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Theorem ioorcl 18932
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 3389 . . 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 18928 . . . . 5  |-  F : ran  (,) --> (  <_  i^i  ( RR*  X.  RR* )
)
43ffvelrni 5664 . . . 4  |-  ( A  e.  ran  (,)  ->  ( F `  A )  e.  (  <_  i^i  ( RR*  X.  RR* )
) )
54adantr 451 . . 3  |-  ( ( A  e.  ran  (,)  /\  ( vol * `  A )  e.  RR )  ->  ( F `  A )  e.  (  <_  i^i  ( RR*  X. 
RR* ) ) )
61, 5sseldi 3178 . 2  |-  ( ( A  e.  ran  (,)  /\  ( vol * `  A )  e.  RR )  ->  ( F `  A )  e.  <_  )
72ioorval 18929 . . . . . 6  |-  ( A  e.  ran  (,)  ->  ( F `  A )  =  if ( A  =  (/) ,  <. 0 ,  0 >. ,  <. sup ( A ,  RR* ,  `'  <  ) ,  sup ( A ,  RR* ,  <  )
>. ) )
87adantr 451 . . . . 5  |-  ( ( A  e.  ran  (,)  /\  ( vol * `  A )  e.  RR )  ->  ( F `  A )  =  if ( A  =  (/) , 
<. 0 ,  0
>. ,  <. sup ( A ,  RR* ,  `'  <  ) ,  sup ( A ,  RR* ,  <  )
>. ) )
9 iftrue 3571 . . . . 5  |-  ( A  =  (/)  ->  if ( A  =  (/) ,  <. 0 ,  0 >. , 
<. sup ( A ,  RR* ,  `'  <  ) ,  sup ( A ,  RR* ,  <  ) >.
)  =  <. 0 ,  0 >. )
108, 9sylan9eq 2335 . . . 4  |-  ( ( ( A  e.  ran  (,) 
/\  ( vol * `  A )  e.  RR )  /\  A  =  (/) )  ->  ( F `  A )  =  <. 0 ,  0 >. )
11 0re 8838 . . . . 5  |-  0  e.  RR
12 opelxpi 4721 . . . . 5  |-  ( ( 0  e.  RR  /\  0  e.  RR )  -> 
<. 0 ,  0
>.  e.  ( RR  X.  RR ) )
1311, 11, 12mp2an 653 . . . 4  |-  <. 0 ,  0 >.  e.  ( RR  X.  RR )
1410, 13syl6eqel 2371 . . 3  |-  ( ( ( A  e.  ran  (,) 
/\  ( vol * `  A )  e.  RR )  /\  A  =  (/) )  ->  ( F `  A )  e.  ( RR  X.  RR ) )
15 ioof 10741 . . . . . 6  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
16 ffn 5389 . . . . . 6  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
17 ovelrn 5996 . . . . . 6  |-  ( (,) 
Fn  ( RR*  X.  RR* )  ->  ( A  e. 
ran  (,)  <->  E. a  e.  RR*  E. b  e.  RR*  A  =  ( a (,) b ) ) )
1815, 16, 17mp2b 9 . . . . 5  |-  ( A  e.  ran  (,)  <->  E. a  e.  RR*  E. b  e. 
