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Theorem ovolunnul 22077
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 22055 . . . . . 6  |-  ( A 
C_  RR  ->  ( vol* `  A )  e.  RR* )
213ad2ant1 1015 . . . . 5  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  -> 
( vol* `  A )  e.  RR* )
3 simp1 994 . . . . . . 7  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  ->  A  C_  RR )
4 simp2 995 . . . . . . 7  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  ->  B  C_  RR )
53, 4unssd 3666 . . . . . 6  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  -> 
( A  u.  B
)  C_  RR )
6 ovolcl 22055 . . . . . 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 9642 . . . . 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 659 . . . 4  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  -> 
( ( vol* `  A )  <  ( vol* `  ( A  u.  B ) )  <->  -.  ( vol* `  ( A  u.  B
) )  <_  ( vol* `  A ) ) )
103adantr 463 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  /\  ( vol* `  A
)  <  ( vol* `  ( A  u.  B ) ) )  ->  A  C_  RR )
11 mnfxr 11326 . . . . . . . . 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 463 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  /\  ( vol* `  A
)  <  ( vol* `  ( A  u.  B ) ) )  ->  ( vol* `  ( A  u.  B
) )  e.  RR* )
15 ovolge0 22058 . . . . . . . . . . 11  |-  ( A 
C_  RR  ->  0  <_ 
( vol* `  A ) )
16153ad2ant1 1015 . . . . . . . . . 10  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  -> 
0  <_  ( vol* `  A ) )
17 ge0gtmnf 11376 . . . . . . . . . 10  |-  ( ( ( vol* `  A )  e.  RR*  /\  0  <_  ( vol* `  A ) )  -> -oo  <  ( vol* `  A )
)
182, 16, 17syl2anc 659 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  -> -oo  <  ( vol* `  A ) )
1918adantr 463 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  /\  ( vol* `  A
)  <  ( vol* `  ( A  u.  B ) ) )  -> -oo  <  ( vol* `  A )
)
20 simpr 459 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  /\  ( vol* `  A
)  <  ( vol* `  ( A  u.  B ) ) )  ->  ( vol* `  A )  <  ( vol* `  ( A  u.  B ) ) )
21 xrre2 11374 . . . . . . . 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 1234 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  /\  ( vol* `  A
)  <  ( vol* `  ( A  u.  B ) ) )  ->  ( vol* `  A )  e.  RR )
234adantr 463 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  /\  ( vol* `  A
)  <  ( vol* `  ( A  u.  B ) ) )  ->  B  C_  RR )
24 simpl3 999 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  /\  ( vol* `  A
)  <  ( vol* `  ( A  u.  B ) ) )  ->  ( vol* `  B )  =  0 )
25 0re 9585 . . . . . . . 8  |-  0  e.  RR
2624, 25syl6eqel 2550 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  /\  ( vol* `  A
)  <  ( vol* `  ( A  u.  B ) ) )  ->  ( vol* `  B )  e.  RR )
27 ovolun 22076 . . . . . . 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 1227 . . . . . 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 6286 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  /\  ( vol* `  A
)  <  ( vol* `  ( A  u.  B ) ) )  ->  ( ( vol* `  A )  +  ( vol* `  B ) )  =  ( ( vol* `  A )  +  0 ) )
3022recnd 9611 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  /\  ( vol* `  A
)  <  ( vol* `  ( A  u.  B ) ) )  ->  ( vol* `  A )  e.  CC )
3130addid1d 9769 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  /\  ( vol* `  A
)  <  ( vol* `  ( A  u.  B ) ) )  ->  ( ( vol* `  A )  +  0 )  =  ( vol* `  A ) )
3229, 31eqtrd 2495 . . . . . 6  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  /\  ( vol* `  A
)  <  ( vol* `  ( A  u.  B ) ) )  ->  ( ( vol* `  A )  +  ( vol* `  B ) )  =  ( vol* `  A ) )
3328, 32breqtrd 4463 . . . . 5  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  /\  ( vol* `  A
)  <  ( vol* `  ( A  u.  B ) ) )  ->  ( vol* `  ( A  u.  B
) )  <_  ( vol* `  A ) )
3433ex 432 . . . 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 3653 . . 3  |-  A  C_  ( A  u.  B
)
38 ovolss 22062 . . 3  |-  ( ( A  C_  ( A  u.  B )  /\  ( A  u.  B )  C_  RR )  ->  ( vol* `  A )  <_  ( vol* `  ( A  u.  B
) ) )
3937, 5, 38sylancr 661 . 2  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol* `  B )  =  0 )  -> 
( vol* `  A )  <_  ( vol* `  ( A  u.  B ) ) )
40 xrletri3 11361 . . 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 659 . 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 920 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    u. cun 3459    C_ wss 3461   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   RRcr 9480   0cc0 9481    + caddc 9484   -oocmnf 9615   RR*cxr 9616    < clt 9617    <_ cle 9618   vol*covol 22040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-q 11184  df-rp 11222  df-ioo 11536  df-ico 11538  df-fz 11676  df-fl 11910  df-seq 12090  df-exp 12149  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-ovol 22042
This theorem is referenced by:  mblfinlem2  30292
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