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Theorem icombl 21057
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 3512 . . . . 5  |-  ( ( B [,) +oo )  u.  ( A [,) B
) )  =  ( ( A [,) B
)  u.  ( B [,) +oo ) )
2 rexr 9441 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  RR* )
32ad2antrr 725 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  A  e.  RR* )
4 simplr 754 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  B  e.  RR* )
5 pnfxr 11104 . . . . . . 7  |- +oo  e.  RR*
65a1i 11 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  -> +oo  e.  RR* )
7 xrltle 11138 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  ->  A  <_  B ) )
82, 7sylan 471 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A  <  B  ->  A  <_  B )
)
98imp 429 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  A  <_  B
)
10 pnfge 11122 . . . . . . 7  |-  ( B  e.  RR*  ->  B  <_ +oo )
114, 10syl 16 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  B  <_ +oo )
12 icoun 11421 . . . . . 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 1226 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( A [,) B )  u.  ( B [,) +oo ) )  =  ( A [,) +oo )
)
141, 13syl5eq 2487 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,) +oo )  u.  ( A [,) B
) )  =  ( A [,) +oo )
)
15 ssun1 3531 . . . . . 6  |-  ( B [,) +oo )  C_  ( ( B [,) +oo )  u.  ( A [,) B ) )
1615, 14syl5sseq 3416 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B [,) +oo )  C_  ( A [,) +oo ) )
17 incom 3555 . . . . . 6  |-  ( ( B [,) +oo )  i^i  ( A [,) B
) )  =  ( ( A [,) B
)  i^i  ( B [,) +oo ) )
18 icodisj 11422 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\ +oo  e.  RR* )  ->  ( ( A [,) B )  i^i  ( B [,) +oo ) )  =  (/) )
195, 18mp3an3 1303 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A [,) B
)  i^i  ( B [,) +oo ) )  =  (/) )
203, 4, 19syl2anc 661 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( A [,) B )  i^i  ( B [,) +oo ) )  =  (/) )
2117, 20syl5eq 2487 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,) +oo )  i^i  ( A [,) B
) )  =  (/) )
22 uneqdifeq 3779 . . . . 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 661 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( ( B [,) +oo )  u.  ( A [,) B
) )  =  ( A [,) +oo )  <->  ( ( A [,) +oo )  \  ( B [,) +oo ) )  =  ( A [,) B ) ) )
2414, 23mpbid 210 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( A [,) +oo )  \ 
( B [,) +oo ) )  =  ( A [,) B ) )
25 icombl1 21056 . . . . 5  |-  ( A  e.  RR  ->  ( A [,) +oo )  e. 
dom  vol )
2625ad2antrr 725 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( A [,) +oo )  e.  dom  vol )
27 xrleloe 11133 . . . . . . . 8  |-  ( ( B  e.  RR*  /\ +oo  e.  RR* )  ->  ( B  <_ +oo  <->  ( B  < +oo  \/  B  = +oo ) ) )
284, 6, 27syl2anc 661 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  <_ +oo 
<->  ( B  < +oo  \/  B  = +oo ) ) )
2911, 28mpbid 210 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  < +oo  \/  B  = +oo ) )
30 simpr 461 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  A  <  B
)
31 xrre2 11154 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\ +oo  e.  RR* )  /\  ( A  <  B  /\  B  < +oo ) )  ->  B  e.  RR )
3231expr 615 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\ +oo  e.  RR* )  /\  A  <  B )  ->  ( B  < +oo  ->  B  e.  RR ) )
333, 4, 6, 30, 32syl31anc 1221 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  < +oo  ->  B  e.  RR ) )
3433orim1d 835 . . . . . 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 21056 . . . . . 6  |-  ( B  e.  RR  ->  ( B [,) +oo )  e. 
dom  vol )
37 oveq1 6110 . . . . . . . 8  |-  ( B  = +oo  ->  ( B [,) +oo )  =  ( +oo [,) +oo ) )
38 pnfge 11122 . . . . . . . . . 10  |-  ( +oo  e.  RR*  -> +oo  <_ +oo )
395, 38ax-mp 5 . . . . . . . . 9  |- +oo  <_ +oo
40 ico0 11358 . . . . . . . . . 10  |-  ( ( +oo  e.  RR*  /\ +oo  e.  RR* )  ->  (
( +oo [,) +oo )  =  (/)  <-> +oo  <_ +oo )
)
415, 5, 40mp2an 672 . . . . . . . . 9  |-  ( ( +oo [,) +oo )  =  (/)  <-> +oo  <_ +oo )
4239, 41mpbir 209 . . . . . . . 8  |-  ( +oo [,) +oo )  =  (/)
4337, 42syl6eq 2491 . . . . . . 7  |-  ( B  = +oo  ->  ( B [,) +oo )  =  (/) )
44 0mbl 21033 . . . . . . 7  |-  (/)  e.  dom  vol
4543, 44syl6eqel 2531 . . . . . 6  |-  ( B  = +oo  ->  ( B [,) +oo )  e. 
dom  vol )
4636, 45jaoi 379 . . . . 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 21036 . . . 4  |-  ( ( ( A [,) +oo )  e.  dom  vol  /\  ( B [,) +oo )  e.  dom  vol )  -> 
( ( A [,) +oo )  \  ( B [,) +oo ) )  e.  dom  vol )
4926, 47, 48syl2anc 661 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( A [,) +oo )  \ 
( B [,) +oo ) )  e.  dom  vol )
5024, 49eqeltrrd 2518 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( A [,) B )  e.  dom  vol )
51 ico0 11358 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A [,) B
)  =  (/)  <->  B  <_  A ) )
522, 51sylan 471 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( ( A [,) B )  =  (/)  <->  B  <_  A ) )
53 simpr 461 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR* )  ->  B  e.  RR* )
542adantr 465 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR* )  ->  A  e.  RR* )
55 xrlenlt 9454 . . . . . 6  |-  ( ( B  e.  RR*  /\  A  e.  RR* )  ->  ( B  <_  A  <->  -.  A  <  B ) )
5653, 54, 55syl2anc 661 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( B  <_  A  <->  -.  A  <  B ) )
5752, 56bitrd 253 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( ( A [,) B )  =  (/)  <->  -.  A  <  B ) )
5857biimpar 485 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  -.  A  <  B
)  ->  ( A [,) B )  =  (/) )
5958, 44syl6eqel 2531 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  -.  A  <  B
)  ->  ( A [,) B )  e.  dom  vol )
6050, 59pm2.61dan 789 1  |-  ( ( A  e.  RR  /\  B  e.  RR* )  -> 
( A [,) B
)  e.  dom  vol )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    \ cdif 3337    u. cun 3338    i^i cin 3339    C_ wss 3340   (/)c0 3649   class class class wbr 4304   dom cdm 4852  (class class class)co 6103   RRcr 9293   +oocpnf 9427   RR*cxr 9429    < clt 9430    <_ cle 9431   [,)cico 11314   volcvol 20959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-of 6332  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-2o 6933  df-oadd 6936  df-er 7113  df-map 7228  df-pm 7229  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-sup 7703  df-oi 7736  df-card 8121  df-cda 8349  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-n0 10592  df-z 10659  df-uz 10874  df-q 10966  df-rp 11004  df-xadd 11102  df-ioo 11316  df-ico 11318  df-icc 11319  df-fz 11450  df-fzo 11561  df-fl 11654  df-seq 11819  df-exp 11878  df-hash 12116  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-clim 12978  df-rlim 12979  df-sum 13176  df-xmet 17822  df-met 17823  df-ovol 20960  df-vol 20961
This theorem is referenced by:  ioombl  21058
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