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Theorem nulmbl 22031
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 455 . 2  |-  ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  ->  A  C_  RR )
2 elpwi 3936 . . . 4  |-  ( x  e.  ~P RR  ->  x 
C_  RR )
3 inss2 3633 . . . . . . . . . 10  |-  ( x  i^i  A )  C_  A
4 ovolssnul 21983 . . . . . . . . . 10  |-  ( ( ( x  i^i  A
)  C_  A  /\  A  C_  RR  /\  ( vol* `  A )  =  0 )  -> 
( vol* `  ( x  i^i  A ) )  =  0 )
53, 4mp3an1 1309 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  -> 
( vol* `  ( x  i^i  A ) )  =  0 )
65adantr 463 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( vol* `  ( x  i^i  A
) )  =  0 )
76oveq1d 6211 . . . . . . 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 3545 . . . . . . . . . . 11  |-  ( x 
\  A )  C_  x
9 ovolsscl 21982 . . . . . . . . . . 11  |-  ( ( ( x  \  A
)  C_  x  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  \  A ) )  e.  RR )
108, 9mp3an1 1309 . . . . . . . . . 10  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  \  A ) )  e.  RR )
1110adantl 464 . . . . . . . . 9  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( vol* `  ( x  \  A
) )  e.  RR )
1211recnd 9533 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( vol* `  ( x  \  A
) )  e.  CC )
1312addid2d 9692 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( 0  +  ( vol* `  ( x  \  A ) ) )  =  ( vol* `  (
x  \  A )
) )
147, 13eqtrd 2423 . . . . . 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 754 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  x  C_  RR )
16 ovolss 21981 . . . . . . 7  |-  ( ( ( x  \  A
)  C_  x  /\  x  C_  RR )  -> 
( vol* `  ( x  \  A ) )  <_  ( vol* `  x ) )
178, 15, 16sylancr 661 . . . . . 6  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  /\  ( x 
C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( vol* `  ( x  \  A
) )  <_  ( vol* `  x ) )
1814, 17eqbrtrd 4387 . . . . 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 613 . . . 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 472 . . 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 2796 . 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 22023 . 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 662 1  |-  ( ( A  C_  RR  /\  ( vol* `  A )  =  0 )  ->  A  e.  dom  vol )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826   A.wral 2732    \ cdif 3386    i^i cin 3388    C_ wss 3389   ~Pcpw 3927   class class class wbr 4367   dom cdm 4913   ` cfv 5496  (class class class)co 6196   RRcr 9402   0cc0 9403    + caddc 9406    <_ cle 9540   vol*covol 21959   volcvol 21960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-map 7340  df-en 7436  df-dom 7437  df-sdom 7438  df-sup 7816  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-n0 10713  df-z 10782  df-uz 11002  df-q 11102  df-rp 11140  df-ioo 11454  df-ico 11456  df-icc 11457  df-fz 11594  df-fl 11828  df-seq 12011  df-exp 12070  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-ovol 21961  df-vol 21962
This theorem is referenced by:  0mbl  22035  icombl1  22058  ioombl  22060  ovolioo  22063  uniiccmbl  22084  volivth  22101  mbfeqalem  22134  itg10a  22202  itg2uba  22235  itgss3  22306  cntnevol  28355  voliunnfl  30223  volsupnfl  30224  cnambfre  30228  snmbl  31928
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