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Theorem ovolun 22501
Description: The Lebesgue outer measure function is finitely sub-additive. (Unlike the stronger ovoliun 22507, 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 765 . . . 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 767 . . . 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 467 . . . 4  |-  ( ( ( ( A  C_  RR  /\  ( vol* `  A )  e.  RR )  /\  ( B  C_  RR  /\  ( vol* `  B )  e.  RR ) )  /\  x  e.  RR+ )  ->  x  e.  RR+ )
41, 2, 3ovolunlem2 22500 . . 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 2814 . 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 3620 . . . . . 6  |-  ( ( A  C_  RR  /\  B  C_  RR )  <->  ( A  u.  B )  C_  RR )
76biimpi 199 . . . . 5  |-  ( ( A  C_  RR  /\  B  C_  RR )  ->  ( A  u.  B )  C_  RR )
87ad2ant2r 758 . . . 4  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  e.  RR )  /\  ( B  C_  RR  /\  ( vol* `  B )  e.  RR ) )  ->  ( A  u.  B )  C_  RR )
9 ovolcl 22480 . . . 4  |-  ( ( A  u.  B ) 
C_  RR  ->  ( vol* `  ( A  u.  B ) )  e. 
RR* )
108, 9syl 17 . . 3  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  e.  RR )  /\  ( B  C_  RR  /\  ( vol* `  B )  e.  RR ) )  ->  ( vol* `  ( A  u.  B ) )  e.  RR* )
11 readdcl 9648 . . . 4  |-  ( ( ( vol* `  A )  e.  RR  /\  ( vol* `  B )  e.  RR )  ->  ( ( vol* `  A )  +  ( vol* `  B ) )  e.  RR )
1211ad2ant2l 757 . . 3  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  e.  RR )  /\  ( B  C_  RR  /\  ( vol* `  B )  e.  RR ) )  ->  (
( vol* `  A )  +  ( vol* `  B
) )  e.  RR )
13 xralrple 11527 . . 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 671 . 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 240 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 189    /\ wa 375    e. wcel 1898   A.wral 2749    u. cun 3414    C_ wss 3416   class class class wbr 4416   ` cfv 5601  (class class class)co 6315   RRcr 9564    + caddc 9568   RR*cxr 9700    <_ cle 9702   RR+crp 11331   vol*covol 22462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642  ax-pre-sup 9643
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-1st 6820  df-2nd 6821  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-er 7389  df-map 7500  df-en 7596  df-dom 7597  df-sdom 7598  df-sup 7982  df-inf 7983  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-div 10298  df-nn 10638  df-2 10696  df-3 10697  df-n0 10899  df-z 10967  df-uz 11189  df-q 11294  df-rp 11332  df-ioo 11668  df-ico 11670  df-fz 11814  df-fl 12060  df-seq 12246  df-exp 12305  df-cj 13211  df-re 13212  df-im 13213  df-sqrt 13347  df-abs 13348  df-ovol 22465
This theorem is referenced by:  ovolunnul  22502  ovolfiniun  22503  ismbl2  22530  nulmbl2  22539  unmbl  22540  volun  22547  voliunlem2  22553  uniioombllem3  22592  uniioombllem4  22593  volcn  22613  mblfinlem3  32024  mblfinlem4  32025
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