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Theorem ovolunnul 18875
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 18853 . . . . . 6  |-  ( A 
C_  RR  ->  ( vol
* `  A )  e.  RR* )
213ad2ant1 976 . . . . 5  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol
* `  B )  =  0 )  -> 
( vol * `  A )  e.  RR* )
3 simp1 955 . . . . . . 7  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol
* `  B )  =  0 )  ->  A  C_  RR )
4 simp2 956 . . . . . . 7  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol
* `  B )  =  0 )  ->  B  C_  RR )
53, 4unssd 3364 . . . . . 6  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol
* `  B )  =  0 )  -> 
( A  u.  B
)  C_  RR )
6 ovolcl 18853 . . . . . 6  |-  ( ( A  u.  B ) 
C_  RR  ->  ( vol
* `  ( A  u.  B ) )  e. 
RR* )
75, 6syl 15 . . . . 5  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol
* `  B )  =  0 )  -> 
( vol * `  ( A  u.  B
) )  e.  RR* )
8 xrltnle 8907 . . . . 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 642 . . . 4  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol
* `  B )  =  0 )  -> 
( ( vol * `  A )  <  ( vol * `  ( A  u.  B ) )  <->  -.  ( vol * `  ( A  u.  B
) )  <_  ( vol * `  A ) ) )
103adantr 451 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol * `  B )  =  0 )  /\  ( vol * `  A
)  <  ( vol * `
 ( A  u.  B ) ) )  ->  A  C_  RR )
11 mnfxr 10472 . . . . . . . . 9  |-  -oo  e.  RR*
1211a1i 10 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol * `  B )  =  0 )  /\  ( vol * `  A
)  <  ( vol * `
 ( A  u.  B ) ) )  ->  -oo  e.  RR* )
1310, 1syl 15 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol * `  B )  =  0 )  /\  ( vol * `  A
)  <  ( vol * `
 ( A  u.  B ) ) )  ->  ( vol * `  A )  e.  RR* )
147adantr 451 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol * `  B )  =  0 )  /\  ( vol * `  A
)  <  ( vol * `
 ( A  u.  B ) ) )  ->  ( vol * `  ( A  u.  B
) )  e.  RR* )
15 ovolge0 18856 . . . . . . . . . . 11  |-  ( A 
C_  RR  ->  0  <_ 
( vol * `  A ) )
16153ad2ant1 976 . . . . . . . . . 10  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol
* `  B )  =  0 )  -> 
0  <_  ( vol * `
 A ) )
17 ge0gtmnf 10517 . . . . . . . . . 10  |-  ( ( ( vol * `  A )  e.  RR*  /\  0  <_  ( vol * `
 A ) )  ->  -oo  <  ( vol
* `  A )
)
182, 16, 17syl2anc 642 . . . . . . . . 9  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol
* `  B )  =  0 )  ->  -oo  <  ( vol * `  A ) )
1918adantr 451 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol * `  B )  =  0 )  /\  ( vol * `  A
)  <  ( vol * `
 ( A  u.  B ) ) )  ->  -oo  <  ( vol
* `  A )
)
20 simpr 447 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol * `  B )  =  0 )  /\  ( vol * `  A
)  <  ( vol * `
 ( A  u.  B ) ) )  ->  ( vol * `  A )  <  ( vol * `  ( A  u.  B ) ) )
21 xrre2 10515 . . . . . . . 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 1190 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol * `  B )  =  0 )  /\  ( vol * `  A
)  <  ( vol * `
 ( A  u.  B ) ) )  ->  ( vol * `  A )  e.  RR )
234adantr 451 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol * `  B )  =  0 )  /\  ( vol * `  A
)  <  ( vol * `
 ( A  u.  B ) ) )  ->  B  C_  RR )
24 simpl3 960 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol * `  B )  =  0 )  /\  ( vol * `  A
)  <  ( vol * `
 ( A  u.  B ) ) )  ->  ( vol * `  B )  =  0 )
25 0re 8854 . . . . . . . 8  |-  0  e.  RR
2624, 25syl6eqel 2384 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol * `  B )  =  0 )  /\  ( vol * `  A
)  <  ( vol * `
 ( A  u.  B ) ) )  ->  ( vol * `  B )  e.  RR )
27 ovolun 18874 . . . . . . 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 1183 . . . . . 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 5890 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol * `  B )  =  0 )  /\  ( vol * `  A
)  <  ( vol * `
 ( A  u.  B ) ) )  ->  ( ( vol
* `  A )  +  ( vol * `  B ) )  =  ( ( vol * `  A )  +  0 ) )
3022recnd 8877 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol * `  B )  =  0 )  /\  ( vol * `  A
)  <  ( vol * `
 ( A  u.  B ) ) )  ->  ( vol * `  A )  e.  CC )
3130addid1d 9028 . . . . . . 7  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol * `  B )  =  0 )  /\  ( vol * `  A
)  <  ( vol * `
 ( A  u.  B ) ) )  ->  ( ( vol
* `  A )  +  0 )  =  ( vol * `  A ) )
3229, 31eqtrd 2328 . . . . . 6  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol * `  B )  =  0 )  /\  ( vol * `  A
)  <  ( vol * `
 ( A  u.  B ) ) )  ->  ( ( vol
* `  A )  +  ( vol * `  B ) )  =  ( vol * `  A ) )
3328, 32breqtrd 4063 . . . . 5  |-  ( ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol * `  B )  =  0 )  /\  ( vol * `  A
)  <  ( vol * `
 ( A  u.  B ) ) )  ->  ( vol * `  ( A  u.  B
) )  <_  ( vol * `  A ) )
3433ex 423 . . . 4  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol
* `  B )  =  0 )  -> 
( ( vol * `  A )  <  ( vol * `  ( A  u.  B ) )  ->  ( vol * `  ( A  u.  B
) )  <_  ( vol * `  A ) ) )
359, 34sylbird 226 . . 3  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol
* `  B )  =  0 )  -> 
( -.  ( vol
* `  ( A  u.  B ) )  <_ 
( vol * `  A )  ->  ( vol * `  ( A  u.  B ) )  <_  ( vol * `  A ) ) )
3635pm2.18d 103 . 2  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol
* `  B )  =  0 )  -> 
( vol * `  ( A  u.  B
) )  <_  ( vol * `  A ) )
37 ssun1 3351 . . 3  |-  A  C_  ( A  u.  B
)
38 ovolss 18860 . . 3  |-  ( ( A  C_  ( A  u.  B )  /\  ( A  u.  B )  C_  RR )  ->  ( vol * `  A )  <_  ( vol * `  ( A  u.  B
) ) )
3937, 5, 38sylancr 644 . 2  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol
* `  B )  =  0 )  -> 
( vol * `  A )  <_  ( vol * `  ( A  u.  B ) ) )
40 xrletri3 10502 . . 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 642 . 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 888 1  |-  ( ( A  C_  RR  /\  B  C_  RR  /\  ( vol
* `  B )  =  0 )  -> 
( vol * `  ( A  u.  B
) )  =  ( vol * `  A
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    u. cun 3163    C_ wss 3165   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753    + caddc 8756    -oocmnf 8881   RR*cxr 8882    < clt 8883    <_ cle 8884   vol *covol 18838
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-ioo 10676  df-ico 10678  df-fz 10799  df-fl 10941  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-ovol 18840
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