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Theorem icombl 19489
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 3477 . . . . 5  |-  ( ( B [,)  +oo )  u.  ( A [,) B
) )  =  ( ( A [,) B
)  u.  ( B [,)  +oo ) )
2 rexr 9161 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  RR* )
32ad2antrr 708 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  A  e.  RR* )
4 simplr 733 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  B  e.  RR* )
5 pnfxr 10744 . . . . . . 7  |-  +oo  e.  RR*
65a1i 11 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  +oo  e.  RR* )
7 xrltle 10773 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  A  <_  B ) )
82, 7sylan 459 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A  <  B  ->  A  <_  B )
)
98imp 420 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  A  <_  B
)
10 pnfge 10758 . . . . . . 7  |-  ( B  e.  RR*  ->  B  <_  +oo )
114, 10syl 16 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  B  <_  +oo )
12 icoun 11052 . . . . . 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 1193 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( A [,) B )  u.  ( B [,)  +oo ) )  =  ( A [,)  +oo )
)
141, 13syl5eq 2486 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,)  +oo )  u.  ( A [,) B ) )  =  ( A [,)  +oo ) )
15 ssun1 3496 . . . . . 6  |-  ( B [,)  +oo )  C_  (
( B [,)  +oo )  u.  ( A [,) B ) )
1615, 14syl5sseq 3382 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B [,)  +oo )  C_  ( A [,)  +oo ) )
17 incom 3519 . . . . . 6  |-  ( ( B [,)  +oo )  i^i  ( A [,) B
) )  =  ( ( A [,) B
)  i^i  ( B [,)  +oo ) )
18 icodisj 11053 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  +oo  e.  RR* )  ->  ( ( A [,) B )  i^i  ( B [,)  +oo ) )  =  (/) )
195, 18mp3an3 1269 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A [,) B
)  i^i  ( B [,)  +oo ) )  =  (/) )
203, 4, 19syl2anc 644 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( A [,) B )  i^i  ( B [,)  +oo ) )  =  (/) )
2117, 20syl5eq 2486 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,)  +oo )  i^i  ( A [,) B ) )  =  (/) )
22 uneqdifeq 3740 . . . . 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 644 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( ( B [,)  +oo )  u.  ( A [,) B
) )  =  ( A [,)  +oo )  <->  ( ( A [,)  +oo )  \  ( B [,)  +oo ) )  =  ( A [,) B ) ) )
2414, 23mpbid 203 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( A [,)  +oo )  \  ( B [,)  +oo ) )  =  ( A [,) B
) )
25 icombl1 19488 . . . . 5  |-  ( A  e.  RR  ->  ( A [,)  +oo )  e.  dom  vol )
2625ad2antrr 708 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( A [,)  +oo )  e.  dom  vol )
27 xrleloe 10768 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  +oo  e.  RR* )  ->  ( B  <_  +oo  <->  ( B  <  +oo  \/  B  =  +oo ) ) )
284, 6, 27syl2anc 644 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  <_  +oo 
<->  ( B  <  +oo  \/  B  =  +oo )
) )
2911, 28mpbid 203 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  <  +oo  \/  B  =  +oo ) )
30 simpr 449 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  A  <  B
)
31 xrre2 10789 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  +oo  e.  RR* )  /\  ( A  <  B  /\  B  <  +oo ) )  ->  B  e.  RR )
3231expr 600 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  +oo  e.  RR* )  /\  A  <  B )  ->  ( B  <  +oo  ->  B  e.  RR ) )
333, 4, 6, 30, 32syl31anc 1188 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  <  +oo  ->  B  e.  RR ) )
3433orim1d 814 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B  <  +oo  \/  B  =  +oo )  ->  ( B  e.  RR  \/  B  =  +oo ) ) )
3529, 34mpd 15 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  e.  RR  \/  B  = 
+oo ) )
36 icombl1 19488 . . . . . 6  |-  ( B  e.  RR  ->  ( B [,)  +oo )  e.  dom  vol )
