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Theorem ovolicopnf 19381
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 10677 . . . . . . . . 9  |-  +oo  e.  RR*
2 icossre 10955 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  +oo 
e.  RR* )  ->  ( A [,)  +oo )  C_  RR )
31, 2mpan2 653 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A [,)  +oo )  C_  RR )
43adantr 452 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( A [,)  +oo )  C_  RR )
5 ovolge0 19338 . . . . . . 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 0re 9055 . . . . . . . 8  |-  0  e.  RR
8 mnflt 10686 . . . . . . . 8  |-  ( 0  e.  RR  ->  -oo  <  0 )
97, 8ax-mp 8 . . . . . . 7  |-  -oo  <  0
10 ovolcl 19335 . . . . . . . . . 10  |-  ( ( A [,)  +oo )  C_  RR  ->  ( vol * `
 ( A [,)  +oo ) )  e.  RR* )
113, 10syl 16 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( vol * `  ( A [,)  +oo ) )  e. 
RR* )
1211adantr 452 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( vol
* `  ( A [,)  +oo ) )  e. 
RR* )
13 mnfxr 10678 . . . . . . . . 9  |-  -oo  e.  RR*
14 0xr 9095 . . . . . . . . 9  |-  0  e.  RR*
15 xrltletr 10711 . . . . . . . . 9  |-  ( ( 
-oo  e.  RR*  /\  0  e.  RR*  /\  ( vol
* `  ( A [,)  +oo ) )  e. 
RR* )  ->  (
(  -oo  <  0  /\  0  <_  ( vol
* `  ( A [,)  +oo ) ) )  ->  -oo  <  ( vol
* `  ( A [,)  +oo ) ) ) )
1613, 14, 15mp3an12 1269 . . . . . . . 8  |-  ( ( vol * `  ( A [,)  +oo ) )  e. 
RR*  ->  ( (  -oo  <  0  /\  0  <_ 
( vol * `  ( A [,)  +oo )
) )  ->  -oo  <  ( vol * `  ( A [,)  +oo ) ) ) )
1712, 16syl 16 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( ( 
-oo  <  0  /\  0  <_  ( vol * `  ( A [,)  +oo )
) )  ->  -oo  <  ( vol * `  ( A [,)  +oo ) ) ) )
189, 17mpani 658 . . . . . 6  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( 0  <_  ( vol * `  ( A [,)  +oo ) )  ->  -oo  <  ( vol * `  ( A [,)  +oo ) ) ) )
196, 18mpd 15 . . . . 5  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  -oo  <  ( vol * `  ( A [,)  +oo ) ) )
20 simpr 448 . . . . 5  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( vol
* `  ( A [,)  +oo ) )  <  +oo )
21 xrrebnd 10720 . . . . . 6  |-  ( ( vol * `  ( A [,)  +oo ) )  e. 
RR*  ->  ( ( vol
* `  ( A [,)  +oo ) )  e.  RR  <->  (  -oo  <  ( vol * `  ( A [,)  +oo ) )  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo ) ) )
2212, 21syl 16 . . . . 5  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( ( vol * `  ( A [,)  +oo ) )  e.  RR  <->  (  -oo  <  ( vol * `  ( A [,)  +oo ) )  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo ) ) )
2319, 20, 22mpbir2and 889 . . . 4  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( vol
* `  ( A [,)  +oo ) )  e.  RR )
2423ltp1d 9905 . . 3  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( vol
* `  ( A [,)  +oo ) )  < 
( ( vol * `  ( A [,)  +oo ) )  +  1 ) )
25 simpl 444 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  A  e.  RR )
26 peano2re 9203 . . . . . . . . 9  |-  ( ( vol * `  ( A [,)  +oo ) )  e.  RR  ->  ( ( vol * `  ( A [,)  +oo ) )  +  1 )  e.  RR )
2723, 26syl 16 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( ( vol * `  ( A [,)  +oo ) )  +  1 )  e.  RR )
2827, 25readdcld 9079 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( ( ( vol * `  ( A [,)  +oo )
)  +  1 )  +  A )  e.  RR )
297a1i 11 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  0  e.  RR )
3023lep1d 9906 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( vol
* `  ( A [,)  +oo ) )  <_ 
( ( vol * `  ( A [,)  +oo ) )  +  1 ) )
3129, 23, 27, 6, 30letrd 9191 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  0  <_ 
( ( vol * `  ( A [,)  +oo ) )  +  1 ) )
3225, 27addge02d 9579 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( 0  <_  ( ( vol
* `  ( A [,)  +oo ) )  +  1 )  <->  A  <_  ( ( ( vol * `  ( A [,)  +oo ) )  +  1 )  +  A ) ) )
3331, 32mpbid 202 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  A  <_ 
( ( ( vol
* `  ( A [,)  +oo ) )  +  1 )  +  A
) )
34 ovolicc 19380 . . . . . . 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 ) )
3525, 28, 33, 34syl3anc 1184 . . . . . 6  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( vol
* `  ( A [,] ( ( ( vol
* `  ( A [,)  +oo ) )  +  1 )  +  A
) ) )  =  ( ( ( ( vol * `  ( A [,)  +oo ) )  +  1 )  +  A
)  -  A ) )
3627recnd 9078 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( ( vol * `  ( A [,)  +oo ) )  +  1 )  e.  CC )
3725recnd 9078 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  A  e.  CC )
