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Theorem volioc 37946
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 37933 . . . 4  |-  ( vol `  (/) )  =  0
2 oveq2 6316 . . . . . . 7  |-  ( A  =  B  ->  ( A (,] A )  =  ( A (,] B
) )
32eqcomd 2477 . . . . . 6  |-  ( A  =  B  ->  ( A (,] B )  =  ( A (,] A
) )
4 leid 9747 . . . . . . 7  |-  ( A  e.  RR  ->  A  <_  A )
5 rexr 9704 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  RR* )
6 ioc0 11708 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  A  e.  RR* )  ->  (
( A (,] A
)  =  (/)  <->  A  <_  A ) )
75, 5, 6syl2anc 673 . . . . . . 7  |-  ( A  e.  RR  ->  (
( A (,] A
)  =  (/)  <->  A  <_  A ) )
84, 7mpbird 240 . . . . . 6  |-  ( A  e.  RR  ->  ( A (,] A )  =  (/) )
93, 8sylan9eqr 2527 . . . . 5  |-  ( ( A  e.  RR  /\  A  =  B )  ->  ( A (,] B
)  =  (/) )
109fveq2d 5883 . . . 4  |-  ( ( A  e.  RR  /\  A  =  B )  ->  ( vol `  ( A (,] B ) )  =  ( vol `  (/) ) )
11 eqcom 2478 . . . . . . . 8  |-  ( A  =  B  <->  B  =  A )
1211biimpi 199 . . . . . . 7  |-  ( A  =  B  ->  B  =  A )
1312adantl 473 . . . . . 6  |-  ( ( A  e.  RR  /\  A  =  B )  ->  B  =  A )
14 recn 9647 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  CC )
1514adantr 472 . . . . . 6  |-  ( ( A  e.  RR  /\  A  =  B )  ->  A  e.  CC )
1613, 15eqeltrd 2549 . . . . 5  |-  ( ( A  e.  RR  /\  A  =  B )  ->  B  e.  CC )
1716, 13subeq0bd 10066 . . . 4  |-  ( ( A  e.  RR  /\  A  =  B )  ->  ( B  -  A
)  =  0 )
181, 10, 173eqtr4a 2531 . . 3  |-  ( ( A  e.  RR  /\  A  =  B )  ->  ( vol `  ( A (,] B ) )  =  ( B  -  A ) )
19183ad2antl1 1192 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  A  =  B )  ->  ( vol `  ( A (,] B ) )  =  ( B  -  A ) )
20 simpl1 1033 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  -.  A  =  B
)  ->  A  e.  RR )
21 simpl2 1034 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  -.  A  =  B
)  ->  B  e.  RR )
22 simpl3 1035 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  -.  A  =  B
)  ->  A  <_  B )
23 eqcom 2478 . . . . . . 7  |-  ( B  =  A  <->  A  =  B )
2423biimpi 199 . . . . . 6  |-  ( B  =  A  ->  A  =  B )
2524necon3bi 2669 . . . . 5  |-  ( -.  A  =  B  ->  B  =/=  A )
2625adantl 473 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  -.  A  =  B
)  ->  B  =/=  A )
2720, 21, 22, 26leneltd 9806 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  -.  A  =  B
)  ->  A  <  B )
2853ad2ant1 1051 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  e.  RR* )
29 rexr 9704 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  RR* )
30293ad2ant2 1052 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  e.  RR* )
31 simp3 1032 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  <  B )
32 snunioo2 37702 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( A (,) B
)  u.  { B } )  =  ( A (,] B ) )
3328, 30, 31, 32syl3anc 1292 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( A (,) B
)  u.  { B } )  =  ( A (,] B ) )
3433eqcomd 2477 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A (,] B )  =  ( ( A (,) B )  u.  { B } ) )
3534fveq2d 5883 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( vol `  ( A (,] B ) )  =  ( vol `  (
( A (,) B
)  u.  { B } ) ) )
36 ioombl 22597 . . . . . 6  |-  ( A (,) B )  e. 
dom  vol
3736a1i 11 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( A (,) B )  e. 
dom  vol )
38 snmbl 37937 . . . . . 6  |-  ( B  e.  RR  ->  { B }  e.  dom  vol )
39383ad2ant2 1052 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  { B }  e.  dom  vol )
40 ubioo 11693 . . . . . . 7  |-  -.  B  e.  ( A (,) B
)
41 disjsn 4023 . . . . . . 7  |-  ( ( ( A (,) B
)  i^i  { B } )  =  (/)  <->  -.  B  e.  ( A (,) B ) )
4240, 41mpbir 214 . . . . . 6  |-  ( ( A (,) B )  i^i  { B }
)  =  (/)
4342a1i 11 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( A (,) B
)  i^i  { B } )  =  (/) )
44 ioovolcl 22601 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( vol `  ( A (,) B ) )  e.  RR )
45443adant3 1050 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( vol `  ( A (,) B ) )  e.  RR )
46 volsn 37941 . . . . . . 7  |-  ( B  e.  RR  ->  ( vol `  { B }
)  =  0 )
47 0red 9662 . . . . . . 7  |-  ( B  e.  RR  ->  0  e.  RR )
4846, 47eqeltrd 2549 . . . . . 6  |-  ( B  e.  RR  ->  ( vol `  { B }
)  e.  RR )
49483ad2ant2 1052 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( vol `  { B }
)  e.  RR )
50 volun 22577 . . . . 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 1300 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( vol `  ( ( A (,) B )  u. 
{ B } ) )  =  ( ( vol `  ( A (,) B ) )  +  ( vol `  { B } ) ) )
52 simp1 1030 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  e.  RR )
53 simp2 1031 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  e.  RR )
5452, 53, 31ltled 9800 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  <_  B )
55 volioo 37922 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( vol `  ( A (,) B ) )  =  ( B  -  A
) )
5652, 53, 54, 55syl3anc 1292 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( vol `  ( A (,) B ) )  =  ( B  -  A
) )
57463ad2ant2 1052 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( vol `  { B }
)  =  0 )
5856, 57oveq12d 6326 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( vol `  ( A (,) B ) )  +  ( vol `  { B } ) )  =  ( ( B  -  A )  +  0 ) )
5953recnd 9687 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  B  e.  CC )
60143ad2ant1 1051 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  A  e.  CC )
6159, 60subcld 10005 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  e.  CC )
6261addid1d 9851 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( B  -  A
)  +  0 )  =  ( B  -  A ) )
6358, 62eqtrd 2505 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  (
( vol `  ( A (,) B ) )  +  ( vol `  { B } ) )  =  ( B  -  A
) )
6435, 51, 633eqtrd 2509 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( vol `  ( A (,] B ) )  =  ( B  -  A
) )
6520, 21, 27, 64syl3anc 1292 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  /\  -.  A  =  B
)  ->  ( vol `  ( A (,] B
) )  =  ( B  -  A ) )
6619, 65pm2.61dan 808 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641    u. cun 3388    i^i cin 3389   (/)c0 3722   {csn 3959   class class class wbr 4395   dom cdm 4839   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557    + caddc 9560   RR*cxr 9692    < clt 9693    <_ cle 9694    - cmin 9880   (,)cioo 11660   (,]cioc 11661   volcvol 22493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-rlim 13630  df-sum 13830  df-rest 15399  df-topgen 15420  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-top 19998  df-bases 19999  df-topon 20000  df-cmp 20479  df-ovol 22494  df-vol 22496
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator