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Theorem ismbl2 22473
Description: From ovolun 22444, it suffices to show that the measure of  x is at least the sum of the measures of  x  i^i  A and  x  \  A. (Contributed by Mario Carneiro, 15-Jun-2014.)
Assertion
Ref Expression
ismbl2  |-  ( 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 ) ) ) )
Distinct variable group:    x, A

Proof of Theorem ismbl2
StepHypRef Expression
1 ismbl 22472 . 2  |-  ( A  e.  dom  vol  <->  ( A  C_  RR  /\  A. x  e.  ~P  RR ( ( vol* `  x
)  e.  RR  ->  ( vol* `  x
)  =  ( ( vol* `  (
x  i^i  A )
)  +  ( vol* `  ( x  \  A ) ) ) ) ) )
2 elpwi 3989 . . . . 5  |-  ( x  e.  ~P RR  ->  x 
C_  RR )
3 simprr 765 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( vol* `  x )  e.  RR )
4 inss1 3683 . . . . . . . . . . . 12  |-  ( x  i^i  A )  C_  x
5 ovolsscl 22431 . . . . . . . . . . . 12  |-  ( ( ( x  i^i  A
)  C_  x  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  A ) )  e.  RR )
64, 5mp3an1 1348 . . . . . . . . . . 11  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  A ) )  e.  RR )
76adantl 468 . . . . . . . . . 10  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( vol* `  ( x  i^i  A
) )  e.  RR )
8 difss 3593 . . . . . . . . . . . 12  |-  ( x 
\  A )  C_  x
9 ovolsscl 22431 . . . . . . . . . . . 12  |-  ( ( ( x  \  A
)  C_  x  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  \  A ) )  e.  RR )
108, 9mp3an1 1348 . . . . . . . . . . 11  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  \  A ) )  e.  RR )
1110adantl 468 . . . . . . . . . 10  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( vol* `  ( x  \  A
) )  e.  RR )
127, 11readdcld 9672 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( ( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) )  e.  RR )
133, 12letri3d 9779 . . . . . . . 8  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( ( vol* `  x )  =  ( ( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) )  <->  ( ( vol* `  x )  <_  ( ( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) )  /\  (
( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) )  <_  ( vol* `  x ) ) ) )
14 inundif 3874 . . . . . . . . . . 11  |-  ( ( x  i^i  A )  u.  ( x  \  A ) )  =  x
1514fveq2i 5882 . . . . . . . . . 10  |-  ( vol* `  ( (
x  i^i  A )  u.  ( x  \  A
) ) )  =  ( vol* `  x )
16 simprl 763 . . . . . . . . . . . 12  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  x  C_  RR )
174, 16syl5ss 3476 . . . . . . . . . . 11  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( x  i^i 
A )  C_  RR )
188, 16syl5ss 3476 . . . . . . . . . . 11  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( x  \  A )  C_  RR )
19 ovolun 22444 . . . . . . . . . . 11  |-  ( ( ( ( x  i^i 
A )  C_  RR  /\  ( vol* `  ( x  i^i  A ) )  e.  RR )  /\  ( ( x 
\  A )  C_  RR  /\  ( vol* `  ( x  \  A
) )  e.  RR ) )  ->  ( vol* `  ( ( x  i^i  A )  u.  ( x  \  A ) ) )  <_  ( ( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) ) )
2017, 7, 18, 11, 19syl22anc 1266 . . . . . . . . . 10  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( vol* `  ( ( x  i^i 
A )  u.  (
x  \  A )
) )  <_  (
( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) ) )
2115, 20syl5eqbrr 4456 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( vol* `  x )  <_  (
( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) ) )
2221biantrurd 511 . . . . . . . 8  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( ( ( vol* `  (
x  i^i  A )
)  +  ( vol* `  ( x  \  A ) ) )  <_  ( vol* `  x )  <->  ( ( vol* `  x )  <_  ( ( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) )  /\  (
( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) )  <_  ( vol* `  x ) ) ) )
2313, 22bitr4d 260 . . . . . . 7  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( ( vol* `  x )  =  ( ( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) )  <->  ( ( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A
) ) )  <_ 
( vol* `  x ) ) )
2423expr 619 . . . . . 6  |-  ( ( A  C_  RR  /\  x  C_  RR )  ->  (
( vol* `  x )  e.  RR  ->  ( ( vol* `  x )  =  ( ( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) )  <-> 
( ( vol* `  ( x  i^i  A
) )  +  ( vol* `  (
x  \  A )
) )  <_  ( vol* `  x ) ) ) )
2524pm5.74d 251 . . . . 5  |-  ( ( A  C_  RR  /\  x  C_  RR )  ->  (
( ( vol* `  x )  e.  RR  ->  ( vol* `  x )  =  ( ( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) ) )  <->  ( ( vol* `  x )  e.  RR  ->  ( ( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A
) ) )  <_ 
( vol* `  x ) ) ) )
262, 25sylan2 477 . . . 4  |-  ( ( A  C_  RR  /\  x  e.  ~P RR )  -> 
( ( ( vol* `  x )  e.  RR  ->  ( vol* `  x )  =  ( ( vol* `  ( x  i^i  A
) )  +  ( vol* `  (
x  \  A )
) ) )  <->  ( ( vol* `  x )  e.  RR  ->  (
( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) )  <_  ( vol* `  x ) ) ) )
2726ralbidva 2862 . . 3  |-  ( A 
C_  RR  ->  ( A. x  e.  ~P  RR ( ( vol* `  x )  e.  RR  ->  ( vol* `  x )  =  ( ( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) ) )  <->  A. x  e.  ~P  RR ( ( vol* `  x )  e.  RR  ->  ( ( vol* `  ( x  i^i  A
) )  +  ( vol* `  (
x  \  A )
) )  <_  ( vol* `  x ) ) ) )
2827pm5.32i 642 . 2  |-  ( ( A  C_  RR  /\  A. x  e.  ~P  RR ( ( vol* `  x )  e.  RR  ->  ( vol* `  x )  =  ( ( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) ) ) )  <->  ( A  C_  RR  /\  A. x  e.  ~P  RR ( ( vol* `  x
)  e.  RR  ->  ( ( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) )  <_  ( vol* `  x ) ) ) )
291, 28bitri 253 1  |-  ( 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 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1438    e. wcel 1869   A.wral 2776    \ cdif 3434    u. cun 3435    i^i cin 3436    C_ wss 3437   ~Pcpw 3980   class class class wbr 4421   dom cdm 4851   ` cfv 5599  (class class class)co 6303   RRcr 9540    + caddc 9544    <_ cle 9678   vol*covol 22405   volcvol 22407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-pre-sup 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-er 7369  df-map 7480  df-en 7576  df-dom 7577  df-sdom 7578  df-sup 7960  df-inf 7961  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-2 10670  df-3 10671  df-n0 10872  df-z 10940  df-uz 11162  df-q 11267  df-rp 11305  df-ioo 11641  df-ico 11643  df-icc 11644  df-fz 11787  df-fl 12029  df-seq 12215  df-exp 12274  df-cj 13156  df-re 13157  df-im 13158  df-sqrt 13292  df-abs 13293  df-ovol 22408  df-vol 22410
This theorem is referenced by:  nulmbl  22481  nulmbl2  22482  unmbl  22483  ioombl1  22507  uniioombl  22539  ismblfin  31901
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