3836, 37pncand 9376 . . . . . 6  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( ( ( ( vol * `  ( A [,)  +oo ) )  +  1 )  +  A )  -  A )  =  ( ( vol * `  ( A [,)  +oo ) )  +  1 ) )
3935, 38eqtrd 2444 . . . . 5  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( vol
* `  ( A [,] ( ( ( vol
* `  ( A [,)  +oo ) )  +  1 )  +  A
) ) )  =  ( ( vol * `  ( A [,)  +oo ) )  +  1 ) )
40 elicc2 10939 . . . . . . . . . . . 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 ) ) ) )
4125, 28, 40syl2anc 643 . . . . . . . . . . 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 ) ) ) )
4241biimpa 471 . . . . . . . . . 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
) ) )
4342simp1d 969 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo )
)  <  +oo )  /\  x  e.  ( A [,] ( ( ( vol
* `  ( A [,)  +oo ) )  +  1 )  +  A
) ) )  ->  x  e.  RR )
4442simp2d 970 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo )
)  <  +oo )  /\  x  e.  ( A [,] ( ( ( vol
* `  ( A [,)  +oo ) )  +  1 )  +  A
) ) )  ->  A  <_  x )
45 elicopnf 10964 . . . . . . . . . 10  |-  ( A  e.  RR  ->  (
x  e.  ( A [,)  +oo )  <->  ( x  e.  RR  /\  A  <_  x ) ) )
4645ad2antrr 707 . . . . . . . . 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 ) ) )
4743, 44, 46mpbir2and 889 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo )
)  <  +oo )  /\  x  e.  ( A [,] ( ( ( vol
* `  ( A [,)  +oo ) )  +  1 )  +  A
) ) )  ->  x  e.  ( A [,)  +oo ) )
4847ex 424 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( x  e.  ( A [,] ( ( ( vol
* `  ( A [,)  +oo ) )  +  1 )  +  A
) )  ->  x  e.  ( A [,)  +oo ) ) )
4948ssrdv 3322 . . . . . 6  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( A [,] ( ( ( vol * `  ( A [,)  +oo ) )  +  1 )  +  A
) )  C_  ( A [,)  +oo ) )
50 ovolss 19342 . . . . . 6  |-  ( ( ( A [,] (
( ( vol * `  ( A [,)  +oo ) )  +  1 )  +  A ) )  C_  ( A [,)  +oo )  /\  ( A [,)  +oo )  C_  RR )  ->  ( vol * `  ( A [,] (
( ( vol * `  ( A [,)  +oo ) )  +  1 )  +  A ) ) )  <_  ( vol * `  ( A [,)  +oo ) ) )
5149, 4, 50syl2anc 643 . . . . 5  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( vol
* `  ( A [,] ( ( ( vol
* `  ( A [,)  +oo ) )  +  1 )  +  A
) ) )  <_ 
( vol * `  ( A [,)  +oo )
) )
5239, 51eqbrtrrd 4202 . . . 4  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( ( vol * `  ( A [,)  +oo ) )  +  1 )  <_  ( vol * `  ( A [,)  +oo ) ) )
5327, 23lenltd 9183 . . . 4  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( ( ( vol * `  ( A [,)  +oo )
)  +  1 )  <_  ( vol * `  ( A [,)  +oo ) )  <->  -.  ( vol * `  ( A [,)  +oo ) )  < 
( ( vol * `  ( A [,)  +oo ) )  +  1 ) ) )
5452, 53mpbid 202 . . 3  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  -.  ( vol * `  ( A [,)  +oo ) )  < 
( ( vol * `  ( A [,)  +oo ) )  +  1 ) )
5524, 54pm2.65da 560 . 2  |-  ( A  e.  RR  ->  -.  ( vol * `  ( A [,)  +oo ) )  <  +oo )
56 nltpnft 10718 . . 3  |-  ( ( vol * `  ( A [,)  +oo ) )  e. 
RR*  ->  ( ( vol
* `  ( A [,)  +oo ) )  = 
+oo 
<->  -.  ( vol * `  ( A [,)  +oo ) )  <  +oo ) )
5711, 56syl 16 . 2  |-  ( A  e.  RR  ->  (
( vol * `  ( A [,)  +oo )
)  =  +oo  <->  -.  ( vol * `  ( A [,)  +oo ) )  <  +oo ) )
5855, 57mpbird 224 1  |-  ( A  e.  RR  ->  ( vol * `  ( A [,)  +oo ) )  = 
+oo )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    C_ wss 3288   class class class wbr 4180   ` cfv 5421  (class class class)co 6048   RRcr 8953   0cc0 8954   1c1 8955    + caddc 8957    +oocpnf 9081    -oocmnf 9082   RR*cxr 9083    < clt 9084    <_ cle 9085    - cmin 9255   [,)cico 10882   [,]cicc 10883   vol *covol 19320
This theorem is referenced by:  ovolre  19382
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  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-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-fi 7382  df-sup 7412  df-oi 7443  df-card 7790  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-n0 10186  df-z 10247  df-uz 10453  df-q 10539  df-rp 10577  df-xneg 10674  df-xadd 10675  df-xmul 10676  df-ioo 10884  df-ico 10886  df-icc 10887  df-fz 11008  df-fzo 11099  df-seq 11287  df-exp 11346  df-hash 11582  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-clim 12245  df-sum 12443  df-rest 13613  df-topgen 13630  df-psmet 16657  df-xmet 16658  df-met 16659  df-bl 16660  df-mopn 16661  df-top 16926  df-bases 16928  df-topon 16929  df-cmp 17412  df-ovol 19322
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