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Theorem ismbl2 21145
Description: From ovolun 21117, 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 21144 . 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 3980 . . . . 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 3681 . . . . . . . . . . . 12  |-  ( x  i^i  A )  C_  x
5 ovolsscl 21104 . . . . . . . . . . . 12  |-  ( ( ( x  i^i  A
)  C_  x  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  i^i  A ) )  e.  RR )
64, 5mp3an1 1302 . . . . . . . . . . 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 3594 . . . . . . . . . . . 12  |-  ( x 
\  A )  C_  x
9 ovolsscl 21104 . . . . . . . . . . . 12  |-  ( ( ( x  \  A
)  C_  x  /\  x  C_  RR  /\  ( vol* `  x )  e.  RR )  -> 
( vol* `  ( x  \  A ) )  e.  RR )
108, 9mp3an1 1302 . . . . . . . . . . 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 9527 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( ( vol* `  ( x  i^i  A ) )  +  ( vol* `  ( x  \  A ) ) )  e.  RR )
133, 12letri3d 9630 . . . . . . . 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 3868 . . . . . . . . . . 11  |-  ( ( x  i^i  A )  u.  ( x  \  A ) )  =  x
1514fveq2i 5805 . . . . . . . . . 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 3478 . . . . . . . . . . 11  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( x  i^i 
A )  C_  RR )
188, 16syl5ss 3478 . . . . . . . . . . 11  |-  ( ( A  C_  RR  /\  (
x  C_  RR  /\  ( vol* `  x )  e.  RR ) )  ->  ( x  \  A )  C_  RR )
19 ovolun 21117 . . . . . . . . . . 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 1220 . . . . . . . . . 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 4437 . . . . . . . . 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 2844 . . 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 1370    e. wcel 1758   A.wral 2799    \ cdif 3436    u. cun 3437    i^i cin 3438    C_ wss 3439   ~Pcpw 3971   class class class wbr 4403   dom cdm 4951   ` cfv 5529  (class class class)co 6203   RRcr 9395    + caddc 9399    <_ cle 9533   vol*covol 21081   volcvol 21082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-sup 7805  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-n0 10694  df-z 10761  df-uz 10976  df-q 11068  df-rp 11106  df-ioo 11418  df-ico 11420  df-icc 11421  df-fz 11558  df-fl 11762  df-seq 11927  df-exp 11986  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846  df-ovol 21083  df-vol 21084
This theorem is referenced by:  nulmbl  21153  nulmbl2  21154  unmbl  21155  ioombl1  21179  uniioombl  21205  ismblfin  28600
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