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Theorem ovolun 20980
Description: The Lebesgue outer measure function is finitely sub-additive. (Unlike the stronger ovoliun 20986, this does not require any choice principles.) (Contributed by Mario Carneiro, 12-Jun-2014.)
Assertion
Ref Expression
ovolun  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  e.  RR )  /\  ( B  C_  RR  /\  ( vol* `  B )  e.  RR ) )  ->  ( vol* `  ( A  u.  B ) )  <_  ( ( vol* `  A )  +  ( vol* `  B ) ) )

Proof of Theorem ovolun
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpll 753 . . . 4  |-  ( ( ( ( A  C_  RR  /\  ( vol* `  A )  e.  RR )  /\  ( B  C_  RR  /\  ( vol* `  B )  e.  RR ) )  /\  x  e.  RR+ )  ->  ( A  C_  RR  /\  ( vol* `  A )  e.  RR ) )
2 simplr 754 . . . 4  |-  ( ( ( ( A  C_  RR  /\  ( vol* `  A )  e.  RR )  /\  ( B  C_  RR  /\  ( vol* `  B )  e.  RR ) )  /\  x  e.  RR+ )  ->  ( B  C_  RR  /\  ( vol* `  B )  e.  RR ) )
3 simpr 461 . . . 4  |-  ( ( ( ( A  C_  RR  /\  ( vol* `  A )  e.  RR )  /\  ( B  C_  RR  /\  ( vol* `  B )  e.  RR ) )  /\  x  e.  RR+ )  ->  x  e.  RR+ )
41, 2, 3ovolunlem2 20979 . . 3  |-  ( ( ( ( A  C_  RR  /\  ( vol* `  A )  e.  RR )  /\  ( B  C_  RR  /\  ( vol* `  B )  e.  RR ) )  /\  x  e.  RR+ )  ->  ( vol* `  ( A  u.  B ) )  <_  ( ( ( vol* `  A
)  +  ( vol* `  B )
)  +  x ) )
54ralrimiva 2797 . 2  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  e.  RR )  /\  ( B  C_  RR  /\  ( vol* `  B )  e.  RR ) )  ->  A. x  e.  RR+  ( vol* `  ( A  u.  B
) )  <_  (
( ( vol* `  A )  +  ( vol* `  B
) )  +  x
) )
6 unss 3528 . . . . . 6  |-  ( ( A  C_  RR  /\  B  C_  RR )  <->  ( A  u.  B )  C_  RR )
76biimpi 194 . . . . 5  |-  ( ( A  C_  RR  /\  B  C_  RR )  ->  ( A  u.  B )  C_  RR )
87ad2ant2r 746 . . . 4  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  e.  RR )  /\  ( B  C_  RR  /\  ( vol* `  B )  e.  RR ) )  ->  ( A  u.  B )  C_  RR )
9 ovolcl 20959 . . . 4  |-  ( ( A  u.  B ) 
C_  RR  ->  ( vol* `  ( A  u.  B ) )  e. 
RR* )
108, 9syl 16 . . 3  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  e.  RR )  /\  ( B  C_  RR  /\  ( vol* `  B )  e.  RR ) )  ->  ( vol* `  ( A  u.  B ) )  e.  RR* )
11 readdcl 9363 . . . 4  |-  ( ( ( vol* `  A )  e.  RR  /\  ( vol* `  B )  e.  RR )  ->  ( ( vol* `  A )  +  ( vol* `  B ) )  e.  RR )
1211ad2ant2l 745 . . 3  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  e.  RR )  /\  ( B  C_  RR  /\  ( vol* `  B )  e.  RR ) )  ->  (
( vol* `  A )  +  ( vol* `  B
) )  e.  RR )
13 xralrple 11173 . . 3  |-  ( ( ( vol* `  ( A  u.  B
) )  e.  RR*  /\  ( ( vol* `  A )  +  ( vol* `  B
) )  e.  RR )  ->  ( ( vol* `  ( A  u.  B ) )  <_ 
( ( vol* `  A )  +  ( vol* `  B
) )  <->  A. x  e.  RR+  ( vol* `  ( A  u.  B
) )  <_  (
( ( vol* `  A )  +  ( vol* `  B
) )  +  x
) ) )
1410, 12, 13syl2anc 661 . 2  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  e.  RR )  /\  ( B  C_  RR  /\  ( vol* `  B )  e.  RR ) )  ->  (
( vol* `  ( A  u.  B
) )  <_  (
( vol* `  A )  +  ( vol* `  B
) )  <->  A. x  e.  RR+  ( vol* `  ( A  u.  B
) )  <_  (
( ( vol* `  A )  +  ( vol* `  B
) )  +  x
) ) )
155, 14mpbird 232 1  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  e.  RR )  /\  ( B  C_  RR  /\  ( vol* `  B )  e.  RR ) )  ->  ( vol* `  ( A  u.  B ) )  <_  ( ( vol* `  A )  +  ( vol* `  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1756   A.wral 2713    u. cun 3324    C_ wss 3326   class class class wbr 4290   ` cfv 5416  (class class class)co 6089   RRcr 9279    + caddc 9283   RR*cxr 9415    <_ cle 9417   RR+crp 10989   vol*covol 20944
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 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-pre-sup 9358
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 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-er 7099  df-map 7214  df-en 7309  df-dom 7310  df-sdom 7311  df-sup 7689  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-2 10378  df-3 10379  df-n0 10578  df-z 10645  df-uz 10860  df-q 10952  df-rp 10990  df-ioo 11302  df-ico 11304  df-fz 11436  df-fl 11640  df-seq 11805  df-exp 11864  df-cj 12586  df-re 12587  df-im 12588  df-sqr 12722  df-abs 12723  df-ovol 20946
This theorem is referenced by:  ovolunnul  20981  ovolfiniun  20982  ismbl2  21008  nulmbl2  21016  unmbl  21017  volun  21024  voliunlem2  21030  uniioombllem3  21063  uniioombllem4  21064  volcn  21084  mblfinlem3  28427  mblfinlem4  28428
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