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Theorem ovolicopnf 22385
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 11401 . . . . . . . . 9  |- +oo  e.  RR*
2 icossre 11704 . . . . . . . . 9  |-  ( ( A  e.  RR  /\ +oo  e.  RR* )  ->  ( A [,) +oo )  C_  RR )
31, 2mpan2 675 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A [,) +oo )  C_  RR )
43adantr 466 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( A [,) +oo )  C_  RR )
5 ovolge0 22341 . . . . . . 7  |-  ( ( A [,) +oo )  C_  RR  ->  0  <_  ( vol* `  ( A [,) +oo ) ) )
64, 5syl 17 . . . . . 6  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  0  <_  ( vol* `  ( A [,) +oo )
) )
7 mnflt0 11416 . . . . . . 7  |- -oo  <  0
8 ovolcl 22338 . . . . . . . . . 10  |-  ( ( A [,) +oo )  C_  RR  ->  ( vol* `  ( A [,) +oo ) )  e.  RR* )
93, 8syl 17 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( vol* `  ( A [,) +oo ) )  e.  RR* )
109adantr 466 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( vol* `  ( A [,) +oo ) )  e.  RR* )
11 mnfxr 11403 . . . . . . . . 9  |- -oo  e.  RR*
12 0xr 9676 . . . . . . . . 9  |-  0  e.  RR*
13 xrltletr 11443 . . . . . . . . 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 1350 . . . . . . . 8  |-  ( ( vol* `  ( A [,) +oo ) )  e.  RR*  ->  ( ( -oo  <  0  /\  0  <_  ( vol* `  ( A [,) +oo ) ) )  -> -oo  <  ( vol* `  ( A [,) +oo ) ) ) )
1510, 14syl 17 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
( -oo  <  0  /\  0  <_  ( vol* `  ( A [,) +oo ) ) )  -> -oo  <  ( vol* `  ( A [,) +oo ) ) ) )
167, 15mpani 680 . . . . . 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 462 . . . . 5  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( vol* `  ( A [,) +oo ) )  < +oo )
19 xrrebnd 11452 . . . . . 6  |-  ( ( vol* `  ( A [,) +oo ) )  e.  RR*  ->  ( ( vol* `  ( A [,) +oo ) )  e.  RR  <->  ( -oo  <  ( vol* `  ( A [,) +oo )
)  /\  ( vol* `  ( A [,) +oo ) )  < +oo ) ) )
2010, 19syl 17 . . . . 5  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
( vol* `  ( A [,) +oo )
)  e.  RR  <->  ( -oo  <  ( vol* `  ( A [,) +oo )
)  /\  ( vol* `  ( A [,) +oo ) )  < +oo ) ) )
2117, 18, 20mpbir2and 930 . . . 4  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( vol* `  ( A [,) +oo ) )  e.  RR )
2221ltp1d 10526 . . 3  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( vol* `  ( A [,) +oo ) )  <  ( ( vol* `  ( A [,) +oo ) )  +  1 ) )
23 simpl 458 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  A  e.  RR )
24 peano2re 9795 . . . . . . . . 9  |-  ( ( vol* `  ( A [,) +oo ) )  e.  RR  ->  (
( vol* `  ( A [,) +oo )
)  +  1 )  e.  RR )
2521, 24syl 17 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
( vol* `  ( A [,) +oo )
)  +  1 )  e.  RR )
2625, 23readdcld 9659 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A )  e.  RR )
27 0red 9633 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  0  e.  RR )
2821lep1d 10527 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( vol* `  ( A [,) +oo ) )  <_  ( ( vol* `  ( A [,) +oo ) )  +  1 ) )
2927, 21, 25, 6, 28letrd 9781 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  0  <_  ( ( vol* `  ( A [,) +oo ) )  +  1 ) )
3023, 25addge02d 10191 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
0  <_  ( ( vol* `  ( A [,) +oo ) )  +  1 )  <->  A  <_  ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) ) )
3129, 30mpbid 213 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  A  <_  ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A
) )
32 ovolicc 22384 . . . . . . 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 1264 . . . . . 6  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( vol* `  ( A [,] ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) ) )  =  ( ( ( ( vol* `  ( A [,) +oo )
)  +  1 )  +  A )  -  A ) )
3425recnd 9658 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
( vol* `  ( A [,) +oo )
)  +  1 )  e.  CC )
3523recnd 9658 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  A  e.  CC )
