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Theorem ovolicopnf 18883
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 10455 . . . . . . . . 9  |-  +oo  e.  RR*
2 icossre 10730 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  +oo 
e.  RR* )  ->  ( A [,)  +oo )  C_  RR )
31, 2mpan2 652 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A [,)  +oo )  C_  RR )
43adantr 451 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( A [,)  +oo )  C_  RR )
5 ovolge0 18840 . . . . . . 7  |-  ( ( A [,)  +oo )  C_  RR  ->  0  <_  ( vol * `  ( A [,)  +oo ) ) )
64, 5syl 15 . . . . . 6  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  0  <_ 
( vol * `  ( A [,)  +oo )
) )
7 0re 8838 . . . . . . . 8  |-  0  e.  RR
8 mnflt 10464 . . . . . . . 8  |-  ( 0  e.  RR  ->  -oo  <  0 )
97, 8ax-mp 8 . . . . . . 7  |-  -oo  <  0
10 ovolcl 18837 . . . . . . . . . 10  |-  ( ( A [,)  +oo )  C_  RR  ->  ( vol * `
 ( A [,)  +oo ) )  e.  RR* )
113, 10syl 15 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( vol * `  ( A [,)  +oo ) )  e. 
RR* )
1211adantr 451 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( vol
* `  ( A [,)  +oo ) )  e. 
RR* )
13 mnfxr 10456 . . . . . . . . 9  |-  -oo  e.  RR*
14 0xr 8878 . . . . . . . . 9  |-  0  e.  RR*
15 xrltletr 10488 . . . . . . . . 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 1267 . . . . . . . 8  |-  ( ( vol * `  ( A [,)  +oo ) )  e. 
RR*  ->  ( (  -oo  <  0  /\  0  <_ 
( vol * `  ( A [,)  +oo )
) )  ->  -oo  <  ( vol * `  ( A [,)  +oo ) ) ) )
1712, 16syl 15 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( ( 
-oo  <  0  /\  0  <_  ( vol * `  ( A [,)  +oo )
) )  ->  -oo  <  ( vol * `  ( A [,)  +oo ) ) ) )
189, 17mpani 657 . . . . . 6  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( 0  <_  ( vol * `  ( A [,)  +oo ) )  ->  -oo  <  ( vol * `  ( A [,)  +oo ) ) ) )
196, 18mpd 14 . . . . 5  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  -oo  <  ( vol * `  ( A [,)  +oo ) ) )
20 simpr 447 . . . . 5  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( vol
* `  ( A [,)  +oo ) )  <  +oo )
21 xrrebnd 10497 . . . . . 6  |-  ( ( vol * `  ( A [,)  +oo ) )  e. 
RR*  ->  ( ( vol
* `  ( A [,)  +oo ) )  e.  RR  <->  (  -oo  <  ( vol * `  ( A [,)  +oo ) )  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo ) ) )
2212, 21syl 15 . . . . 5  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( ( vol * `  ( A [,)  +oo ) )  e.  RR  <->  (  -oo  <  ( vol * `  ( A [,)  +oo ) )  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo ) ) )
2319, 20, 22mpbir2and 888 . . . 4  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( vol
* `  ( A [,)  +oo ) )  e.  RR )
2423ltp1d 9687 . . 3  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( vol
* `  ( A [,)  +oo ) )  < 
( ( vol * `  ( A [,)  +oo ) )  +  1 ) )
25 simpl 443 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  A  e.  RR )
26 peano2re 8985 . . . . . . . . 9  |-  ( ( vol * `  ( A [,)  +oo ) )  e.  RR  ->  ( ( vol * `  ( A [,)  +oo ) )  +  1 )  e.  RR )
2723, 26syl 15 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( ( vol * `  ( A [,)  +oo ) )  +  1 )  e.  RR )
2827, 25readdcld 8862 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( ( ( vol * `  ( A [,)  +oo )
)  +  1 )  +  A )  e.  RR )
297a1i 10 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  0  e.  RR )
3023lep1d 9688 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( vol
* `  ( A [,)  +oo ) )  <_ 
( ( vol * `  ( A [,)  +oo ) )  +  1 ) )
3129, 23, 27, 6, 30letrd 8973 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  0  <_ 
( ( vol * `  ( A [,)  +oo ) )  +  1 ) )
3225, 27addge02d 9361 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( 0  <_  ( ( vol
* `  ( A [,)  +oo ) )  +  1 )  <->  A  <_  ( ( ( vol * `  ( A [,)  +oo ) )  +  1 )  +  A ) ) )
3331, 32mpbid 201 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  A  <_ 
( ( ( vol
