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Theorem ioorcl 19338
Description: The function  F does not always return real numbers, but it does on intervals of finite volume. (Contributed by Mario Carneiro, 26-Mar-2015.)
Hypothesis
Ref Expression
ioorf.1  |-  F  =  ( x  e.  ran  (,)  |->  if ( x  =  (/) ,  <. 0 ,  0
>. ,  <. sup (
x ,  RR* ,  `'  <  ) ,  sup (
x ,  RR* ,  <  )
>. ) )
Assertion
Ref Expression
ioorcl  |-  ( ( A  e.  ran  (,)  /\  ( vol * `  A )  e.  RR )  ->  ( F `  A )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
Distinct variable group:    x, A
Allowed substitution hint:    F( x)

Proof of Theorem ioorcl
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss1 3506 . . 3  |-  (  <_  i^i  ( RR*  X.  RR* )
)  C_  <_
2 ioorf.1 . . . . . 6  |-  F  =  ( x  e.  ran  (,)  |->  if ( x  =  (/) ,  <. 0 ,  0
>. ,  <. sup (
x ,  RR* ,  `'  <  ) ,  sup (
x ,  RR* ,  <  )
>. ) )
32ioorf 19334 . . . . 5  |-  F : ran  (,) --> (  <_  i^i  ( RR*  X.  RR* )
)
43ffvelrni 5810 . . . 4  |-  ( A  e.  ran  (,)  ->  ( F `  A )  e.  (  <_  i^i  ( RR*  X.  RR* )
) )
54adantr 452 . . 3  |-  ( ( A  e.  ran  (,)  /\  ( vol * `  A )  e.  RR )  ->  ( F `  A )  e.  (  <_  i^i  ( RR*  X. 
RR* ) ) )
61, 5sseldi 3291 . 2  |-  ( ( A  e.  ran  (,)  /\  ( vol * `  A )  e.  RR )  ->  ( F `  A )  e.  <_  )
72ioorval 19335 . . . . . 6  |-  ( A  e.  ran  (,)  ->  ( F `  A )  =  if ( A  =  (/) ,  <. 0 ,  0 >. ,  <. sup ( A ,  RR* ,  `'  <  ) ,  sup ( A ,  RR* ,  <  )
>. ) )
87adantr 452 . . . . 5  |-  ( ( A  e.  ran  (,)  /\  ( vol * `  A )  e.  RR )  ->  ( F `  A )  =  if ( A  =  (/) , 
<. 0 ,  0
>. ,  <. sup ( A ,  RR* ,  `'  <  ) ,  sup ( A ,  RR* ,  <  )
>. ) )
9 iftrue 3690 . . . . 5  |-  ( A  =  (/)  ->  if ( A  =  (/) ,  <. 0 ,  0 >. , 
<. sup ( A ,  RR* ,  `'  <  ) ,  sup ( A ,  RR* ,  <  ) >.
)  =  <. 0 ,  0 >. )
108, 9sylan9eq 2441 . . . 4  |-  ( ( ( A  e.  ran  (,) 
/\  ( vol * `  A )  e.  RR )  /\  A  =  (/) )  ->  ( F `  A )  =  <. 0 ,  0 >. )
11 0re 9026 . . . . 5  |-  0  e.  RR
12 opelxpi 4852 . . . . 5  |-  ( ( 0  e.  RR  /\  0  e.  RR )  -> 
<. 0 ,  0
>.  e.  ( RR  X.  RR ) )
1311, 11, 12mp2an 654 . . . 4  |-  <. 0 ,  0 >.  e.  ( RR  X.  RR )
1410, 13syl6eqel 2477 . . 3  |-  ( ( ( A  e.  ran  (,) 
/\  ( vol * `  A )  e.  RR )  /\  A  =  (/) )  ->  ( F `  A )  e.  ( RR  X.  RR ) )
15 ioof 10936 . . . . . 6  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
16 ffn 5533 . . . . . 6  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
17 ovelrn 6163 . . . . . 6  |-  ( (,) 
Fn  ( RR*  X.  RR* )  ->  ( A  e. 
ran  (,)  <->  E. a  e.  RR*  E. b  e.  RR*  A  =  ( a (,) b ) ) )
1815, 16, 17mp2b 10 . . . . 5  |-  ( A  e.  ran  (,)  <->  E. a  e.  RR*  E. b  e. 
