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Theorem ovolicopnf 21026
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 11111 . . . . . . . . 9  |- +oo  e.  RR*
2 icossre 11395 . . . . . . . . 9  |-  ( ( A  e.  RR  /\ +oo  e.  RR* )  ->  ( A [,) +oo )  C_  RR )
31, 2mpan2 671 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A [,) +oo )  C_  RR )
43adantr 465 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( A [,) +oo )  C_  RR )
5 ovolge0 20983 . . . . . . 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 mnflt0 11124 . . . . . . 7  |- -oo  <  0
8 ovolcl 20980 . . . . . . . . . 10  |-  ( ( A [,) +oo )  C_  RR  ->  ( vol* `  ( A [,) +oo ) )  e.  RR* )
93, 8syl 16 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( vol* `  ( A [,) +oo ) )  e.  RR* )
109adantr 465 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( vol* `  ( A [,) +oo ) )  e.  RR* )
11 mnfxr 11113 . . . . . . . . 9  |- -oo  e.  RR*
12 0xr 9449 . . . . . . . . 9  |-  0  e.  RR*
13 xrltletr 11150 . . . . . . . . 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 1304 . . . . . . . 8  |-  ( ( vol* `  ( A [,) +oo ) )  e.  RR*  ->  ( ( -oo  <  0  /\  0  <_  ( vol* `  ( A [,) +oo ) ) )  -> -oo  <  ( vol* `  ( A [,) +oo ) ) ) )
1510, 14syl 16 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
( -oo  <  0  /\  0  <_  ( vol* `  ( A [,) +oo ) ) )  -> -oo  <  ( vol* `  ( A [,) +oo ) ) ) )
167, 15mpani 676 . . . . . 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 461 . . . . 5  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( vol* `  ( A [,) +oo ) )  < +oo )
19 xrrebnd 11159 . . . . . 6  |-  ( ( vol* `  ( A [,) +oo ) )  e.  RR*  ->  ( ( vol* `  ( A [,) +oo ) )  e.  RR  <->  ( -oo  <  ( vol* `  ( A [,) +oo )
)  /\  ( vol* `  ( A [,) +oo ) )  < +oo ) ) )
2010, 19syl 16 . . . . 5  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
( vol* `  ( A [,) +oo )
)  e.  RR  <->  ( -oo  <  ( vol* `  ( A [,) +oo )
)  /\  ( vol* `  ( A [,) +oo ) )  < +oo ) ) )
2117, 18, 20mpbir2and 913 . . . 4  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( vol* `  ( A [,) +oo ) )  e.  RR )
2221ltp1d 10282 . . 3  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( vol* `  ( A [,) +oo ) )  <  ( ( vol* `  ( A [,) +oo ) )  +  1 ) )
23 simpl 457 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  A  e.  RR )
24 peano2re 9561 . . . . . . . . 9  |-  ( ( vol* `  ( A [,) +oo ) )  e.  RR  ->  (
( vol* `  ( A [,) +oo )
)  +  1 )  e.  RR )
2521, 24syl 16 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
( vol* `  ( A [,) +oo )
)  +  1 )  e.  RR )
2625, 23readdcld 9432 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A )  e.  RR )
27 0red 9406 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  0  e.  RR )
2821lep1d 10283 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( vol* `  ( A [,) +oo ) )  <_  ( ( vol* `  ( A [,) +oo ) )  +  1 ) )
2927, 21, 25, 6, 28letrd 9547 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  0  <_  ( ( vol* `  ( A [,) +oo ) )  +  1 ) )
3023, 25addge02d 9947 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
0  <_  ( ( vol* `  ( A [,) +oo ) )  +  1 )  <->  A  <_  ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) ) )
3129, 30mpbid 210 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  A  <_  ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A
) )
32 ovolicc 21025 . . . . . . 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 1218 . . . . . 6  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( vol* `  ( A [,] ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) ) )  =  ( ( ( ( vol* `  ( A [,) +oo )
)  +  1 )  +  A )  -  A ) )
3425recnd 9431 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
( vol* `  ( A [,) +oo )
)  +  1 )  e.  CC )
3523recnd 9431 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  A  e.  CC )
3634, 35pncand 9739 . . . . . 6  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A
)  -  A )  =  ( ( vol* `  ( A [,) +oo ) )  +  1 ) )
3733, 36eqtrd 2475 . . . . 5  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( vol* `  ( A [,] ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) ) )  =  ( ( vol* `  ( A [,) +oo ) )  +  1 ) )
