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Theorem ovolunnul 21005
Description: Adding a nullset does not change the measure of a set. (Contributed by Mario Carneiro, 25-Mar-2015.)
Assertion
Ref Expression
ovolunnul  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  -> 
( vol* `  ( A  u.  B
) )  =  ( vol* `  A
) )

Proof of Theorem ovolunnul
StepHypRef Expression
1 ovolcl 20983 . . . . . 6  |-  ( A 
C_  RR  ->  ( vol* `  A )  e.  RR* )
213ad2ant1 1009 . . . . 5  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  -> 
( vol* `  A )  e.  RR* )
3 simp1 988 . . . . . . 7  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  ->  A  C_  RR )
4 simp2 989 . . . . . . 7  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  ->  B  C_  RR )
53, 4unssd 3553 . . . . . 6  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  -> 
( A  u.  B
)  C_  RR )
6 ovolcl 20983 . . . . . 6  |-  ( ( A  u.  B ) 
C_  RR  ->  ( vol* `  ( A  u.  B ) )  e. 
RR* )
75, 6syl 16 . . . . 5  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  -> 
( vol* `  ( A  u.  B
) )  e.  RR* )
8 xrltnle 9464 . . . . 5  |-  ( ( ( vol* `  A )  e.  RR*  /\  ( vol* `  ( A  u.  B
) )  e.  RR* )  ->  ( ( vol* `  A )  <  ( vol* `  ( A  u.  B
) )  <->  -.  ( vol* `  ( A  u.  B ) )  <_  ( vol* `  A ) ) )
92, 7, 8syl2anc 661 . . . 4  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  -> 
( ( vol* `  A )  <  ( vol* `  ( A  u.  B ) )  <->  -.  ( vol* `  ( A  u.  B
) )  <_  ( vol* `  A ) ) )
103adantr 465 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  /\  ( vol* `  A
)  <  ( vol* `  ( A  u.  B ) ) )  ->  A  C_  RR )
11 mnfxr 11115 . . . . . . . . 9  |- -oo  e.  RR*
1211a1i 11 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  /\  ( vol* `  A
)  <  ( vol* `  ( A  u.  B ) ) )  -> -oo  e.  RR* )
1310, 1syl 16 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  /\  ( vol* `  A
)  <  ( vol* `  ( A  u.  B ) ) )  ->  ( vol* `  A )  e.  RR* )
147adantr 465 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  /\  ( vol* `  A
)  <  ( vol* `  ( A  u.  B ) ) )  ->  ( vol* `  ( A  u.  B
) )  e.  RR* )
15 ovolge0 20986 . . . . . . . . . . 11  |-  ( A 
C_  RR  ->  0  <_ 
( vol* `  A ) )
16153ad2ant1 1009 . . . . . . . . . 10  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  -> 
0  <_  ( vol* `  A ) )
17 ge0gtmnf 11165 . . . . . . . . . 10  |-  ( ( ( vol* `  A )  e.  RR*  /\  0  <_  ( vol* `  A ) )  -> -oo  <  ( vol* `  A )
)
182, 16, 17syl2anc 661 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  -> -oo  <  ( vol* `  A ) )
1918adantr 465 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  /\  ( vol* `  A
)  <  ( vol* `  ( A  u.  B ) ) )  -> -oo  <  ( vol* `  A )
)
20 simpr 461 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  /\  ( vol* `  A
)  <  ( vol* `  ( A  u.  B ) ) )  ->  ( vol* `  A )  <  ( vol* `  ( A  u.  B ) ) )
21 xrre2 11163 . . . . . . . 8  |-  ( ( ( -oo  e.  RR*  /\  ( vol* `  A )  e.  RR*  /\  ( vol* `  ( A  u.  B
) )  e.  RR* )  /\  ( -oo  <  ( vol* `  A
)  /\  ( vol* `  A )  < 
( vol* `  ( A  u.  B
) ) ) )  ->  ( vol* `  A )  e.  RR )
2212, 13, 14, 19, 20, 21syl32anc 1226 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  /\  ( vol* `  A
)  <  ( vol* `  ( A  u.  B ) ) )  ->  ( vol* `  A )  e.  RR )
234adantr 465 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  /\  ( vol* `  A
)  <  ( vol* `  ( A  u.  B ) ) )  ->  B  C_  RR )
24 simpl3 993 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  /\  ( vol* `  A
)  <  ( vol* `  ( A  u.  B ) ) )  ->  ( vol* `  B )  =  0 )
25 0re 9407 . . . . . . . 8  |-  0  e.  RR
2624, 25syl6eqel 2531 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  /\  ( vol* `  A
