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Theorem nulmbl 18909
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 443 . 2  |-  ( ( A  C_  RR  /\  ( vol * `  A )  =  0 )  ->  A  C_  RR )
2 elpwi 3646 . . . 4  |-  ( x  e.  ~P RR  ->  x 
C_  RR )
3 inss2 3403 . . . . . . . . . 10  |-  ( x  i^i  A )  C_  A
4 ovolssnul 18862 . . . . . . . . . 10  |-  ( ( ( x  i^i  A
)  C_  A  /\  A  C_  RR  /\  ( vol * `  A )  =  0 )  -> 
( vol * `  ( x  i^i  A ) )  =  0 )
53, 4mp3an1 1264 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  ( vol * `  A )  =  0 )  -> 
( vol * `  ( x  i^i  A ) )  =  0 )
65adantr 451 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  ( vol * `  A )  =  0 )  /\  ( x 
C_  RR  /\  ( vol * `  x )  e.  RR ) )  ->  ( vol * `  ( x  i^i  A
) )  =  0 )
76oveq1d 5889 . . . . . . 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 3316 . . . . . . . . . . 11  |-  ( x 
\  A )  C_  x
9 ovolsscl 18861 . . . . . . . . . . 11  |-  ( ( ( x  \  A
)  C_  x  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  \  A ) )  e.  RR )
108, 9mp3an1 1264 . . . . . . . . . 10  |-  ( ( x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  \  A ) )  e.  RR )
1110adantl 452 . . . . . . . . 9  |-  ( ( ( A  C_  RR  /\  ( vol * `  A )  =  0 )  /\  ( x 
C_  RR  /\  ( vol * `  x )  e.  RR ) )  ->  ( vol * `  ( x  \  A
) )  e.  RR )
1211recnd 8877 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  ( vol * `  A )  =  0 )  /\  ( x 
C_  RR  /\  ( vol * `  x )  e.  RR ) )  ->  ( vol * `  ( x  \  A
) )  e.  CC )
1312addid2d 9029 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  ( vol * `  A )  =  0 )  /\  ( x 
C_  RR  /\  ( vol * `  x )  e.  RR ) )  ->  ( 0  +  ( vol * `  ( x  \  A ) ) )  =  ( vol * `  (
x  \  A )
) )
147, 13eqtrd 2328 . . . . . 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 732 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  ( vol * `  A )  =  0 )  /\  ( x 
C_  RR  /\  ( vol * `  x )  e.  RR ) )  ->  x  C_  RR )
16 ovolss 18860 . . . . . . 7  |-  ( ( ( x  \  A
)  C_  x  /\  x  C_  RR )  -> 
( vol * `  ( x  \  A ) )  <_  ( vol * `
 x ) )
178, 15, 16sylancr 644 . . . . . 6  |-  ( ( ( A  C_  RR  /\  ( vol * `  A )  =  0 )  /\  ( x 
C_  RR  /\  ( vol * `  x )  e.  RR ) )  ->  ( vol * `  ( x  \  A
) )  <_  ( vol * `  x ) )
1814, 17eqbrtrd 4059 . . . . 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 598 . . . 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 460 . . 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 2639 . 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 18902 . 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 645 1  |-  ( ( A  C_  RR  /\  ( vol * `  A )  =  0 )  ->  A  e.  dom  vol )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    \ cdif 3162    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   class class class wbr 4039   dom cdm 4705   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753    + caddc 8756    <_ cle 8884   vol *covol 18838   volcvol 18839
This theorem is referenced by:  0mbl  18913  icombl1  18936  ioombl  18938  ovolioo  18941  uniiccmbl  18961  volivth  18978  mbfeqalem  19013  itg10a  19081  itg2uba  19114  itgss3  19185  cntnevol  23191
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-ioo 10676  df-ico 10678  df-icc 10679  df-fz 10799  df-fl 10941  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-ovol 18840  df-vol 18841
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