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Theorem volioc 40188
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 40175 . . . 4 |- ( vol ` (/) ) = 0
2 oveq2 6658 . . . . . . 7 |- ( A = B -> ( A (,] A ) = ( A (,] B ) )
32eqcomd 2628 . . . . . 6 |- ( A = B -> ( A (,] B ) = ( A (,] A ) )
4 leid 10133 . . . . . . 7 |- ( A e. RR -> A <_ A )
5 rexr 10085 . . . . . . . 8 |- ( A e. RR -> A e. RR* )
6 ioc0 12222 . . . . . . . 8 |- ( ( A e. RR* /\ A e. RR* ) -> ( ( A (,] A ) = (/) <-> A <_ A ) )
75, 5, 6syl2anc 693 . . . . . . 7 |- ( A e. RR -> ( ( A (,] A ) = (/) <-> A <_ A ) )
84, 7mpbird 247 . . . . . 6 |- ( A e. RR -> ( A (,] A ) = (/) )
93, 8sylan9eqr 2678 . . . . 5 |- ( ( A e. RR /\ A = B ) -> ( A (,] B ) = (/) )
109fveq2d 6195 . . . 4 |- ( ( A e. RR /\ A = B ) -> ( vol ` ( A (,] B ) ) = ( vol ` (/) ) )
11 eqcom 2629 . . . . . . . 8 |- ( A = B <-> B = A )
1211biimpi 206 . . . . . . 7 |- ( A = B -> B = A )
1312adantl 482 . . . . . 6 |- ( ( A e. RR /\ A = B ) -> B = A )
14 recn 10026 . . . . . . 7 |- ( A e. RR -> A e. CC )
1514adantr 481 . . . . . 6 |- ( ( A e. RR /\ A = B ) -> A e. CC )
1613, 15eqeltrd 2701 . . . . 5 |- ( ( A e. RR /\ A = B ) -> B e. CC )
1716, 13subeq0bd 10456 . . . 4 |- ( ( A e. RR /\ A = B ) -> ( B - A ) = 0 )
181, 10, 173eqtr4a 2682 . . 3 |- ( ( A e. RR /\ A = B ) -> ( vol ` ( A (,] B ) ) = ( B - A ) )
19183ad2antl1 1223 . 2 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ A = B ) -> ( vol ` ( A (,] B ) ) = ( B - A ) )
20 simpl1 1064 . . 3 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ -. A = B ) -> A e. RR )
21 simpl2 1065 . . 3 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ -. A = B ) -> B e. RR )
22 simpl3 1066 . . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ -. A = B ) -> A <_ B )
23 eqcom 2629 . . . . . . 7 |- ( B = A <-> A = B )
2423biimpi 206 . . . . . 6 |- ( B = A -> A = B )
2524necon3bi 2820 . . . . 5 |- ( -. A = B -> B =/= A )
2625adantl 482 . . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ -. A = B ) -> B =/= A )
2720, 21, 22, 26leneltd 10191 . . 3 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ -. A = B ) -> A < B )
2853ad2ant1 1082 . . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> A e. RR* )
29 rexr 10085 . . . . . . . 8 |- ( B e. RR -> B e. RR* )
30293ad2ant2 1083 . . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> B e. RR* )
31 simp3 1063 . . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> A < B )
32 snunioo2 39731 . . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( A (,) B ) u. { B } ) = ( A (,] B ) )
3328, 30, 31, 32syl3anc 1326 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( ( A (,) B ) u. { B } ) = ( A (,] B ) )
3433eqcomd 2628 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( A (,] B ) = ( ( A (,) B ) u. { B } ) )
3534fveq2d 6195 . . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( vol ` ( A (,] B ) ) = ( vol ` ( ( A (,) B ) u. { B } ) ) )
36 ioombl 23333 . . . . . 6 |- ( A (,) B ) e. dom vol
3736a1i 11 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( A (,) B ) e. dom vol )
38 snmbl 40179 . . . . . 6 |- ( B e. RR -> { B } e. dom vol )
39383ad2ant2 1083 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> { B } e. dom vol )
40 ubioo 12207 . . . . . . 7 |- -. B e. ( A (,) B )
41 disjsn 4246 . . . . . . 7 |- ( ( ( A (,) B ) i^i { B } ) = (/) <-> -. B e. ( A (,) B ) )
4240, 41mpbir 221 . . . . . 6 |- ( ( A (,) B ) i^i { B } ) = (/)
4342a1i 11 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( ( A (,) B ) i^i { B } ) = (/) )
44 ioovolcl 23338 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( vol ` ( A (,) B ) ) e. RR )
45443adant3 1081 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( vol ` ( A (,) B ) ) e. RR )
46 volsn 40183 . . . . . . 7 |- ( B e. RR -> ( vol ` { B } ) = 0 )
47 0red 10041 . . . . . . 7 |- ( B e. RR -> 0 e. RR )
4846, 47eqeltrd 2701 . . . . . 6 |- ( B e. RR -> ( vol ` { B } ) e. RR )
49483ad2ant2 1083 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( vol ` { B } ) e. RR )
50 volun 23313 . . . . 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 1334 . . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( vol ` ( ( A (,) B ) u. { B } ) ) = ( ( vol ` ( A (,) B ) ) + ( vol ` { B } ) ) )
52 simp1 1061 . . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> A e. RR )
53 simp2 1062 . . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> B e. RR )
5452, 53, 31ltled 10185 . . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> A <_ B )
55 volioo 23337 . . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( vol ` ( A (,) B ) ) = ( B - A ) )
5652, 53, 54, 55syl3anc 1326 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( vol ` ( A (,) B ) ) = ( B - A ) )
57463ad2ant2 1083 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( vol ` { B } ) = 0 )
5856, 57oveq12d 6668 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( ( vol ` ( A (,) B ) ) + ( vol ` { B } ) ) = ( ( B - A ) + 0 ) )
5953recnd 10068 . . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> B e. CC )
60143ad2ant1 1082 . . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> A e. CC )
6159, 60subcld 10392 . . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( B - A ) e. CC )
6261addid1d 10236 . . . . 5 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( ( B - A ) + 0 ) = ( B - A ) )
6358, 62eqtrd 2656 . . . 4 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( ( vol ` ( A (,) B ) ) + ( vol ` { B } ) ) = ( B - A ) )
6435, 51, 633eqtrd 2660 . . 3 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( vol ` ( A (,] B ) ) = ( B - A ) )
6520, 21, 27, 64syl3anc 1326 . 2 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ -. A = B ) -> ( vol ` ( A (,] B ) ) = ( B - A ) )
6619, 65pm2.61dan 832 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 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   class class class wbr 4653   dom cdm 5114   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   (,)cioo 12175   (,]cioc 12176   volcvol 23232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cmp 21190  df-ovol 23233  df-vol 23234
This theorem is referenced by: (None)
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