)  <  ( vol* `  ( A  u.  B ) ) )  ->  ( vol* `  B )  e.  RR )
27 ovolun 21004 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  ( vol* `  A )  e.  RR )  /\  ( B  C_  RR  /\  ( vol* `  B )  e.  RR ) )  ->  ( vol* `  ( A  u.  B ) )  <_  ( ( vol* `  A )  +  ( vol* `  B ) ) )
2810, 22, 23, 26, 27syl22anc 1219 . . . . . 6  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  /\  ( vol* `  A
)  <  ( vol* `  ( A  u.  B ) ) )  ->  ( vol* `  ( A  u.  B
) )  <_  (
( vol* `  A )  +  ( vol* `  B
) ) )
2924oveq2d 6128 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  /\  ( vol* `  A
)  <  ( vol* `  ( A  u.  B ) ) )  ->  ( ( vol* `  A )  +  ( vol* `  B ) )  =  ( ( vol* `  A )  +  0 ) )
3022recnd 9433 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  /\  ( vol* `  A
)  <  ( vol* `  ( A  u.  B ) ) )  ->  ( vol* `  A )  e.  CC )
3130addid1d 9590 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  /\  ( vol* `  A
)  <  ( vol* `  ( A  u.  B ) ) )  ->  ( ( vol* `  A )  +  0 )  =  ( vol* `  A ) )
3229, 31eqtrd 2475 . . . . . 6  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  /\  ( vol* `  A
)  <  ( vol* `  ( A  u.  B ) ) )  ->  ( ( vol* `  A )  +  ( vol* `  B ) )  =  ( vol* `  A ) )
3328, 32breqtrd 4337 . . . . 5  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  /\  ( vol* `  A
)  <  ( vol* `  ( A  u.  B ) ) )  ->  ( vol* `  ( A  u.  B
) )  <_  ( vol* `  A ) )
3433ex 434 . . . 4  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  -> 
( ( vol* `  A )  <  ( vol* `  ( A  u.  B ) )  ->  ( vol* `  ( A  u.  B
) )  <_  ( vol* `  A ) ) )
359, 34sylbird 235 . . 3  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  -> 
( -.  ( vol* `  ( A  u.  B ) )  <_ 
( vol* `  A )  ->  ( vol* `  ( A  u.  B ) )  <_  ( vol* `  A ) ) )
3635pm2.18d 111 . 2  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  -> 
( vol* `  ( A  u.  B
) )  <_  ( vol* `  A ) )
37 ssun1 3540 . . 3  |-  A  C_  ( A  u.  B
)
38 ovolss 20990 . . 3  |-  ( ( A  C_  ( A  u.  B )  /\  ( A  u.  B )  C_  RR )  ->  ( vol* `  A )  <_  ( vol* `  ( A  u.  B
) ) )
3937, 5, 38sylancr 663 . 2  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  -> 
( vol* `  A )  <_  ( vol* `  ( A  u.  B ) ) )
40 xrletri3 11150 . . 3  |-  ( ( ( vol* `  ( A  u.  B
) )  e.  RR*  /\  ( vol* `  A )  e.  RR* )  ->  ( ( vol* `  ( A  u.  B ) )  =  ( vol* `  A )  <->  ( ( vol* `  ( A  u.  B ) )  <_  ( vol* `  A )  /\  ( vol* `  A )  <_  ( vol* `  ( A  u.  B
) ) ) ) )
417, 2, 40syl2anc 661 . 2  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  -> 
( ( vol* `  ( A  u.  B
) )  =  ( vol* `  A
)  <->  ( ( vol* `  ( A  u.  B ) )  <_ 
( vol* `  A )  /\  ( vol* `  A )  <_  ( vol* `  ( A  u.  B
) ) ) ) )
4236, 39, 41mpbir2and 913 1  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  -> 
( vol* `  ( A  u.  B
) )  =  ( vol* `  A
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    u. cun 3347    C_ wss 3349   class class class wbr 4313   ` cfv 5439  (class class class)co 6112   RRcr 9302   0cc0 9303    + caddc 9306   -oocmnf 9437   RR*cxr 9438    < clt 9439    <_ cle 9440   vol*covol 20968
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 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-er 7122  df-map 7237  df-en 7332  df-dom 7333  df-sdom 7334  df-sup 7712  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-n0 10601  df-z 10668  df-uz 10883  df-q 10975  df-rp 11013  df-ioo 11325  df-ico 11327  df-fz 11459  df-fl 11663  df-seq 11828  df-exp 11887  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-ovol 20970
This theorem is referenced by:  mblfinlem2  28455
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