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Theorem ovolun 22035
Description: The Lebesgue outer measure function is finitely sub-additive. (Unlike the stronger ovoliun 22041, 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 755 . . . 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 22034 . . 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 2871 . 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 3674 . . . . . 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 22014 . . . 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 9592 . . . 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 11429 . . 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 1819   A.wral 2807    u. cun 3469    C_ wss 3471   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   RRcr 9508    + caddc 9512   RR*cxr 9644    <_ cle 9646   RR+crp 11245   vol*covol 21999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-ioo 11558  df-ico 11560  df-fz 11698  df-fl 11931  df-seq 12110  df-exp 12169  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-ovol 22001
This theorem is referenced by:  ovolunnul  22036  ovolfiniun  22037  ismbl2  22063  nulmbl2  22072  unmbl  22073  volun  22080  voliunlem2  22086  uniioombllem3  22119  uniioombllem4  22120  volcn  22140  mblfinlem3  30215  mblfinlem4  30216
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