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Theorem ismbl2 20985
Description: From ovolun 20957, 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 20984 . 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 3864 . . . . 5  |-  ( x  e.  ~P RR  ->  x 
C_  RR )
3 simprr 756 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( vol* `  x )  e.  RR )
4 inss1 3565 . . . . . . . . . . . 12  |-  ( x  i^i  A )  C_  x
5 ovolsscl 20944 . . . . . . . . . . . 12  |-  ( ( ( x  i^i  A
)  C_  x  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  A ) )  e.  RR )
64, 5mp3an1 1301 . . . . . . . . . . 11  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  A ) )  e.  RR )
76adantl 466 . . . . . . . . . 10  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( vol* `  ( x  i^i  A
) )  e.  RR )
8 difss 3478 . . . . . . . . . . . 12  |-  ( x 
\  A )  C_  x
9 ovolsscl 20944 . . . . . . . . . . . 12  |-  ( ( ( x  \  A
)  C_  x  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  \  A ) )  e.  RR )
108, 9mp3an1 1301 . . . . . . . . . . 11  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  \  A ) )  e.  RR )
1110adantl 466 . . . . . . . . . 10  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( vol* `  ( x  \  A
) )  e.  RR )
127, 11readdcld 9405 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( ( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) )  e.  RR )
133, 12letri3d 9508 . . . . . . . 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 3752 . . . . . . . . . . 11  |-  ( ( x  i^i  A )  u.  ( x  \  A ) )  =  x
1514fveq2i 5689 . . . . . . . . . 10  |-  ( vol* `  ( (
x  i^i  A )  u.  ( x  \  A
) ) )  =  ( vol* `  x )
16 simprl 755 . . . . . . . . . . . 12  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  x  C_  RR )
174, 16syl5ss 3362 . . . . . . . . . . 11  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( x  i^i 
A )  C_  RR )
188, 16syl5ss 3362 . . . . . . . . . . 11  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( x  \  A )  C_  RR )
19 ovolun 20957 . . . . . . . . . . 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 1219 . . . . . . . . . 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 4321 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( vol* `  x )  <_  (
( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) ) )
2221biantrurd 508 . . . . . . . 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 256 . . . . . . 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 615 . . . . . 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 247 . . . . 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 474 . . . 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 2726 . . 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 637 . 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 249 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 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2710    \ cdif 3320    u. cun 3321    i^i cin 3322    C_ wss 3323   ~Pcpw 3855   class class class wbr 4287   dom cdm 4835   ` cfv 5413  (class class class)co 6086   RRcr 9273    + caddc 9277    <_ cle 9411   vol*covol 20921   volcvol 20922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-q 10946  df-rp 10984  df-ioo 11296  df-ico 11298  df-icc 11299  df-fz 11430  df-fl 11634  df-seq 11799  df-exp 11858  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-ovol 20923  df-vol 20924
This theorem is referenced by:  nulmbl  20992  nulmbl2  20993  unmbl  20994  ioombl1  21018  uniioombl  21044  ismblfin  28385
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