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Theorem nulmbl 20917
Description: A nullset is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.)
Assertion
Ref Expression
nulmbl  |-  ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  ->  A  e.  dom  vol )

Proof of Theorem nulmbl
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpl 454 . 2  |-  ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  ->  A  C_  RR )
2 elpwi 3866 . . . 4  |-  ( x  e.  ~P RR  ->  x 
C_  RR )
3 inss2 3568 . . . . . . . . . 10  |-  ( x  i^i  A )  C_  A
4 ovolssnul 20870 . . . . . . . . . 10  |-  ( ( ( x  i^i  A
)  C_  A  /\  A  C_  RR  /\  ( vol* `  A )  =  0 )  -> 
( vol* `  ( x  i^i  A ) )  =  0 )
53, 4mp3an1 1296 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  -> 
( vol* `  ( x  i^i  A ) )  =  0 )
65adantr 462 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( vol* `  ( x  i^i  A
) )  =  0 )
76oveq1d 6105 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( ( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) )  =  ( 0  +  ( vol* `  ( x  \  A ) ) ) )
8 difss 3480 . . . . . . . . . . 11  |-  ( x 
\  A )  C_  x
9 ovolsscl 20869 . . . . . . . . . . 11  |-  ( ( ( x  \  A
)  C_  x  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  \  A ) )  e.  RR )
108, 9mp3an1 1296 . . . . . . . . . 10  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  \  A ) )  e.  RR )
1110adantl 463 . . . . . . . . 9  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( vol* `  ( x  \  A
) )  e.  RR )
1211recnd 9408 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( vol* `  ( x  \  A
) )  e.  CC )
1312addid2d 9566 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( 0  +  ( vol* `  ( x  \  A ) ) )  =  ( vol* `  (
x  \  A )
) )
147, 13eqtrd 2473 . . . . . 6  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( ( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) )  =  ( vol* `  (
x  \  A )
) )
15 simprl 750 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  x  C_  RR )
16 ovolss 20868 . . . . . . 7  |-  ( ( ( x  \  A
)  C_  x  /\  x  C_  RR )  -> 
( vol* `  ( x  \  A ) )  <_  ( vol* `  x ) )
178, 15, 16sylancr 658 . . . . . 6  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( vol* `  ( x  \  A
) )  <_  ( vol* `  x ) )
1814, 17eqbrtrd 4309 . . . . 5  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( ( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) )  <_  ( vol* `  x ) )
1918expr 612 . . . 4  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  /\  x  C_  RR )  ->  ( ( vol* `  x
)  e.  RR  ->  ( ( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) )  <_  ( vol* `  x ) ) )
202, 19sylan2 471 . . 3  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  /\  x  e. 
~P RR )  -> 
( ( vol* `  x )  e.  RR  ->  ( ( vol* `  ( x  i^i  A
) )  +  ( vol* `  (
x  \  A )
) )  <_  ( vol* `  x ) ) )
2120ralrimiva 2797 . 2  |-  ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  ->  A. x  e.  ~P  RR ( ( vol* `  x )  e.  RR  ->  ( ( vol* `  ( x  i^i  A
) )  +  ( vol* `  (
x  \  A )
) )  <_  ( vol* `  x ) ) )
22 ismbl2 20910 . 2  |-  ( A  e.  dom  vol  <->  ( A  C_  RR  /\  A. x  e.  ~P  RR ( ( vol* `  x
)  e.  RR  ->  ( ( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) )  <_  ( vol* `  x ) ) ) )
231, 21, 22sylanbrc 659 1  |-  ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  ->  A  e.  dom  vol )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713    \ cdif 3322    i^i cin 3324    C_ wss 3325   ~Pcpw 3857   class class class wbr 4289   dom cdm 4836   ` cfv 5415  (class class class)co 6090   RRcr 9277   0cc0 9278    + caddc 9281    <_ cle 9415   vol*covol 20846   volcvol 20847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-q 10950  df-rp 10988  df-ioo 11300  df-ico 11302  df-icc 11303  df-fz 11434  df-fl 11638  df-seq 11803  df-exp 11862  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-ovol 20848  df-vol 20849
This theorem is referenced by:  0mbl  20921  icombl1  20944  ioombl  20946  ovolioo  20949  uniiccmbl  20970  volivth  20987  mbfeqalem  21020  itg10a  21088  itg2uba  21121  itgss3  21192  cntnevol  26562  voliunnfl  28344  volsupnfl  28345  cnambfre  28349
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