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Theorem ovolicopnf 18981
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 10544 . . . . . . . . 9  |-  +oo  e.  RR*
2 icossre 10819 . . . . . . . . 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 18938 . . . . . . 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 8925 . . . . . . . 8  |-  0  e.  RR
8 mnflt 10553 . . . . . . . 8  |-  ( 0  e.  RR  ->  -oo  <  0 )
97, 8ax-mp 8 . . . . . . 7  |-  -oo  <  0
10 ovolcl 18935 . . . . . . . . . 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 10545 . . . . . . . . 9  |-  -oo  e.  RR*
14 0xr 8965 . . . . . . . . 9  |-  0  e.  RR*
15 xrltletr 10577 . . . . . . . . 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 10586 . . . . . 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 9774 . . 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 9072 . . . . . . . . 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 8949 . . . . . . 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 9775 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( vol
* `  ( A [,)  +oo ) )  <_ 
( ( vol * `  ( A [,)  +oo ) )  +  1 ) )
3129, 23, 27, 6, 30letrd 9060 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  0  <_ 
( ( vol * `  ( A [,)  +oo ) )  +  1 ) )
3225, 27addge02d 9448 . . . . . . . 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 18980 . . . . . . 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 8948 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( ( vol * `  ( A [,)  +oo ) )  +  1 )  e.  CC )
3725recnd 8948 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  A  e.  CC )
3836, 37pncand 9245 . . . . . 6  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( ( ( ( vol * `  ( A [,)  +oo ) )  +  1 )  +  A )  -  A )  =  ( ( vol * `  ( A [,)  +oo ) )  +  1 ) )
3935, 38eqtrd 2390 . . . . 5  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( vol
* `  ( A [,] ( ( ( vol
* `  ( A [,)  +oo ) )  +  1 )  +  A
) ) )  =  ( ( vol * `  ( A [,)  +oo ) )  +  1 ) )
40 elicc2 10804 . . . . . . . . . . . 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 10828 . . . . . . . . . 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 3261 . . . . . 6  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( A [,] ( ( ( vol * `  ( A [,)  +oo ) )  +  1 )  +  A
) )  C_  ( A [,)  +oo ) )
50 ovolss 18942 . . . . . 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 4124 . . . 4  |-  ( ( A  e.  RR  /\  ( vol * `  ( A [,)  +oo ) )  <  +oo )  ->  ( ( vol * `  ( A [,)  +oo ) )  +  1 )  <_  ( vol * `  ( A [,)  +oo ) ) )
5327, 23lenltd 9052 . . . 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 10584 . . 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 1642    e. wcel 1710    C_ wss 3228   class class class wbr 4102   ` cfv 5334  (class class class)co 5942   RRcr 8823   0cc0 8824   1c1 8825    + caddc 8827    +oocpnf 8951    -oocmnf 8952   RR*cxr 8953    < clt 8954    <_ cle 8955    - cmin 9124   [,)cico 10747   [,]cicc 10748   vol *covol 18920
This theorem is referenced by:  ovolre  18982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-inf2 7429  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-se 4432  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-isom 5343  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-oadd 6567  df-er 6744  df-map 6859  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-fi 7252  df-sup 7281  df-oi 7312  df-card 7659  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-n0 10055  df-z 10114  df-uz 10320  df-q 10406  df-rp 10444  df-xneg 10541  df-xadd 10542  df-xmul 10543  df-ioo 10749  df-ico 10751  df-icc 10752  df-fz 10872  df-fzo 10960  df-seq 11136  df-exp 11195  df-hash 11428  df-cj 11674  df-re 11675  df-im 11676  df-sqr 11810  df-abs 11811  df-clim 12052  df-sum 12250  df-rest 13420  df-topgen 13437  df-xmet 16469  df-met 16470  df-bl 16471  df-mopn 16472  df-top 16736  df-bases 16738  df-topon 16739  df-cmp 17214  df-ovol 18922
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