38 elicc2 11379 . . . . . . . . . . . 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 661 . . . . . . . . . . 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 484 . . . . . . . . . 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 1000 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo )
)  < +oo )  /\  x  e.  ( A [,] ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) ) )  ->  x  e.  RR )
4240simp2d 1001 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo )
)  < +oo )  /\  x  e.  ( A [,] ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) ) )  ->  A  <_  x
)
43 elicopnf 11404 . . . . . . . . . 10  |-  ( A  e.  RR  ->  (
x  e.  ( A [,) +oo )  <->  ( x  e.  RR  /\  A  <_  x ) ) )
4443ad2antrr 725 . . . . . . . . 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 913 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo )
)  < +oo )  /\  x  e.  ( A [,] ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) ) )  ->  x  e.  ( A [,) +oo )
)
4645ex 434 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
x  e.  ( A [,] ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) )  ->  x  e.  ( A [,) +oo ) ) )
4746ssrdv 3381 . . . . . 6  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( A [,] ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) )  C_  ( A [,) +oo )
)
48 ovolss 20987 . . . . . 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 661 . . . . 5  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  ( vol* `  ( A [,] ( ( ( vol* `  ( A [,) +oo ) )  +  1 )  +  A ) ) )  <_  ( vol* `  ( A [,) +oo ) ) )
5037, 49eqbrtrrd 4333 . . . 4  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
( vol* `  ( A [,) +oo )
)  +  1 )  <_  ( vol* `  ( A [,) +oo ) ) )
5125, 21lenltd 9539 . . . 4  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  (
( ( vol* `  ( A [,) +oo ) )  +  1 )  <_  ( vol* `  ( A [,) +oo ) )  <->  -.  ( vol* `  ( A [,) +oo ) )  <  ( ( vol* `  ( A [,) +oo ) )  +  1 ) ) )
5250, 51mpbid 210 . . 3  |-  ( ( A  e.  RR  /\  ( vol* `  ( A [,) +oo ) )  < +oo )  ->  -.  ( vol* `  ( A [,) +oo ) )  <  ( ( vol* `  ( A [,) +oo ) )  +  1 ) )
5322, 52pm2.65da 576 . 2  |-  ( A  e.  RR  ->  -.  ( vol* `  ( A [,) +oo ) )  < +oo )
54 nltpnft 11157 . . 3  |-  ( ( vol* `  ( A [,) +oo ) )  e.  RR*  ->  ( ( vol* `  ( A [,) +oo ) )  = +oo  <->  -.  ( vol* `  ( A [,) +oo ) )  < +oo ) )
559, 54syl 16 . 2  |-  ( A  e.  RR  ->  (
( vol* `  ( A [,) +oo )
)  = +oo  <->  -.  ( vol* `  ( A [,) +oo ) )  < +oo ) )
5653, 55mpbird 232 1  |-  ( A  e.  RR  ->  ( vol* `  ( A [,) +oo ) )  = +oo )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    C_ wss 3347   class class class wbr 4311   ` cfv 5437  (class class class)co 6110   RRcr 9300   0cc0 9301   1c1 9302    + caddc 9304   +oocpnf 9434   -oocmnf 9435   RR*cxr 9436    < clt 9437    <_ cle 9438    - cmin 9614   [,)cico 11321   [,]cicc 11322   vol*covol 20965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-inf2 7866  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378  ax-pre-sup 9379
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-int 4148  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-se 4699  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-isom 5446  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-om 6496  df-1st 6596  df-2nd 6597  df-recs 6851  df-rdg 6885  df-1o 6939  df-oadd 6943  df-er 7120  df-map 7235  df-en 7330  df-dom 7331  df-sdom 7332  df-fin 7333  df-fi 7680  df-sup 7710  df-oi 7743  df-card 8128  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-div 10013  df-nn 10342  df-2 10399  df-3 10400  df-n0 10599  df-z 10666  df-uz 10881  df-q 10973  df-rp 11011  df-xneg 11108  df-xadd 11109  df-xmul 11110  df-ioo 11323  df-ico 11325  df-icc 11326  df-fz 11457  df-fzo 11568  df-seq 11826  df-exp 11885  df-hash 12123  df-cj 12607  df-re 12608  df-im 12609  df-sqr 12743  df-abs 12744  df-clim 12985  df-sum 13183  df-rest 14380  df-topgen 14401  df-psmet 17828  df-xmet 17829  df-met 17830  df-bl 17831  df-mopn 17832  df-top 18522  df-bases 18524  df-topon 18525  df-cmp 19009  df-ovol 20967
This theorem is referenced by:  ovolre  21027
  Copyright terms: Public domain W3C validator