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Theorem ismbl2 22559
Description: From ovolun 22530, 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 22558 . 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 3951 . . . . 5  |-  ( x  e.  ~P RR  ->  x 
C_  RR )
3 simprr 774 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( vol* `  x )  e.  RR )
4 inss1 3643 . . . . . . . . . . . 12  |-  ( x  i^i  A )  C_  x
5 ovolsscl 22517 . . . . . . . . . . . 12  |-  ( ( ( x  i^i  A
)  C_  x  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  A ) )  e.  RR )
64, 5mp3an1 1377 . . . . . . . . . . 11  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  A ) )  e.  RR )
76adantl 473 . . . . . . . . . 10  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( vol* `  ( x  i^i  A
) )  e.  RR )
8 difss 3549 . . . . . . . . . . . 12  |-  ( x 
\  A )  C_  x
9 ovolsscl 22517 . . . . . . . . . . . 12  |-  ( ( ( x  \  A
)  C_  x  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  \  A ) )  e.  RR )
108, 9mp3an1 1377 . . . . . . . . . . 11  |-  ( ( x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  \  A ) )  e.  RR )
1110adantl 473 . . . . . . . . . 10  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( vol* `  ( x  \  A
) )  e.  RR )
127, 11readdcld 9688 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( ( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) )  e.  RR )
133, 12letri3d 9794 . . . . . . . 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 3836 . . . . . . . . . . 11  |-  ( ( x  i^i  A )  u.  ( x  \  A ) )  =  x
1514fveq2i 5882 . . . . . . . . . 10  |-  ( vol* `  ( (
x  i^i  A )  u.  ( x  \  A
) ) )  =  ( vol* `  x )
16 simprl 772 . . . . . . . . . . . 12  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  x  C_  RR )
174, 16syl5ss 3429 . . . . . . . . . . 11  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( x  i^i 
A )  C_  RR )
188, 16syl5ss 3429 . . . . . . . . . . 11  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( x  \  A )  C_  RR )
19 ovolun 22530 . . . . . . . . . . 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 1293 . . . . . . . . . 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 4430 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( vol* `  x )  <_  (
( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) ) )
2221biantrurd 516 . . . . . . . 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 264 . . . . . . 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 626 . . . . . 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 255 . . . . 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 482 . . . 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 2828 . . 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 649 . 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 257 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 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756    \ cdif 3387    u. cun 3388    i^i cin 3389    C_ wss 3390   ~Pcpw 3942   class class class wbr 4395   dom cdm 4839   ` cfv 5589  (class class class)co 6308   RRcr 9556    + caddc 9560    <_ cle 9694   vol*covol 22491   volcvol 22493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-ioo 11664  df-ico 11666  df-icc 11667  df-fz 11811  df-fl 12061  df-seq 12252  df-exp 12311  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-ovol 22494  df-vol 22496
This theorem is referenced by:  nulmbl  22567  nulmbl2  22568  unmbl  22569  ioombl1  22594  uniioombl  22626  ismblfin  32045  ismbl3  37961
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