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Theorem icombl 21801
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 3648 . . . . 5  |-  ( ( B [,) +oo )  u.  ( A [,) B
) )  =  ( ( A [,) B
)  u.  ( B [,) +oo ) )
2 rexr 9640 . . . . . . 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 11322 . . . . . . 7  |- +oo  e.  RR*
65a1i 11 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  -> +oo  e.  RR* )
7 xrltle 11356 . . . . . . . 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 11340 . . . . . . 7  |-  ( B  e.  RR*  ->  B  <_ +oo )
114, 10syl 16 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  B  <_ +oo )
12 icoun 11645 . . . . . 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 1236 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( A [,) B )  u.  ( B [,) +oo ) )  =  ( A [,) +oo )
)
141, 13syl5eq 2520 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,) +oo )  u.  ( A [,) B
) )  =  ( A [,) +oo )
)
15 ssun1 3667 . . . . . 6  |-  ( B [,) +oo )  C_  ( ( B [,) +oo )  u.  ( A [,) B ) )
1615, 14syl5sseq 3552 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B [,) +oo )  C_  ( A [,) +oo ) )
17 incom 3691 . . . . . 6  |-  ( ( B [,) +oo )  i^i  ( A [,) B
) )  =  ( ( A [,) B
)  i^i  ( B [,) +oo ) )
18 icodisj 11646 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\ +oo  e.  RR* )  ->  ( ( A [,) B )  i^i  ( B [,) +oo ) )  =  (/) )
195, 18mp3an3 1313 . . . . . . 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 2520 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,) +oo )  i^i  ( A [,) B
) )  =  (/) )
22 uneqdifeq 3915 . . . . 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 21800 . . . . 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 11351 . . . . . . . 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 11372 . . . . . . . . 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 1231 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B  < +oo  ->  B  e.  RR ) )
3433orim1d 837 . . . . . 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 21800 . . . . . 6  |-  ( B  e.  RR  ->  ( B [,) +oo )  e. 
dom  vol )
37 oveq1 6292 . . . . . . . 8  |-  ( B  = +oo  ->  ( B [,) +oo )  =  ( +oo [,) +oo ) )
38 pnfge 11340 . . . . . . . . . 10  |-  ( +oo  e.  RR*  -> +oo  <_ +oo )
395, 38ax-mp 5 . . . . . . . . 9  |- +oo  <_ +oo
40 ico0 11576 . . . . . . . . . 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 2524 . . . . . . 7  |-  ( B  = +oo  ->  ( B [,) +oo )  =  (/) )
44 0mbl 21777 . . . . . . 7  |-  (/)  e.  dom  vol
4543, 44syl6eqel 2563 . . . . . 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 21780 . . . 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 2556 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( A [,) B )  e.  dom  vol )
51 ico0 11576 . . . . . 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 9653 . . . . . 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 2563 . 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 973    = wceq 1379    e. wcel 1767    \ cdif 3473    u. cun 3474    i^i cin 3475    C_ wss 3476   (/)c0 3785   class class class wbr 4447   dom cdm 4999  (class class class)co 6285   RRcr 9492   +oocpnf 9626   RR*cxr 9628    < clt 9629    <_ cle 9630   [,)cico 11532   volcvol 21702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-inf2 8059  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-2o 7132  df-oadd 7135  df-er 7312  df-map 7423  df-pm 7424  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-sup 7902  df-oi 7936  df-card 8321  df-cda 8549  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11084  df-q 11184  df-rp 11222  df-xadd 11320  df-ioo 11534  df-ico 11536  df-icc 11537  df-fz 11674  df-fzo 11794  df-fl 11898  df-seq 12077  df-exp 12136  df-hash 12375  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035  df-clim 13277  df-rlim 13278  df-sum 13475  df-xmet 18223  df-met 18224  df-ovol 21703  df-vol 21704
This theorem is referenced by:  ioombl  21802
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