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Theorem volioc 37859
Description: The measure of left open, right closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
volioc  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( vol `  ( A (,] B ) )  =  ( B  -  A
) )

Proof of Theorem volioc
StepHypRef Expression
1 vol0 37846 . . . 4  |-  ( vol `  (/) )  =  0
2 oveq2 6303 . . . . . . 7  |-  ( A  =  B  ->  ( A (,] A )  =  ( A (,] B
) )
32eqcomd 2459 . . . . . 6  |-  ( A  =  B  ->  ( A (,] B )  =  ( A (,] A
) )
4 leid 9734 . . . . . . 7  |-  ( A  e.  RR  ->  A  <_  A )
5 rexr 9691 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  RR* )
6 ioc0 11690 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  A  e.  RR* )  ->  (
( A (,] A
)  =  (/)  <->  A  <_  A ) )
75, 5, 6syl2anc 667 . . . . . . 7  |-  ( A  e.  RR  ->  (
( A (,] A
)  =  (/)  <->  A  <_  A ) )
84, 7mpbird 236 . . . . . 6  |-  ( A  e.  RR  ->  ( A (,] A )  =  (/) )
93, 8sylan9eqr 2509 . . . . 5  |-  ( ( A  e.  RR  /\  A  =  B )  ->  ( A (,] B
)  =  (/) )
109fveq2d 5874 . . . 4  |-  ( ( A  e.  RR  /\  A  =  B )  ->  ( vol `  ( A (,] B ) )  =  ( vol `  (/) ) )
11 eqcom 2460 . . . . . . . 8  |-  ( A  =  B  <->  B  =  A )
1211biimpi 198 . . . . . . 7  |-  ( A  =  B  ->  B  =  A )
1312adantl 468 . . . . . 6  |-  ( ( A  e.  RR  /\  A  =  B )  ->  B  =  A )
14 recn 9634 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  CC )
1514adantr 467 . . . . . 6  |-  ( ( A  e.  RR  /\  A  =  B )  ->  A  e.  CC )
1613, 15eqeltrd 2531 . . . . 5  |-  ( ( A  e.  RR  /\  A  =  B )  ->  B  e.  CC )
1716, 13subeq0bd 10052 . . . 4  |-  ( ( A  e.  RR  /\  A  =  B )  ->  ( B  -  A
)  =  0 )
181, 10, 173eqtr4a 2513 . . 3  |-  ( ( A  e.  RR  /\  A  =  B )  ->  ( vol `  ( A (,] B ) )  =  ( B  -  A ) )
19183ad2antl1 1171 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  A  =  B )  ->  ( vol `  ( A (,] B ) )  =  ( B  -  A ) )
20 simpl1 1012 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  -.  A  =  B
)  ->  A  e.  RR )
21 simpl2 1013 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  -.  A  =  B
)  ->  B  e.  RR )
22 simpl3 1014 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  -.  A  =  B
)  ->  A  <_  B )
23 eqcom 2460 . . . . . . 7  |-  ( B  =  A  <->  A  =  B )
2423biimpi 198 . . . . . 6  |-  ( B  =  A  ->  A  =  B )
2524necon3bi 2652 . . . . 5  |-  ( -.  A  =  B  ->  B  =/=  A )
2625adantl 468 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  -.  A  =  B
)  ->  B  =/=  A )
2720, 21, 22, 26leneltd 9794 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  -.  A  =  B
)  ->  A  <  B )
2853ad2ant1 1030 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  e.  RR* )
29 rexr 9691 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  RR* )
30293ad2ant2 1031 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  e.  RR* )
31 simp3 1011 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  <  B )
32 snunioo2 37616 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( A (,) B
)  u.  { B } )  =  ( A (,] B ) )
3328, 30, 31, 32syl3anc 1269 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( A (,) B
)  u.  { B } )  =  ( A (,] B ) )
3433eqcomd 2459 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A (,] B )  =  ( ( A (,) B )  u.  { B } ) )
3534fveq2d 5874 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( vol `  ( A (,] B ) )  =  ( vol `  (
( A (,) B
)  u.  { B } ) ) )
36 ioombl 22530 . . . . . 6  |-  ( A (,) B )  e. 
dom  vol
3736a1i 11 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A (,) B )  e. 
