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Theorem nulmbl 21145
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 457 . 2  |-  ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  ->  A  C_  RR )
2 elpwi 3972 . . . 4  |-  ( x  e.  ~P RR  ->  x 
C_  RR )
3 inss2 3674 . . . . . . . . . 10  |-  ( x  i^i  A )  C_  A
4 ovolssnul 21097 . . . . . . . . . 10  |-  ( ( ( x  i^i  A
)  C_  A  /\  A  C_  RR  /\  ( vol* `  A )  =  0 )  -> 
( vol* `  ( x  i^i  A ) )  =  0 )
53, 4mp3an1 1302 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  -> 
( vol* `  ( x  i^i  A ) )  =  0 )
65adantr 465 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( vol* `  ( x  i^i  A
) )  =  0 )
76oveq1d 6210 . . . . . . 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 3586 . . . . . . . . . . 11  |-  ( x 
\  A )  C_  x
9 ovolsscl 21096 . . . . . . . . . . 11  |-  ( ( ( x  \  A
)  C_  x  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  \  A ) )  e.  RR )
108, 9mp3an1 1302 . . . . . . . . . 10  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  \  A ) )  e.  RR )
1110adantl 466 . . . . . . . . 9  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( vol* `  ( x  \  A
) )  e.  RR )
1211recnd 9518 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( vol* `  ( x  \  A
) )  e.  CC )
1312addid2d 9676 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( 0  +  ( vol* `  ( x  \  A ) ) )  =  ( vol* `  (
x  \  A )
) )
147, 13eqtrd 2493 . . . . . 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 755 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  x  C_  RR )
16 ovolss 21095 . . . . . . 7  |-  ( ( ( x  \  A
)  C_  x  /\  x  C_  RR )  -> 
( vol* `  ( x  \  A ) )  <_  ( vol* `  x ) )
178, 15, 16sylancr 663 . . . . . 6  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( vol* `  ( x  \  A
) )  <_  ( vol* `  x ) )
1814, 17eqbrtrd 4415 . . . . 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 615 . . . 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 474 . . 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 2827 . 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 21137 . 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 664 1  |-  ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  ->  A  e.  dom  vol )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2796    \ cdif 3428    i^i cin 3430    C_ wss 3431   ~Pcpw 3963   class class class wbr 4395   dom cdm 4943   ` cfv 5521  (class class class)co 6195   RRcr 9387   0cc0 9388    + caddc 9391    <_ cle 9525   vol*covol 21073   volcvol 21074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-er 7206  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-sup 7797  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-n0 10686  df-z 10753  df-uz 10968  df-q 11060  df-rp 11098  df-ioo 11410  df-ico 11412  df-icc 11413  df-fz 11550  df-fl 11754  df-seq 11919  df-exp 11978  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837  df-abs 12838  df-ovol 21075  df-vol 21076
This theorem is referenced by:  0mbl  21149  icombl1  21172  ioombl  21174  ovolioo  21177  uniiccmbl  21198  volivth  21215  mbfeqalem  21248  itg10a  21316  itg2uba  21349  itgss3  21420  cntnevol  26782  voliunnfl  28578  volsupnfl  28579  cnambfre  28583
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