* `  ( A [,)  +oo ) )  +  1 )  +  A
) )
34 ovolicc 18882 . . . . . . 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 1182 . . . . . 6  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( vol
* `  ( A [,] ( ( ( vol
* `  ( A [,)  +oo ) )  +  1 )  +  A
) ) )  =  ( ( ( ( vol * `  ( A [,)  +oo ) )  +  1 )  +  A
)  -  A ) )
3627recnd 8861 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( ( vol * `  ( A [,)  +oo ) )  +  1 )  e.  CC )
3725recnd 8861 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  A  e.  CC )
3836, 37pncand 9158 . . . . . 6  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( ( ( ( vol * `  ( A [,)  +oo ) )  +  1 )  +  A )  -  A )  =  ( ( vol * `  ( A [,)  +oo ) )  +  1 ) )
3935, 38eqtrd 2315 . . . . 5  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( vol
* `  ( A [,] ( ( ( vol
* `  ( A [,)  +oo ) )  +  1 )  +  A
) ) )  =  ( ( vol * `  ( A [,)  +oo ) )  +  1 ) )
40 elicc2 10715 . . . . . . . . . . . 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 642 . . . . . . . . . . 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 470 . . . . . . . . . 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 967 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo )
)  <  +oo )  /\  x  e.  ( A [,] ( ( ( vol
* `  ( A [,)  +oo ) )  +  1 )  +  A
) ) )  ->  x  e.  RR )
4442simp2d 968 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo )
)  <  +oo )  /\  x  e.  ( A [,] ( ( ( vol
* `  ( A [,)  +oo ) )  +  1 )  +  A
) ) )  ->  A  <_  x )
45 elicopnf 10739 . . . . . . . . . 10  |-  ( A  e.  RR  ->  (
x  e.  ( A [,)  +oo )  <->  ( x  e.  RR  /\  A  <_  x ) ) )
4645ad2antrr 706 . . . . . . . . 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 888 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo )
)  <  +oo )  /\  x  e.  ( A [,] ( ( ( vol
* `  ( A [,)  +oo ) )  +  1 )  +  A
) ) )  ->  x  e.  ( A [,)  +oo ) )
4847ex 423 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( x  e.  ( A [,] ( ( ( vol
* `  ( A [,)  +oo ) )  +  1 )  +  A
) )  ->  x  e.  ( A [,)  +oo ) ) )
4948ssrdv 3185 . . . . . 6  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( A [,] ( ( ( vol * `  ( A [,)  +oo ) )  +  1 )  +  A
) )  C_  ( A [,)  +oo ) )
50 ovolss 18844 . . . . . 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 642 . . . . 5  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( vol
* `  ( A [,] ( ( ( vol
* `  ( A [,)  +oo ) )  +  1 )  +  A
) ) )  <_ 
( vol * `  ( A [,)  +oo )
) )
5239, 51eqbrtrrd 4045 . . . 4  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( ( vol * `  ( A [,)  +oo ) )  +  1 )  <_  ( vol * `  ( A [,)  +oo ) ) )
5327, 23lenltd 8965 . . . 4  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( ( ( vol * `  ( A [,)  +oo )
)  +  1 )  <_  ( vol * `  ( A [,)  +oo ) )  <->  -.  ( vol * `  ( A [,)  +oo ) )  < 
( ( vol * `  ( A [,)  +oo ) )  +  1 ) ) )
5452, 53mpbid 201 . . 3  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  -.  ( vol * `  ( A [,)  +oo ) )  < 
( ( vol * `  ( A [,)  +oo ) )  +  1 ) )
5524, 54pm2.65da 559 . 2  |-  ( A  e.  RR  ->  -.  ( vol * `  ( A [,)  +oo ) )  <  +oo )
56 nltpnft 10495 . . 3  |-  ( ( vol * `  ( A [,)  +oo ) )  e. 
RR*  ->  ( ( vol
* `  ( A [,)  +oo ) )  = 
+oo 
<->  -.  ( vol * `  ( A [,)  +oo ) )  <  +oo ) )
5711, 56syl 15 . 2  |-  ( A  e.  RR  ->  (
( vol * `  ( A [,)  +oo )
)  =  +oo  <->  -.  ( vol * `  ( A [,)  +oo ) )  <  +oo ) )
5855, 57mpbird 223 1  |-  ( A  e.  RR  ->  ( vol * `  ( A [,)  +oo ) )  = 
+oo )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    +oocpnf 8864    -oocmnf 8865   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037   [,)cico 10658   [,]cicc 10659   vol *covol 18822
This theorem is referenced by:  ovolre  18884
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-rest 13327  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-cmp 17114  df-ovol 18824
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