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Theorem icombl 19136
Description: A closed-below, open-above real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)
Assertion
Ref Expression
icombl  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A [,) B
)  e.  dom  vol )

Proof of Theorem icombl
StepHypRef Expression
1 uncom 3407 . . . . 5  |-  ( ( B [,)  +oo )  u.  ( A [,) B
) )  =  ( ( A [,) B
)  u.  ( B [,)  +oo ) )
2 rexr 9024 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  RR* )
32ad2antrr 706 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  A  e.  RR* )
4 simplr 731 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  B  e.  RR* )
5 pnfxr 10606 . . . . . . 7  |-  +oo  e.  RR*
65a1i 10 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  +oo  e.  RR* )
7 xrltle 10635 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  A  <_  B ) )
82, 7sylan 457 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A  <  B  ->  A  <_  B )
)
98imp 418 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  A  <_  B
)
10 pnfge 10620 . . . . . . 7  |-  ( B  e.  RR*  ->  B  <_  +oo )
114, 10syl 15 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  B  <_  +oo )
12 icoun 10913 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  +oo  e.  RR* )  /\  ( A  <_  B  /\  B  <_  +oo ) )  -> 
( ( A [,) B )  u.  ( B [,)  +oo ) )  =  ( A [,)  +oo ) )
133, 4, 6, 9, 11, 12syl32anc 1191 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( A [,) B )  u.  ( B [,)  +oo ) )  =  ( A [,)  +oo )
)
141, 13syl5eq 2410 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,)  +oo )  u.  ( A [,) B ) )  =  ( A [,)  +oo ) )
15 ssun1 3426 . . . . . 6  |-  ( B [,)  +oo )  C_  (
( B [,)  +oo )  u.  ( A [,) B ) )
1615, 14syl5sseq 3312 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B [,)  +oo )  C_  ( A [,)  +oo ) )
17 incom 3449 . . . . . 6  |-  ( ( B [,)  +oo )  i^i  ( A [,) B
) )  =  ( ( A [,) B
)  i^i  ( B [,)  +oo ) )
18 icodisj 10914 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  +oo  e.  RR* )  ->  ( ( A [,) B )  i^i  ( B [,)  +oo ) )  =  (/) )
195, 18mp3an3 1267 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A [,) B
)  i^i  ( B [,)  +oo ) )  =  (/) )
203, 4, 19syl2anc 642 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( A [,) B )  i^i  ( B [,)  +oo ) )  =  (/) )
2117, 20syl5eq 2410 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,)  +oo )  i^i  ( A [,) B ) )  =  (/) )
22 uneqdifeq 3631 . . . . 5  |-  ( ( ( B [,)  +oo )  C_  ( A [,)  +oo )  /\  ( ( B [,)  +oo )  i^i  ( A [,) B
) )  =  (/) )  ->  ( ( ( B [,)  +oo )  u.  ( A [,) B
) )  =  ( A [,)  +oo )  <->  ( ( A [,)  +oo )  \  ( B [,)  +oo ) )  =  ( A [,) B ) ) )
2316, 21, 22syl2anc 642 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( ( B [,)  +oo )  u.  ( A [,) B
) )  =  ( A [,)  +oo )  <->  ( ( A [,)  +oo )  \  ( B [,)  +oo ) )  =  ( A [,) B ) ) )
2414, 23mpbid 201 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( A [,)  +oo )  \  ( B [,)  +oo ) )  =  ( A [,) B
) )
25 icombl1 19135 . . . . 5  |-  ( A  e.  RR  ->  ( A [,)  +oo )  e.  dom  vol )
2625ad2antrr 706 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( A [,)  +oo )  e.  dom  vol )
27 xrleloe 10630 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  +oo  e.  RR* )  ->  ( B  <_  +oo  <->  ( B  <  +oo  \/  B  =  +oo ) ) )
284, 6, 27syl2anc 642 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  <_  +oo 
<->  ( B  <  +oo  \/  B  =  +oo )
) )
2911, 28mpbid 201 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  <  +oo  \/  B  =  +oo ) )
30 simpr 447 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  A  <  B
)
31 xrre2 10651 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  +oo  e.  RR* )  /\  ( A  <  B  /\  B  <  +oo ) )  ->  B  e.  RR )
3231expr 598 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  +oo  e.  RR* )  /\  A  <  B )  ->  ( B  <  +oo  ->  B  e.  RR ) )
