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Theorem ovolicopnf 21762
Description: The measure of a right-unbounded interval. (Contributed by Mario Carneiro, 14-Jun-2014.)
Assertion
Ref Expression
ovolicopnf  |-  ( A  e.  RR  ->  ( vol* `  ( A [,) +oo ) )  = +oo )

Proof of Theorem ovolicopnf
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pnfxr 11322 . . . . . . . . 9  |- +oo  e.  RR*
2 icossre 11606 . . . . . . . . 9  |-  ( ( A  e.  RR  /\ +oo  e.  RR* )  ->  ( A [,) +oo )  C_  RR )
31, 2mpan2 671 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A [,) +oo )  C_  RR )
43adantr 465 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( A [,) +oo )  C_  RR )
5 ovolge0 21719 . . . . . . 7  |-  ( ( A [,) +oo )  C_  RR  ->  0  <_  ( vol* `  ( A [,) +oo ) ) )
64, 5syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  0  <_  ( vol* `  ( A [,) +oo )
) )
7 mnflt0 11335 . . . . . . 7  |- -oo  <  0
8 ovolcl 21716 . . . . . . . . . 10  |-  ( ( A [,) +oo )  C_  RR  ->  ( vol* `  ( A [,) +oo ) )  e.  RR* )
93, 8syl 16 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( vol* `  ( A [,) +oo ) )  e.  RR* )
109adantr 465 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( vol* `  ( A [,) +oo ) )  e.  RR* )
11 mnfxr 11324 . . . . . . . . 9  |- -oo  e.  RR*
12 0xr 9641 . . . . . . . . 9  |-  0  e.  RR*
13 xrltletr 11361 . . . . . . . . 9  |-  ( ( -oo  e.  RR*  /\  0  e.  RR*  /\  ( vol* `  ( A [,) +oo ) )  e. 
RR* )  ->  (
( -oo  <  0  /\  0  <_  ( vol* `  ( A [,) +oo ) ) )  -> -oo  <  ( vol* `  ( A [,) +oo ) ) ) )
1411, 12, 13mp3an12 1314 . . . . . . . 8  |-  ( ( vol* `  ( A [,) +oo ) )  e.  RR*  ->  ( ( -oo  <  0  /\  0  <_  ( vol* `  ( A [,) +oo ) ) )  -> -oo  <  ( vol* `  ( A [,) +oo ) ) ) )
1510, 14syl 16 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
( -oo  <  0  /\  0  <_  ( vol* `  ( A [,) +oo ) ) )  -> -oo  <  ( vol* `  ( A [,) +oo ) ) ) )
167, 15mpani 676 . . . . . 6  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
0  <_  ( vol* `  ( A [,) +oo ) )  -> -oo  <  ( vol* `  ( A [,) +oo ) ) ) )
176, 16mpd 15 . . . . 5  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  -> -oo  <  ( vol* `  ( A [,) +oo ) ) )
18 simpr 461 . . . . 5  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( vol* `  ( A [,) +oo ) )  < +oo )
19 xrrebnd 11370 . . . . . 6  |-  ( ( vol* `  ( A [,) +oo ) )  e.  RR*  ->  ( ( vol* `  ( A [,) +oo ) )  e.  RR  <->  ( -oo  <  ( vol* `  ( A [,) +oo )
)  /\  ( vol* `  ( A [,) +oo ) )  < +oo ) ) )
2010, 19syl 16 . . . . 5  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
( vol* `  ( A [,) +oo )
)  e.  RR  <->  ( -oo  <  ( vol* `  ( A [,) +oo )
)  /\  ( vol* `  ( A [,) +oo ) )  < +oo ) ) )
2117, 18, 20mpbir2and 920 . . . 4  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( vol* `  ( A [,) +oo ) )  e.  RR )
2221ltp1d 10477 . . 3  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( vol* `  ( A [,) +oo ) )  <  ( ( vol* `  ( A [,) +oo ) )  +  1 ) )
23 simpl 457 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  A  e.  RR )
24 peano2re 9753 . . . . . . . . 9  |-  ( ( vol* `  ( A [,) +oo ) )  e.  RR  ->  (
( vol* `  ( A [,) +oo )
)  +  1 )  e.  RR )
2521, 24syl 16 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
( vol* `  ( A [,) +oo )
)  +  1 )  e.  RR )
2625, 23readdcld 9624 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A )  e.  RR )
27 0red 9598 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  0  e.  RR )
2821lep1d 10478 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( vol* `  ( A [,) +oo ) )  <_  ( ( vol* `  ( A [,) +oo ) )  +  1 ) )
2927, 21, 25, 6, 28letrd 9739 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  0  <_  ( ( vol* `  ( A [,) +oo ) )  +  1 ) )
3023, 25addge02d 10142 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
0  <_  ( ( vol* `  ( A [,) +oo ) )  +  1 )  <->  A  <_  ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) ) )
3129, 30mpbid 210 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  A  <_  ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A
) )
32 ovolicc 21761 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A
)  e.  RR  /\  A  <_  ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) )  -> 
( vol* `  ( A [,] ( ( ( vol* `  ( A [,) +oo )
)  +  1 )  +  A ) ) )  =  ( ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A )  -  A ) )
3323, 26, 31, 32syl3anc 1228 . . . . . 6  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( vol* `  ( A [,] ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) ) )  =  ( ( ( ( vol* `  ( A [,) +oo )
)  +  1 )  +  A )  -  A ) )
3425recnd 9623 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
( vol* `  ( A [,) +oo )
)  +  1 )  e.  CC )
3523recnd 9623 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  A  e.  CC )
3634, 35pncand 9932 . . . . . 6  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A