3634, 35pncand 9976 . . . . . 6  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A
)  -  A )  =  ( ( vol* `  ( A [,) +oo ) )  +  1 ) )
3733, 36eqtrd 2461 . . . . 5  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( vol* `  ( A [,] ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) ) )  =  ( ( vol* `  ( A [,) +oo ) )  +  1 ) )
38 elicc2 11688 . . . . . . . . . . . 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 665 . . . . . . . . . . 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 486 . . . . . . . . . 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 1017 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo )
)  < +oo )  /\  x  e.  ( A [,] ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) ) )  ->  x  e.  RR )
4240simp2d 1018 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo )
)  < +oo )  /\  x  e.  ( A [,] ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) ) )  ->  A  <_  x
)
43 elicopnf 11719 . . . . . . . . . 10  |-  ( A  e.  RR  ->  (
x  e.  ( A [,) +oo )  <->  ( x  e.  RR  /\  A  <_  x ) ) )
4443ad2antrr 730 . . . . . . . . 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 930 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo )
)  < +oo )  /\  x  e.  ( A [,] ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) ) )  ->  x  e.  ( A [,) +oo )
)
4645ex 435 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
x  e.  ( A [,] ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) )  ->  x  e.  ( A [,) +oo ) ) )
4746ssrdv 3467 . . . . . 6  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( A [,] ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) )  C_  ( A [,) +oo )
)
48 ovolss 22345 . . . . . 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 665 . . . . 5  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( vol* `  ( A [,] ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) ) )  <_  ( vol* `  ( A [,) +oo ) ) )
5037, 49eqbrtrrd 4439 . . . 4  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
( vol* `  ( A [,) +oo )
)  +  1 )  <_  ( vol* `  ( A [,) +oo ) ) )
5125, 21lenltd 9770 . . . 4  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
( ( vol* `  ( A [,) +oo ) )  +  1 )  <_  ( vol* `  ( A [,) +oo ) )  <->  -.  ( vol* `  ( A [,) +oo ) )  <  ( ( vol* `  ( A [,) +oo ) )  +  1 ) ) )
5250, 51mpbid 213 . . 3  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  -.  ( vol* `  ( A [,) +oo ) )  <  ( ( vol* `  ( A [,) +oo ) )  +  1 ) )
5322, 52pm2.65da 578 . 2  |-  ( A  e.  RR  ->  -.  ( vol* `  ( A [,) +oo ) )  < +oo )
54 nltpnft 11450 . . 3  |-  ( ( vol* `  ( A [,) +oo ) )  e.  RR*  ->  ( ( vol* `  ( A [,) +oo ) )  = +oo  <->  -.  ( vol* `  ( A [,) +oo ) )  < +oo ) )
559, 54syl 17 . 2  |-  ( A  e.  RR  ->  (
( vol* `  ( A [,) +oo )
)  = +oo  <->  -.  ( vol* `  ( A [,) +oo ) )  < +oo ) )
5653, 55mpbird 235 1  |-  ( A  e.  RR  ->  ( vol* `  ( A [,) +oo ) )  = +oo )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867    C_ wss 3433   class class class wbr 4417   ` cfv 5592  (class class class)co 6296   RRcr 9527   0cc0 9528   1c1 9529    + caddc 9531   +oocpnf 9661   -oocmnf 9662   RR*cxr 9663    < clt 9664    <_ cle 9665    - cmin 9849   [,)cico 11626   [,]cicc 11627   vol*covol 22320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-map 7473  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-fi 7922  df-sup 7953  df-inf 7954  df-oi 8016  df-card 8363  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-n0 10859  df-z 10927  df-uz 11149  df-q 11254  df-rp 11292  df-xneg 11398  df-xadd 11399  df-xmul 11400  df-ioo 11628  df-ico 11630  df-icc 11631  df-fz 11772  df-fzo 11903  df-seq 12200  df-exp 12259  df-hash 12502  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-clim 13519  df-sum 13720  df-rest 15281  df-topgen 15302  df-psmet 18903  df-xmet 18904  df-met 18905  df-bl 18906  df-mopn 18907  df-top 19858  df-bases 19859  df-topon 19860  df-cmp 20339  df-ovol 22323
This theorem is referenced by:  ovolre  22386
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