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Theorem ismbl2 19425
Description: From ovolun 19397, 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 19424 . 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 3809 . . . . 5  |-  ( x  e.  ~P RR  ->  x 
C_  RR )
3 simprr 735 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol * `  x )  e.  RR ) )  ->  ( vol * `  x )  e.  RR )
4 inss1 3563 . . . . . . . . . . . 12  |-  ( x  i^i  A )  C_  x
5 ovolsscl 19384 . . . . . . . . . . . 12  |-  ( ( ( x  i^i  A
)  C_  x  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  i^i  A ) )  e.  RR )
64, 5mp3an1 1267 . . . . . . . . . . 11  |-  ( ( x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  i^i  A ) )  e.  RR )
76adantl 454 . . . . . . . . . 10  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol * `  x )  e.  RR ) )  ->  ( vol * `  ( x  i^i  A
) )  e.  RR )
8 difss 3476 . . . . . . . . . . . 12  |-  ( x 
\  A )  C_  x
9 ovolsscl 19384 . . . . . . . . . . . 12  |-  ( ( ( x  \  A
)  C_  x  /\  x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  \  A ) )  e.  RR )
108, 9mp3an1 1267 . . . . . . . . . . 11  |-  ( ( x  C_  RR  /\  ( vol * `  x )  e.  RR )  -> 
( vol * `  ( x  \  A ) )  e.  RR )
1110adantl 454 . . . . . . . . . 10  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol * `  x )  e.  RR ) )  ->  ( vol * `  ( x  \  A
) )  e.  RR )
127, 11readdcld 9117 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol * `  x )  e.  RR ) )  ->  ( ( vol
* `  ( x  i^i  A ) )  +  ( vol * `  ( x  \  A ) ) )  e.  RR )
133, 12letri3d 9217 . . . . . . . 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 3708 . . . . . . . . . . 11  |-  ( ( x  i^i  A )  u.  ( x  \  A ) )  =  x
1514fveq2i 5733 . . . . . . . . . 10  |-  ( vol
* `  ( (
x  i^i  A )  u.  ( x  \  A
) ) )  =  ( vol * `  x )
16 simprl 734 . . . . . . . . . . . 12  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol * `  x )  e.  RR ) )  ->  x  C_  RR )
174, 16syl5ss 3361 . . . . . . . . . . 11  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol * `  x )  e.  RR ) )  ->  ( x  i^i 
A )  C_  RR )
188, 16syl5ss 3361 . . . . . . . . . . 11  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol * `  x )  e.  RR ) )  ->  ( x  \  A )  C_  RR )
19 ovolun 19397 . . . . . . . . . . 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 1186 . . . . . . . . . 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 4248 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol * `  x )  e.  RR ) )  ->  ( vol * `  x )  <_  (
( vol * `  ( x  i^i  A ) )  +  ( vol
* `  ( x  \  A ) ) ) )
2221biantrurd 496 . . . . . . . 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 249 . . . . . . 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 600 . . . . . 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 240 . . . . 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 462 . . . 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 2723 . . 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 620 . 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 242 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 set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707    \ cdif 3319    u. cun 3320    i^i cin 3321    C_ wss 3322   ~Pcpw 3801   class class class wbr 4214   dom cdm 4880   ` cfv 5456  (class class class)co 6083   RRcr 8991    + caddc 8995    <_ cle 9123   vol *covol 19361   volcvol 19362
This theorem is referenced by:  nulmbl  19432  nulmbl2  19433  unmbl  19434  ioombl1  19458  uniioombl  19483  ismblfin  26249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-q 10577  df-rp 10615  df-ioo 10922  df-ico 10924  df-icc 10925  df-fz 11046  df-fl 11204  df-seq 11326  df-exp 11385  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-ovol 19363  df-vol 19364
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