)  -  A )  =  ( ( vol* `  ( A [,) +oo ) )  +  1 ) )
3733, 36eqtrd 2508 . . . . 5  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( vol* `  ( A [,] ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) ) )  =  ( ( vol* `  ( A [,) +oo ) )  +  1 ) )
38 elicc2 11590 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A
)  e.  RR )  ->  ( x  e.  ( A [,] (
( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) )  <->  ( x  e.  RR  /\  A  <_  x  /\  x  <_  (
( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) ) ) )
3923, 26, 38syl2anc 661 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
x  e.  ( A [,] ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) )  <->  ( x  e.  RR  /\  A  <_  x  /\  x  <_  (
( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) ) ) )
4039biimpa 484 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo )
)  < +oo )  /\  x  e.  ( A [,] ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) ) )  ->  ( x  e.  RR  /\  A  <_  x  /\  x  <_  (
( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) ) )
4140simp1d 1008 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo )
)  < +oo )  /\  x  e.  ( A [,] ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) ) )  ->  x  e.  RR )
4240simp2d 1009 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo )
)  < +oo )  /\  x  e.  ( A [,] ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) ) )  ->  A  <_  x
)
43 elicopnf 11621 . . . . . . . . . 10  |-  ( A  e.  RR  ->  (
x  e.  ( A [,) +oo )  <->  ( x  e.  RR  /\  A  <_  x ) ) )
4443ad2antrr 725 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo )
)  < +oo )  /\  x  e.  ( A [,] ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) ) )  ->  ( x  e.  ( A [,) +oo ) 
<->  ( x  e.  RR  /\  A  <_  x )
) )
4541, 42, 44mpbir2and 920 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo )
)  < +oo )  /\  x  e.  ( A [,] ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) ) )  ->  x  e.  ( A [,) +oo )
)
4645ex 434 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
x  e.  ( A [,] ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) )  ->  x  e.  ( A [,) +oo ) ) )
4746ssrdv 3510 . . . . . 6  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( A [,] ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) )  C_  ( A [,) +oo )
)
48 ovolss 21723 . . . . . 6  |-  ( ( ( A [,] (
( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) )  C_  ( A [,) +oo )  /\  ( A [,) +oo )  C_  RR )  ->  ( vol* `  ( A [,] ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A
) ) )  <_ 
( vol* `  ( A [,) +oo )
) )
4947, 4, 48syl2anc 661 . . . . 5  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( vol* `  ( A [,] ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) ) )  <_  ( vol* `  ( A [,) +oo ) ) )
5037, 49eqbrtrrd 4469 . . . 4  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
( vol* `  ( A [,) +oo )
)  +  1 )  <_  ( vol* `  ( A [,) +oo ) ) )
5125, 21lenltd 9731 . . . 4  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
( ( vol* `  ( A [,) +oo ) )  +  1 )  <_  ( vol* `  ( A [,) +oo ) )  <->  -.  ( vol* `  ( A [,) +oo ) )  <  ( ( vol* `  ( A [,) +oo ) )  +  1 ) ) )
5250, 51mpbid 210 . . 3  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  -.  ( vol* `  ( A [,) +oo ) )  <  ( ( vol* `  ( A [,) +oo ) )  +  1 ) )
5322, 52pm2.65da 576 . 2  |-  ( A  e.  RR  ->  -.  ( vol* `  ( A [,) +oo ) )  < +oo )
54 nltpnft 11368 . . 3  |-  ( ( vol* `  ( A [,) +oo ) )  e.  RR*  ->  ( ( vol* `  ( A [,) +oo ) )  = +oo  <->  -.  ( vol* `  ( A [,) +oo ) )  < +oo ) )
559, 54syl 16 . 2  |-  ( A  e.  RR  ->  (
( vol* `  ( A [,) +oo )
)  = +oo  <->  -.  ( vol* `  ( A [,) +oo ) )  < +oo ) )
5653, 55mpbird 232 1  |-  ( A  e.  RR  ->  ( vol* `  ( A [,) +oo ) )  = +oo )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    C_ wss 3476   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   RRcr 9492   0cc0 9493   1c1 9494    + caddc 9496   +oocpnf 9626   -oocmnf 9627   RR*cxr 9628    < clt 9629    <_ cle 9630    - cmin 9806   [,)cico 11532   [,]cicc 11533   vol*covol 21701
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-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-map 7423  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-fi 7872  df-sup 7902  df-oi 7936  df-card 8321  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-xneg 11319  df-xadd 11320  df-xmul 11321  df-ioo 11534  df-ico 11536  df-icc 11537  df-fz 11674  df-fzo 11794  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-sum 13475  df-rest 14681  df-topgen 14702  df-psmet 18222  df-xmet 18223  df-met 18224  df-bl 18225  df-mopn 18226  df-top 19206  df-bases 19208  df-topon 19209  df-cmp 19693  df-ovol 21703
This theorem is referenced by:  ovolre  21763
  Copyright terms: Public domain W3C validator