 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 19381 . . . . . 6  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  x )  =  ( ( vol * `  ( x  i^i  A ) )  +  ( vol
* `  ( x  \  A ) ) ) )
15 indifcom 3546 . . . . . . . . 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 3445 . . . . . . . . . 10  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( x  \  A
)  C_  RR )
18 sseqin2 3520 . . . . . . . . . 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 2450 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( x  i^i  ( RR  \  A ) )  =  ( x  \  A ) )
2120fveq2d 5691 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  i^i  ( RR  \  A ) ) )  =  ( vol
* `  ( x  \  A ) ) )
22 difin 3538 . . . . . . . . . 10  |-  ( x 
\  ( x  i^i  ( RR  \  A
) ) )  =  ( x  \  ( RR  \  A ) )
2320difeq2d 3425 . . . . . . . . . 10  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( x  \  (
x  i^i  ( RR  \  A ) ) )  =  ( x  \ 
( x  \  A
) ) )
2422, 23syl5eqr 2450 . . . . . . . . 9  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( x  \  ( RR  \  A ) )  =  ( x  \ 
( x  \  A
) ) )
25 dfin4 3541 . . . . . . . . 9  |-  ( x  i^i  A )  =  ( x  \  (
x  \  A )
)
2624, 25syl6eqr 2454 . . . . . . . 8  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( x  \  ( RR  \  A ) )  =  ( x  i^i 
A ) )
2726fveq2d 5691 . . . . . . 7  |-  ( ( A  e.  dom  vol  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  \  ( RR  \  A ) ) )  =  ( vol
* `  ( x  i^i  A ) ) )
2821, 27oveq12d 6058 . . . . . 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 2446 . . . . 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 2749 . 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 19375 . 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 1721   A.wral 2666    \ cdif 3277    i^i cin 3279    C_ wss 3280   ~Pcpw 3759   dom cdm 4837   ` cfv 5413  (class class class)co 6040   RRcr 8945    + caddc 8949   vol *covol 19312   volcvol 19313
This theorem is referenced by:  rembl  19388  inmbl  19389  difmbl  19390  iccmbl  19413  itg2uba  19588  itg2monolem1  19595  itg2cnlem1  19606  itg2cnlem2  19607  dmvlsiga  24465
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-ico 10878  df-icc 10879  df-fz 11000  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-ovol 19314  df-vol 19315
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