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Theorem icombl 18921
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 3319 . . . . 5  |-  ( ( B [,)  +oo )  u.  ( A [,) B
) )  =  ( ( A [,) B
)  u.  ( B [,)  +oo ) )
2 rexr 8877 . . . . . . 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 10455 . . . . . . 7  |-  +oo  e.  RR*
65a1i 10 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  +oo  e.  RR* )
7 xrltle 10483 . . . . . . . 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 10469 . . . . . . 7  |-  ( B  e.  RR*  ->  B  <_  +oo )
114, 10syl 15 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  B  <_  +oo )
12 icoun 10760 . . . . . 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 1190 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( A [,) B )  u.  ( B [,)  +oo ) )  =  ( A [,)  +oo )
)
141, 13syl5eq 2327 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,)  +oo )  u.  ( A [,) B ) )  =  ( A [,)  +oo ) )
15 ssun1 3338 . . . . . 6  |-  ( B [,)  +oo )  C_  (
( B [,)  +oo )  u.  ( A [,) B ) )
1615, 14syl5sseq 3226 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B [,)  +oo )  C_  ( A [,)  +oo ) )
17 incom 3361 . . . . . 6  |-  ( ( B [,)  +oo )  i^i  ( A [,) B
) )  =  ( ( A [,) B
)  i^i  ( B [,)  +oo ) )
18 icodisj 10761 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  +oo  e.  RR* )  ->  ( ( A [,) B )  i^i  ( B [,)  +oo ) )  =  (/) )
195, 18mp3an3 1266 . . . . . . 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 2327 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,)  +oo )  i^i  ( A [,) B ) )  =  (/) )
22 uneqdifeq 3542 . . . . 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 18920 . . . . 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 10478 . . . . . . . 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 10499 . . . . . . . . 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 1185 . . . . . . 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 18920 . . . . . 6  |-  ( B  e.  RR  ->  ( B [,)  +oo )  e.  dom  vol )
37 oveq1 5865 . . . . . . . 8  |-  ( B  =  +oo  ->  ( B [,)  +oo )  =  ( 
+oo [,)  +oo ) )
38 pnfge 10469 . . . . . . . . . 10  |-  (  +oo  e.  RR*  ->  +oo  <_  +oo )
395, 38ax-mp 8 . . . . . . . . 9  |-  +oo  <_  +oo
40 ico0 10702 . . . . . . . . . 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 2331 . . . . . . 7  |-  ( B  =  +oo  ->  ( B [,)  +oo )  =  (/) )
44 0mbl 18897 . . . . . . 7  |-  (/)  e.  dom  vol
4543, 44syl6eqel 2371 . . . . . 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 18900 . . . 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 2358 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( A [,) B )  e.  dom  vol )
51 ico0 10702 . . . . . 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 8890 . . . . . 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 2371 . 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 934    = wceq 1623    e. wcel 1684    \ cdif 3149    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   class class class wbr 4023   dom cdm 4689  (class class class)co 5858   RRcr 8736    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868   [,)cico 10658   volcvol 18823
This theorem is referenced by:  ioombl  18922
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xadd 10453  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-xmet 16373  df-met 16374  df-ovol 18824  df-vol 18825
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