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Theorem ovolicopnf 19425
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 10718 . . . . . . . . 9  |-  +oo  e.  RR*
2 icossre 10996 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  +oo 
e.  RR* )  ->  ( A [,)  +oo )  C_  RR )
31, 2mpan2 654 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A [,)  +oo )  C_  RR )
43adantr 453 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( A [,)  +oo )  C_  RR )
5 ovolge0 19382 . . . . . . 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 9096 . . . . . . . 8  |-  0  e.  RR
8 mnflt 10727 . . . . . . . 8  |-  ( 0  e.  RR  ->  -oo  <  0 )
97, 8ax-mp 5 . . . . . . 7  |-  -oo  <  0
10 ovolcl 19379 . . . . . . . . . 10  |-  ( ( A [,)  +oo )  C_  RR  ->  ( vol * `
 ( A [,)  +oo ) )  e.  RR* )
113, 10syl 16 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( vol * `  ( A [,)  +oo ) )  e. 
RR* )
1211adantr 453 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( vol
* `  ( A [,)  +oo ) )  e. 
RR* )
13 mnfxr 10719 . . . . . . . . 9  |-  -oo  e.  RR*
14 0xr 9136 . . . . . . . . 9  |-  0  e.  RR*
15 xrltletr 10752 . . . . . . . . 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 1270 . . . . . . . 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 659 . . . . . 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 449 . . . . 5  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( vol
* `  ( A [,)  +oo ) )  <  +oo )
21 xrrebnd 10761 . . . . . 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 890 . . . 4  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( vol
* `  ( A [,)  +oo ) )  e.  RR )
2423ltp1d 9946 . . 3  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( vol
* `  ( A [,)  +oo ) )  < 
( ( vol * `  ( A [,)  +oo ) )  +  1 ) )
25 simpl 445 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  A  e.  RR )
26 peano2re 9244 . . . . . . . . 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 9120 . . . . . . 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 9947 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( vol
* `  ( A [,)  +oo ) )  <_ 
( ( vol * `  ( A [,)  +oo ) )  +  1 ) )
3129, 23, 27, 6, 30letrd 9232 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  0  <_ 
( ( vol * `  ( A [,)  +oo ) )  +  1 ) )
3225, 27addge02d 9620 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( 0  <_  ( ( vol
* `  ( A [,)  +oo ) )  +  1 )  <->  A  <_  ( ( ( vol * `  ( A [,)  +oo ) )  +  1 )  +  A ) ) )
3331, 32mpbid 203 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  A  <_ 
( ( ( vol
* `  ( A [,)  +oo ) )  +  1 )  +  A
) )
34 ovolicc 19424 . . . . . . 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 1185 . . . . . 6  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( vol
* `  ( A [,] ( ( ( vol
* `  ( A [,)  +oo ) )  +  1 )  +  A
) ) )  =  ( ( ( ( vol * `  ( A [,)  +oo ) )  +  1 )  +  A
)  -  A ) )
3627recnd 9119 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( ( vol * `  ( A [,)  +oo ) )  +  1 )  e.  CC )
3725recnd 9119 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  A  e.  CC )
3836, 37pncand 9417 . . . . . 6  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( ( ( ( vol * `  ( A [,)  +oo ) )  +  1 )  +  A )  -  A )  =  ( ( vol * `  ( A [,)  +oo ) )  +  1 ) )
3935, 38eqtrd 2470 . . . . 5  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( vol
* `  ( A [,] ( ( ( vol
* `  ( A [,)  +oo ) )  +  1 )  +  A
) ) )  =  ( ( vol * `  ( A [,)  +oo ) )  +  1 ) )
40 elicc2 10980 . . . . . . . . . . . 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 644 . . . . . . . . . . 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 472 . . . . . . . . . 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 970 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo )
)  <  +oo )  /\  x  e.  ( A [,] ( ( ( vol
* `  ( A [,)  +oo ) )  +  1 )  +  A
) ) )  ->  x  e.  RR )
4442simp2d 971 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo )
)  <  +oo )  /\  x  e.  ( A [,] ( ( ( vol
* `  ( A [,)  +oo ) )  +  1 )  +  A
) ) )  ->  A  <_  x )
45 elicopnf 11005 . . . . . . . . . 10  |-  ( A  e.  RR  ->  (
x  e.  ( A [,)  +oo )  <->  ( x  e.  RR  /\  A  <_  x ) ) )
4645ad2antrr 708 . . . . . . . . 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 890 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo )
)  <  +oo )  /\  x  e.  ( A [,] ( ( ( vol
* `  ( A [,)  +oo ) )  +  1 )  +  A
) ) )  ->  x  e.  ( A [,)  +oo ) )
4847ex 425 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( x  e.  ( A [,] ( ( ( vol
* `  ( A [,)  +oo ) )  +  1 )  +  A
) )  ->  x  e.  ( A [,)  +oo ) ) )
4948ssrdv 3356 . . . . . 6  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( A [,] ( ( ( vol * `  ( A [,)  +oo ) )  +  1 )  +  A
) )  C_  ( A [,)  +oo ) )
50 ovolss 19386 . . . . . 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 644 . . . . 5  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( vol
* `  ( A [,] ( ( ( vol
* `  ( A [,)  +oo ) )  +  1 )  +  A
) ) )  <_ 
( vol * `  ( A [,)  +oo )
) )
5239, 51eqbrtrrd 4237 . . . 4  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( ( vol * `  ( A [,)  +oo ) )  +  1 )  <_  ( vol * `  ( A [,)  +oo ) ) )
5327, 23lenltd 9224 . . . 4  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( ( ( vol * `  ( A [,)  +oo )
)  +  1 )  <_  ( vol * `  ( A [,)  +oo ) )  <->  -.  ( vol * `  ( A [,)  +oo ) )  < 
( ( vol * `  ( A [,)  +oo ) )  +  1 ) ) )
5452, 53mpbid 203 . . 3  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  -.  ( vol * `  ( A [,)  +oo ) )  < 
( ( vol * `  ( A [,)  +oo ) )  +  1 ) )
5524, 54pm2.65da 561 . 2  |-  ( A  e.  RR  ->  -.  ( vol * `  ( A [,)  +oo ) )  <  +oo )
56 nltpnft 10759 . . 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 225 1  |-  ( A  e.  RR  ->  ( vol * `  ( A [,)  +oo ) )  = 
+oo )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    C_ wss 3322   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   RRcr 8994   0cc0 8995   1c1 8996    + caddc 8998    +oocpnf 9122    -oocmnf 9123   RR*cxr 9124    < clt 9125    <_ cle 9126    - cmin 9296   [,)cico 10923   [,]cicc 10924   vol *covol 19364
This theorem is referenced by:  ovolre  19426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-fi 7419  df-sup 7449  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-q 10580  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-ioo 10925  df-ico 10927  df-icc 10928  df-fz 11049  df-fzo 11141  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-sum 12485  df-rest 13655  df-topgen 13672  df-psmet 16699  df-xmet 16700  df-met 16701  df-bl 16702  df-mopn 16703  df-top 16968  df-bases 16970  df-topon 16971  df-cmp 17455  df-ovol 19366
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