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Theorem icombl 19419
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 3459 . . . . 5  |-  ( ( B [,)  +oo )  u.  ( A [,) B
) )  =  ( ( A [,) B
)  u.  ( B [,)  +oo ) )
2 rexr 9094 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  RR* )
32ad2antrr 707 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  A  e.  RR* )
4 simplr 732 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  B  e.  RR* )
5 pnfxr 10677 . . . . . . 7  |-  +oo  e.  RR*
65a1i 11 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  +oo  e.  RR* )
7 xrltle 10706 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  A  <_  B ) )
82, 7sylan 458 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A  <  B  ->  A  <_  B )
)
98imp 419 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  A  <_  B
)
10 pnfge 10691 . . . . . . 7  |-  ( B  e.  RR*  ->  B  <_  +oo )
114, 10syl 16 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  B  <_  +oo )
12 icoun 10985 . . . . . 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 1192 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( A [,) B )  u.  ( B [,)  +oo ) )  =  ( A [,)  +oo )
)
141, 13syl5eq 2456 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,)  +oo )  u.  ( A [,) B ) )  =  ( A [,)  +oo ) )
15 ssun1 3478 . . . . . 6  |-  ( B [,)  +oo )  C_  (
( B [,)  +oo )  u.  ( A [,) B ) )
1615, 14syl5sseq 3364 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B [,)  +oo )  C_  ( A [,)  +oo ) )
17 incom 3501 . . . . . 6  |-  ( ( B [,)  +oo )  i^i  ( A [,) B
) )  =  ( ( A [,) B
)  i^i  ( B [,)  +oo ) )
18 icodisj 10986 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  +oo  e.  RR* )  ->  ( ( A [,) B )  i^i  ( B [,)  +oo ) )  =  (/) )
195, 18mp3an3 1268 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A [,) B
)  i^i  ( B [,)  +oo ) )  =  (/) )
203, 4, 19syl2anc 643 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( A [,) B )  i^i  ( B [,)  +oo ) )  =  (/) )
2117, 20syl5eq 2456 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,)  +oo )  i^i  ( A [,) B ) )  =  (/) )
22 uneqdifeq 3684 . . . . 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 643 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( ( B [,)  +oo )  u.  ( A [,) B
) )  =  ( A [,)  +oo )  <->  ( ( A [,)  +oo )  \  ( B [,)  +oo ) )  =  ( A [,) B ) ) )
2414, 23mpbid 202 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( A [,)  +oo )  \  ( B [,)  +oo ) )  =  ( A [,) B
) )
25 icombl1 19418 . . . . 5  |-  ( A  e.  RR  ->  ( A [,)  +oo )  e.  dom  vol )
2625ad2antrr 707 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( A [,)  +oo )  e.  dom  vol )
27 xrleloe 10701 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  +oo  e.  RR* )  ->  ( B  <_  +oo  <->  ( B  <  +oo  \/  B  =  +oo ) ) )
284, 6, 27syl2anc 643 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  <_  +oo 
<->  ( B  <  +oo  \/  B  =  +oo )
) )
2911, 28mpbid 202 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  <  +oo  \/  B  =  +oo ) )
30 simpr 448 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  A  <  B
)
31 xrre2 10722 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  +oo  e.  RR* )  /\  ( A  <  B  /\  B  <  +oo ) )  ->  B  e.  RR )
3231expr 599 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  +oo  e.  RR* )  /\  A  <  B )  ->  ( B  <  +oo  ->  B  e.  RR ) )
333, 4, 6, 30, 32syl31anc 1187 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  <  +oo  ->  B  e.  RR ) )
3433orim1d 813 . . . . . 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 19418 . . . . . 6  |-  ( B  e.  RR  ->  ( B [,)  +oo )  e.  dom  vol )
37 oveq1 6055 . . . . . . . 8  |-  ( B  =  +oo  ->  ( B [,)  +oo )  =  ( 
+oo [,)  +oo ) )
38 pnfge 10691 . . . . . . . . . 10  |-  (  +oo  e.  RR*  ->  +oo  <_  +oo )
395, 38ax-mp 8 . . . . . . . . 9  |-  +oo  <_  +oo
40 ico0 10926 . . . . . . . . . 10  |-  ( ( 
+oo  e.  RR*  /\  +oo  e.  RR* )  ->  (
(  +oo [,)  +oo )  =  (/)  <->  +oo  <_  +oo ) )
415, 5, 40mp2an 654 . . . . . . . . 9  |-  ( ( 
+oo [,)  +oo )  =  (/) 
<-> 
+oo  <_  +oo )
4239, 41mpbir 201 . . . . . . . 8  |-  (  +oo [,) 
+oo )  =  (/)
4337, 42syl6eq 2460 . . . . . . 7  |-  ( B  =  +oo  ->  ( B [,)  +oo )  =  (/) )
44 0mbl 19395 . . . . . . 7  |-  (/)  e.  dom  vol
4543, 44syl6eqel 2500 . . . . . 6  |-  ( B  =  +oo  ->  ( B [,)  +oo )  e.  dom  vol )
4636, 45jaoi 369 . . . . 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 19398 . . . 4  |-  ( ( ( A [,)  +oo )  e.  dom  vol  /\  ( B [,)  +oo )  e.  dom  vol )  -> 
( ( A [,)  +oo )  \  ( B [,)  +oo ) )  e. 
dom  vol )
4926, 47, 48syl2anc 643 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( A [,)  +oo )  \  ( B [,)  +oo ) )  e. 
dom  vol )
5024, 49eqeltrrd 2487 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( A [,) B )  e.  dom  vol )
51 ico0 10926 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A [,) B
)  =  (/)  <->  B  <_  A ) )
522, 51sylan 458 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( ( A [,) B )  =  (/)  <->  B  <_  A ) )
53 simpr 448 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR* )  ->  B  e.  RR* )
542adantr 452 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR* )  ->  A  e.  RR* )
55 xrlenlt 9107 . . . . . 6  |-  ( ( B  e.  RR*  /\  A  e.  RR* )  ->  ( B  <_  A  <->  -.  A  <  B ) )
5653, 54, 55syl2anc 643 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( B  <_  A  <->  -.  A  <  B ) )
5752, 56bitrd 245 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( ( A [,) B )  =  (/)  <->  -.  A  <  B ) )
5857biimpar 472 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  -.  A  <  B
)  ->  ( A [,) B )  =  (/) )
5958, 44syl6eqel 2500 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  -.  A  <  B
)  ->  ( A [,) B )  e.  dom  vol )
6050, 59pm2.61dan 767 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 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    \ cdif 3285    u. cun 3286    i^i cin 3287    C_ wss 3288   (/)c0 3596   class class class wbr 4180   dom cdm 4845  (class class class)co 6048   RRcr 8953    +oocpnf 9081   RR*cxr 9083    < clt 9084    <_ cle 9085   [,)cico 10882   volcvol 19321
This theorem is referenced by:  ioombl  19420
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-map 6987  df-pm 6988  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-oi 7443  df-card 7790  df-cda 8012  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-n0 10186  df-z 10247  df-uz 10453  df-q 10539  df-rp 10577  df-xadd 10675  df-ioo 10884  df-ico 10886  df-icc 10887  df-fz 11008  df-fzo 11099  df-fl 11165  df-seq 11287  df-exp 11346  df-hash 11582  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-clim 12245  df-rlim 12246  df-sum 12443  df-xmet 16658  df-met 16659  df-ovol 19322  df-vol 19323
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