dom  vol )
38 snmbl 37850 . . . . . 6  |-  ( B  e.  RR  ->  { B }  e.  dom  vol )
39383ad2ant2 1031 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  { B }  e.  dom  vol )
40 ubioo 11675 . . . . . . 7  |-  -.  B  e.  ( A (,) B
)
41 disjsn 4034 . . . . . . 7  |-  ( ( ( A (,) B
)  i^i  { B } )  =  (/)  <->  -.  B  e.  ( A (,) B ) )
4240, 41mpbir 213 . . . . . 6  |-  ( ( A (,) B )  i^i  { B }
)  =  (/)
4342a1i 11 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( A (,) B
)  i^i  { B } )  =  (/) )
44 ioovolcl 22534 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( vol `  ( A (,) B ) )  e.  RR )
45443adant3 1029 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( vol `  ( A (,) B ) )  e.  RR )
46 volsn 37854 . . . . . . 7  |-  ( B  e.  RR  ->  ( vol `  { B }
)  =  0 )
47 0red 9649 . . . . . . 7  |-  ( B  e.  RR  ->  0  e.  RR )
4846, 47eqeltrd 2531 . . . . . 6  |-  ( B  e.  RR  ->  ( vol `  { B }
)  e.  RR )
49483ad2ant2 1031 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( vol `  { B }
)  e.  RR )
50 volun 22510 . . . . 5  |-  ( ( ( ( A (,) B )  e.  dom  vol 
/\  { B }  e.  dom  vol  /\  (
( A (,) B
)  i^i  { B } )  =  (/) )  /\  ( ( vol `  ( A (,) B
) )  e.  RR  /\  ( vol `  { B } )  e.  RR ) )  ->  ( vol `  ( ( A (,) B )  u. 
{ B } ) )  =  ( ( vol `  ( A (,) B ) )  +  ( vol `  { B } ) ) )
5137, 39, 43, 45, 49, 50syl32anc 1277 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( vol `  ( ( A (,) B )  u. 
{ B } ) )  =  ( ( vol `  ( A (,) B ) )  +  ( vol `  { B } ) ) )
52 simp1 1009 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  e.  RR )
53 simp2 1010 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  e.  RR )
5452, 53, 31ltled 9788 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  <_  B )
55 volioo 37835 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( vol `  ( A (,) B ) )  =  ( B  -  A
) )
5652, 53, 54, 55syl3anc 1269 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( vol `  ( A (,) B ) )  =  ( B  -  A
) )
57463ad2ant2 1031 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( vol `  { B }
)  =  0 )
5856, 57oveq12d 6313 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( vol `  ( A (,) B ) )  +  ( vol `  { B } ) )  =  ( ( B  -  A )  +  0 ) )
5953recnd 9674 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  e.  CC )
60143ad2ant1 1030 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  e.  CC )
6159, 60subcld 9991 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  e.  CC )
6261addid1d 9838 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( B  -  A
)  +  0 )  =  ( B  -  A ) )
6358, 62eqtrd 2487 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( vol `  ( A (,) B ) )  +  ( vol `  { B } ) )  =  ( B  -  A
) )
6435, 51, 633eqtrd 2491 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( vol `  ( A (,] B ) )  =  ( B  -  A
) )
6520, 21, 27, 64syl3anc 1269 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  -.  A  =  B
)  ->  ( vol `  ( A (,] B
) )  =  ( B  -  A ) )
6619, 65pm2.61dan 801 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( vol `  ( A (,] B ) )  =  ( B  -  A
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889    =/= wne 2624    u. cun 3404    i^i cin 3405   (/)c0 3733   {csn 3970   class class class wbr 4405   dom cdm 4837   ` cfv 5585  (class class class)co 6295   CCcc 9542   RRcr 9543   0cc0 9544    + caddc 9547   RR*cxr 9679    < clt 9680    <_ cle 9681    - cmin 9865   (,)cioo 11642   (,]cioc 11643   volcvol 22427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fi 7930  df-sup 7961  df-inf 7962  df-oi 8030  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-q 11272  df-rp 11310  df-xneg 11416  df-xadd 11417  df-xmul 11418  df-ioo 11646  df-ioc 11647  df-ico 11648  df-icc 11649  df-fz 11792  df-fzo 11923  df-fl 12035  df-seq 12221  df-exp 12280  df-hash 12523  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-clim 13564  df-rlim 13565  df-sum 13765  df-rest 15333  df-topgen 15354  df-psmet 18974  df-xmet 18975  df-met 18976  df-bl 18977  df-mopn 18978  df-top 19933  df-bases 19934  df-topon 19935  df-cmp 20414  df-ovol 22428  df-vol 22430
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator