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Theorem icombl 22100
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 3644 . . . . 5  |-  ( ( B [,) +oo )  u.  ( A [,) B
) )  =  ( ( A [,) B
)  u.  ( B [,) +oo ) )
2 rexr 9656 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  RR* )
32ad2antrr 725 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  A  e.  RR* )
4 simplr 755 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  B  e.  RR* )
5 pnfxr 11346 . . . . . . 7  |- +oo  e.  RR*
65a1i 11 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  -> +oo  e.  RR* )
7 xrltle 11380 . . . . . . . 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 11364 . . . . . . 7  |-  ( B  e.  RR*  ->  B  <_ +oo )
114, 10syl 16 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  B  <_ +oo )
12 icoun 11669 . . . . . 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 2510 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,) +oo )  u.  ( A [,) B
) )  =  ( A [,) +oo )
)
15 ssun1 3663 . . . . . 6  |-  ( B [,) +oo )  C_  ( ( B [,) +oo )  u.  ( A [,) B ) )
1615, 14syl5sseq 3547 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( B [,) +oo )  C_  ( A [,) +oo ) )
17 incom 3687 . . . . . 6  |-  ( ( B [,) +oo )  i^i  ( A [,) B
) )  =  ( ( A [,) B
)  i^i  ( B [,) +oo ) )
18 icodisj 11670 . . . . . . . 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 2510 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( ( B [,) +oo )  i^i  ( A [,) B
) )  =  (/) )
22 uneqdifeq 3919 . . . . 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 22099 . . . . 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 11375 . . . . . . . 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 11396 . . . . . . . . 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 839 . . . . . 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 22099 . . . . . 6  |-  ( B  e.  RR  ->  ( B [,) +oo )  e. 
dom  vol )
37 oveq1 6303 . . . . . . . 8  |-  ( B  = +oo  ->  ( B [,) +oo )  =  ( +oo [,) +oo ) )
38 pnfge 11364 . . . . . . . . . 10  |-  ( +oo  e.  RR*  -> +oo  <_ +oo )
395, 38ax-mp 5 . . . . . . . . 9  |- +oo  <_ +oo
40 ico0 11600 . . . . . . . . . 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 2514 . . . . . . 7  |-  ( B  = +oo  ->  ( B [,) +oo )  =  (/) )
44 0mbl 22076 . . . . . . 7  |-  (/)  e.  dom  vol
4543, 44syl6eqel 2553 . . . . . 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 22079 . . . 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 2546 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  A  <  B )  ->  ( A [,) B )  e.  dom  vol )
51 ico0 11600 . . . . . 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 9669 . . . . . 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 2553 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR* )  /\  -.  A  <  B
)  ->  ( A [,) B )  e.  dom  vol )
6050, 59pm2.61dan 791 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 1395    e. wcel 1819    \ cdif 3468    u. cun 3469    i^i cin 3470    C_ wss 3471   (/)c0 3793   class class class wbr 4456   dom cdm 5008  (class class class)co 6296   RRcr 9508   +oocpnf 9642   RR*cxr 9644    < clt 9645    <_ cle 9646   [,)cico 11556   volcvol 22001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-xadd 11344  df-ioo 11558  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-fl 11932  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323  df-rlim 13324  df-sum 13521  df-xmet 18539  df-met 18540  df-ovol 22002  df-vol 22003
This theorem is referenced by:  ioombl  22101
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