RR*  A  =  (
a (,) b ) )
192ioorinv2 19336 . . . . . . . . . 10  |-  ( ( a (,) b )  =/=  (/)  ->  ( F `  ( a (,) b
) )  =  <. a ,  b >. )
2019adantl 453 . . . . . . . . 9  |-  ( ( ( vol * `  ( a (,) b
) )  e.  RR  /\  ( a (,) b
)  =/=  (/) )  -> 
( F `  (
a (,) b ) )  =  <. a ,  b >. )
21 ioorcl2 19333 . . . . . . . . . . 11  |-  ( ( ( a (,) b
)  =/=  (/)  /\  ( vol * `  ( a (,) b ) )  e.  RR )  -> 
( a  e.  RR  /\  b  e.  RR ) )
2221ancoms 440 . . . . . . . . . 10  |-  ( ( ( vol * `  ( a (,) b
) )  e.  RR  /\  ( a (,) b
)  =/=  (/) )  -> 
( a  e.  RR  /\  b  e.  RR ) )
23 opelxpi 4852 . . . . . . . . . 10  |-  ( ( a  e.  RR  /\  b  e.  RR )  -> 
<. a ,  b >.  e.  ( RR  X.  RR ) )
2422, 23syl 16 . . . . . . . . 9  |-  ( ( ( vol * `  ( a (,) b
) )  e.  RR  /\  ( a (,) b
)  =/=  (/) )  ->  <. a ,  b >.  e.  ( RR  X.  RR ) )
2520, 24eqeltrd 2463 . . . . . . . 8  |-  ( ( ( vol * `  ( a (,) b
) )  e.  RR  /\  ( a (,) b
)  =/=  (/) )  -> 
( F `  (
a (,) b ) )  e.  ( RR 
X.  RR ) )
26 fveq2 5670 . . . . . . . . . . 11  |-  ( A  =  ( a (,) b )  ->  ( vol * `  A )  =  ( vol * `  ( a (,) b
) ) )
2726eleq1d 2455 . . . . . . . . . 10  |-  ( A  =  ( a (,) b )  ->  (
( vol * `  A )  e.  RR  <->  ( vol * `  (
a (,) b ) )  e.  RR ) )
28 neeq1 2560 . . . . . . . . . 10  |-  ( A  =  ( a (,) b )  ->  ( A  =/=  (/)  <->  ( a (,) b )  =/=  (/) ) )
2927, 28anbi12d 692 . . . . . . . . 9  |-  ( A  =  ( a (,) b )  ->  (
( ( vol * `  A )  e.  RR  /\  A  =/=  (/) )  <->  ( ( vol * `  ( a (,) b ) )  e.  RR  /\  (
a (,) b )  =/=  (/) ) ) )
30 fveq2 5670 . . . . . . . . . 10  |-  ( A  =  ( a (,) b )  ->  ( F `  A )  =  ( F `  ( a (,) b
) ) )
3130eleq1d 2455 . . . . . . . . 9  |-  ( A  =  ( a (,) b )  ->  (
( F `  A
)  e.  ( RR 
X.  RR )  <->  ( F `  ( a (,) b
) )  e.  ( RR  X.  RR ) ) )
3229, 31imbi12d 312 . . . . . . . 8  |-  ( A  =  ( a (,) b )  ->  (
( ( ( vol
* `  A )  e.  RR  /\  A  =/=  (/) )  ->  ( F `
 A )  e.  ( RR  X.  RR ) )  <->  ( (
( vol * `  ( a (,) b
) )  e.  RR  /\  ( a (,) b
)  =/=  (/) )  -> 
( F `  (
a (,) b ) )  e.  ( RR 
X.  RR ) ) ) )
3325, 32mpbiri 225 . . . . . . 7  |-  ( A  =  ( a (,) b )  ->  (
( ( vol * `  A )  e.  RR  /\  A  =/=  (/) )  -> 
( F `  A
)  e.  ( RR 
X.  RR ) ) )
3433a1i 11 . . . . . 6  |-  ( ( a  e.  RR*  /\  b  e.  RR* )  ->  ( A  =  ( a (,) b )  ->  (
( ( vol * `  A )  e.  RR  /\  A  =/=  (/) )  -> 
( F `  A
)  e.  ( RR 
X.  RR ) ) ) )
3534rexlimivv 2780 . . . . 5  |-  ( E. a  e.  RR*  E. b  e.  RR*  A  =  ( a (,) b )  ->  ( ( ( vol * `  A
)  e.  RR  /\  A  =/=  (/) )  ->  ( F `  A )  e.  ( RR  X.  RR ) ) )
3618, 35sylbi 188 . . . 4  |-  ( A  e.  ran  (,)  ->  ( ( ( vol * `  A )  e.  RR  /\  A  =/=  (/) )  -> 
( F `  A
)  e.  ( RR 
X.  RR ) ) )
3736impl 604 . . 3  |-  ( ( ( A  e.  ran  (,) 
/\  ( vol * `  A )  e.  RR )  /\  A  =/=  (/) )  -> 
( F `  A
)  e.  ( RR 
X.  RR ) )
3814, 37pm2.61dane 2630 . 2  |-  ( ( A  e.  ran  (,)  /\  ( vol * `  A )  e.  RR )  ->  ( F `  A )  e.  ( RR  X.  RR ) )
39 elin 3475 . 2  |-  ( ( F `  A )  e.  (  <_  i^i  ( RR  X.  RR ) )  <->  ( ( F `  A )  e.  <_  /\  ( F `  A )  e.  ( RR  X.  RR ) ) )
406, 38, 39sylanbrc 646 1  |-  ( ( A  e.  ran  (,)  /\  ( vol * `  A )  e.  RR )  ->  ( F `  A )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2552   E.wrex 2652    i^i cin 3264   (/)c0 3573   ifcif 3684   ~Pcpw 3744   <.cop 3762    e. cmpt 4209    X. cxp 4818   `'ccnv 4819   ran crn 4821    Fn wfn 5391   -->wf 5392   ` cfv 5396  (class class class)co 6022   supcsup 7382   RRcr 8924   0cc0 8925   RR*cxr 9054    < clt 9055    <_ cle 9056   (,)cioo 10850   vol *covol 19228
This theorem is referenced by:  uniioombllem2  19344
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-of 6246  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-2o 6663  df-oadd 6666  df-er 6843  df-map 6958  df-pm 6959  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-fi 7353  df-sup 7383  df-oi 7414  df-card 7761  df-cda 7983  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-n0 10156  df-z 10217  df-uz 10423  df-q 10509  df-rp 10547  df-xneg 10644  df-xadd 10645  df-xmul 10646  df-ioo 10854  df-ico 10856  df-icc 10857  df-fz 10978  df-fzo 11068  df-fl 11131  df-seq 11253  df-exp 11312  df-hash 11548  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-clim 12211  df-rlim 12212  df-sum 12409  df-rest 13579  df-topgen 13596  df-xmet 16621  df-met 16622  df-bl 16623  df-mopn 16624  df-top 16888  df-bases 16890  df-topon 16891  df-cmp 17374  df-ovol 19230  df-vol 19231
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