333, 4, 6, 30, 32syl31anc 1186 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  <  +oo  ->  B  e.  RR ) )
3433orim1d 812 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B  <  +oo  \/  B  =  +oo )  ->  ( B  e.  RR  \/  B  =  +oo ) ) )
3529, 34mpd 14 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  e.  RR  \/  B  = 
+oo ) )
36 icombl1 19135 . . . . . 6  |-  ( B  e.  RR  ->  ( B [,)  +oo )  e.  dom  vol )
37 oveq1 5988 . . . . . . . 8  |-  ( B  =  +oo  ->  ( B [,)  +oo )  =  ( 
+oo [,)  +oo ) )
38 pnfge 10620 . . . . . . . . . 10  |-  (  +oo  e.  RR*  ->  +oo  <_  +oo )
395, 38ax-mp 8 . . . . . . . . 9  |-  +oo  <_  +oo
40 ico0 10855 . . . . . . . . . 10  |-  ( ( 
+oo  e.  RR*  /\  +oo  e.  RR* )  ->  (
(  +oo [,)  +oo )  =  (/)  <->  +oo  <_  +oo ) )
415, 5, 40mp2an 653 . . . . . . . . 9  |-  ( ( 
+oo [,)  +oo )  =  (/) 
<-> 
+oo  <_  +oo )
4239, 41mpbir 200 . . . . . . . 8  |-  (  +oo [,) 
+oo )  =  (/)
4337, 42syl6eq 2414 . . . . . . 7  |-  ( B  =  +oo  ->  ( B [,)  +oo )  =  (/) )
44 0mbl 19112 . . . . . . 7  |-  (/)  e.  dom  vol
4543, 44syl6eqel 2454 . . . . . 6  |-  ( B  =  +oo  ->  ( B [,)  +oo )  e.  dom  vol )
4636, 45jaoi 368 . . . . 5  |-  ( ( B  e.  RR  \/  B  =  +oo )  -> 
( B [,)  +oo )  e.  dom  vol )
4735, 46syl 15 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B [,)  +oo )  e.  dom  vol )
48 difmbl 19115 . . . 4  |-  ( ( ( A [,)  +oo )  e.  dom  vol  /\  ( B [,)  +oo )  e.  dom  vol )  -> 
( ( A [,)  +oo )  \  ( B [,)  +oo ) )  e. 
dom  vol )
4926, 47, 48syl2anc 642 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( A [,)  +oo )  \  ( B [,)  +oo ) )  e. 
dom  vol )
5024, 49eqeltrrd 2441 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( A [,) B )  e.  dom  vol )
51 ico0 10855 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A [,) B
)  =  (/)  <->  B  <_  A ) )
522, 51sylan 457 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( ( A [,) B )  =  (/)  <->  B  <_  A ) )
53 simpr 447 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR* )  ->  B  e.  RR* )
542adantr 451 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR* )  ->  A  e.  RR* )
55 xrlenlt 9037 . . . . . 6  |-  ( ( B  e.  RR*  /\  A  e.  RR* )  ->  ( B  <_  A  <->  -.  A  <  B ) )
5653, 54, 55syl2anc 642 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( B  <_  A  <->  -.  A  <  B ) )
5752, 56bitrd 244 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( ( A [,) B )  =  (/)  <->  -.  A  <  B ) )
5857biimpar 471 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  -.  A  <  B
)  ->  ( A [,) B )  =  (/) )
5958, 44syl6eqel 2454 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  -.  A  <  B
)  ->  ( A [,) B )  e.  dom  vol )
6050, 59pm2.61dan 766 1  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A [,) B
)  e.  dom  vol )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    \ cdif 3235    u. cun 3236    i^i cin 3237    C_ wss 3238   (/)c0 3543   class class class wbr 4125   dom cdm 4792  (class class class)co 5981   RRcr 8883    +oocpnf 9011   RR*cxr 9013    < clt 9014    <_ cle 9015   [,)cico 10811   volcvol 19038
This theorem is referenced by:  ioombl  19137
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-se 4456  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-of 6205  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-2o 6622  df-oadd 6625  df-er 6802  df-map 6917  df-pm 6918  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-sup 7341  df-oi 7372  df-card 7719  df-cda 7941  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-2 9951  df-3 9952  df-n0 10115  df-z 10176  df-uz 10382  df-q 10468  df-rp 10506  df-xadd 10604  df-ioo 10813  df-ico 10815  df-icc 10816  df-fz 10936  df-fzo 11026  df-fl 11089  df-seq 11211  df-exp 11270  df-hash 11506  df-cj 11791  df-re 11792  df-im 11793  df-sqr 11927  df-abs 11928  df-clim 12169  df-rlim 12170  df-sum 12367  df-xmet 16586  df-met 16587  df-ovol 19039  df-vol 19040
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