37 oveq1 6117 . . . . . . . 8  |-  ( B  =  +oo  ->  ( B [,)  +oo )  =  ( 
+oo [,)  +oo ) )
38 pnfge 10758 . . . . . . . . . 10  |-  (  +oo  e.  RR*  ->  +oo  <_  +oo )
395, 38ax-mp 5 . . . . . . . . 9  |-  +oo  <_  +oo
40 ico0 10993 . . . . . . . . . 10  |-  ( ( 
+oo  e.  RR*  /\  +oo  e.  RR* )  ->  (
(  +oo [,)  +oo )  =  (/)  <->  +oo  <_  +oo ) )
415, 5, 40mp2an 655 . . . . . . . . 9  |-  ( ( 
+oo [,)  +oo )  =  (/) 
<-> 
+oo  <_  +oo )
4239, 41mpbir 202 . . . . . . . 8  |-  (  +oo [,) 
+oo )  =  (/)
4337, 42syl6eq 2490 . . . . . . 7  |-  ( B  =  +oo  ->  ( B [,)  +oo )  =  (/) )
44 0mbl 19465 . . . . . . 7  |-  (/)  e.  dom  vol
4543, 44syl6eqel 2530 . . . . . 6  |-  ( B  =  +oo  ->  ( B [,)  +oo )  e.  dom  vol )
4636, 45jaoi 370 . . . . 5  |-  ( ( B  e.  RR  \/  B  =  +oo )  -> 
( B [,)  +oo )  e.  dom  vol )
4735, 46syl 16 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B [,)  +oo )  e.  dom  vol )
48 difmbl 19468 . . . 4  |-  ( ( ( A [,)  +oo )  e.  dom  vol  /\  ( B [,)  +oo )  e.  dom  vol )  -> 
( ( A [,)  +oo )  \  ( B [,)  +oo ) )  e. 
dom  vol )
4926, 47, 48syl2anc 644 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( A [,)  +oo )  \  ( B [,)  +oo ) )  e. 
dom  vol )
5024, 49eqeltrrd 2517 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( A [,) B )  e.  dom  vol )
51 ico0 10993 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A [,) B
)  =  (/)  <->  B  <_  A ) )
522, 51sylan 459 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( ( A [,) B )  =  (/)  <->  B  <_  A ) )
53 simpr 449 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR* )  ->  B  e.  RR* )
542adantr 453 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR* )  ->  A  e.  RR* )
55 xrlenlt 9174 . . . . . 6  |-  ( ( B  e.  RR*  /\  A  e.  RR* )  ->  ( B  <_  A  <->  -.  A  <  B ) )
5653, 54, 55syl2anc 644 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( B  <_  A  <->  -.  A  <  B ) )
5752, 56bitrd 246 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( ( A [,) B )  =  (/)  <->  -.  A  <  B ) )
5857biimpar 473 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  -.  A  <  B
)  ->  ( A [,) B )  =  (/) )
5958, 44syl6eqel 2530 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  -.  A  <  B
)  ->  ( A [,) B )  e.  dom  vol )
6050, 59pm2.61dan 768 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 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727    \ cdif 3303    u. cun 3304    i^i cin 3305    C_ wss 3306   (/)c0 3613   class class class wbr 4237   dom cdm 4907  (class class class)co 6110   RRcr 9020    +oocpnf 9148   RR*cxr 9150    < clt 9151    <_ cle 9152   [,)cico 10949   volcvol 19391
This theorem is referenced by:  ioombl  19490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-inf2 7625  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098  ax-pre-sup 9099
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-se 4571  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-isom 5492  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-of 6334  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-1o 6753  df-2o 6754  df-oadd 6757  df-er 6934  df-map 7049  df-pm 7050  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-sup 7475  df-oi 7508  df-card 7857  df-cda 8079  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-div 9709  df-nn 10032  df-2 10089  df-3 10090  df-n0 10253  df-z 10314  df-uz 10520  df-q 10606  df-rp 10644  df-xadd 10742  df-ioo 10951  df-ico 10953  df-icc 10954  df-fz 11075  df-fzo 11167  df-fl 11233  df-seq 11355  df-exp 11414  df-hash 11650  df-cj 11935  df-re 11936  df-im 11937  df-sqr 12071  df-abs 12072  df-clim 12313  df-rlim 12314  df-sum 12511  df-xmet 16726  df-met 16727  df-ovol 19392  df-vol 19393
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