RR*  A  =  (
a (,) b ) )
192ioorinv2 18930 . . . . . . . . . 10  |-  ( ( a (,) b )  =/=  (/)  ->  ( F `  ( a (,) b
) )  =  <. a ,  b >. )
2019adantl 452 . . . . . . . . 9  |-  ( ( ( vol * `  ( a (,) b
) )  e.  RR  /\  ( a (,) b
)  =/=  (/) )  -> 
( F `  (
a (,) b ) )  =  <. a ,  b >. )
21 ioorcl2 18927 . . . . . . . . . . 11  |-  ( ( ( a (,) b
)  =/=  (/)  /\  ( vol * `  ( a (,) b ) )  e.  RR )  -> 
( a  e.  RR  /\  b  e.  RR ) )
2221ancoms 439 . . . . . . . . . 10  |-  ( ( ( vol * `  ( a (,) b
) )  e.  RR  /\  ( a (,) b
)  =/=  (/) )  -> 
( a  e.  RR  /\  b  e.  RR ) )
23 opelxpi 4721 . . . . . . . . . 10  |-  ( ( a  e.  RR  /\  b  e.  RR )  -> 
<. a ,  b >.  e.  ( RR  X.  RR ) )
2422, 23syl 15 . . . . . . . . 9  |-  ( ( ( vol * `  ( a (,) b
) )  e.  RR  /\  ( a (,) b
)  =/=  (/) )  ->  <. a ,  b >.  e.  ( RR  X.  RR ) )
2520, 24eqeltrd 2357 . . . . . . . 8  |-  ( ( ( vol * `  ( a (,) b
) )  e.  RR  /\  ( a (,) b
)  =/=  (/) )  -> 
( F `  (
a (,) b ) )  e.  ( RR 
X.  RR ) )
26 fveq2 5525 . . . . . . . . . . 11  |-  ( A  =  ( a (,) b )  ->  ( vol * `  A )  =  ( vol * `  ( a (,) b
) ) )
2726eleq1d 2349 . . . . . . . . . 10  |-  ( A  =  ( a (,) b )  ->  (
( vol * `  A )  e.  RR  <->  ( vol * `  (
a (,) b ) )  e.  RR ) )
28 neeq1 2454 . . . . . . . . . 10  |-  ( A  =  ( a (,) b )  ->  ( A  =/=  (/)  <->  ( a (,) b )  =/=  (/) ) )
2927, 28anbi12d 691 . . . . . . . . 9  |-  ( A  =  ( a (,) b )  ->  (
( ( vol * `  A )  e.  RR  /\  A  =/=  (/) )  <->  ( ( vol * `  ( a (,) b ) )  e.  RR  /\  (
a (,) b )  =/=  (/) ) ) )
30 fveq2 5525 . . . . . . . . . 10  |-  ( A  =  ( a (,) b )  ->  ( F `  A )  =  ( F `  ( a (,) b
) ) )
3130eleq1d 2349 . . . . . . . . 9  |-  ( A  =  ( a (,) b )  ->  (
( F `  A
)  e.  ( RR 
X.  RR )  <->  ( F `  ( a (,) b
) )  e.  ( RR  X.  RR ) ) )
3229, 31imbi12d 311 . . . . . . . 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 224 . . . . . . 7  |-  ( A  =  ( a (,) b )  ->  (
( ( vol * `  A )  e.  RR  /\  A  =/=  (/) )  -> 
( F `  A
)  e.  ( RR 
X.  RR ) ) )
3433a1i 10 . . . . . 6  |-  ( ( a  e.  RR*  /\  b  e.  RR* )  ->  ( A  =  ( a (,) b )  ->  (
( ( vol * `  A )  e.  RR  /\  A  =/=  (/) )  -> 
( F `  A
)  e.  ( RR 
X.  RR ) ) ) )
3534rexlimivv 2672 . . . . 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 187 . . . 4  |-  ( A  e.  ran  (,)  ->  ( ( ( vol * `  A )  e.  RR  /\  A  =/=  (/) )  -> 
( F `  A
)  e.  ( RR 
X.  RR ) ) )
3736impl 603 . . 3  |-  ( ( ( A  e.  ran  (,) 
/\  ( vol * `  A )  e.  RR )  /\  A  =/=  (/) )  -> 
( F `  A
)  e.  ( RR 
X.  RR ) )
3814, 37pm2.61dane 2524 . 2  |-  ( ( A  e.  ran  (,)  /\  ( vol * `  A )  e.  RR )  ->  ( F `  A )  e.  ( RR  X.  RR ) )
39 elin 3358 . 2  |-  ( ( F `  A )  e.  (  <_  i^i  ( RR  X.  RR ) )  <->  ( ( F `  A )  e.  <_  /\  ( F `  A )  e.  ( RR  X.  RR ) ) )
406, 38, 39sylanbrc 645 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 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    i^i cin 3151   (/)c0 3455   ifcif 3565   ~Pcpw 3625   <.cop 3643    e. cmpt 4077    X. cxp 4687   `'ccnv 4688   ran crn 4690    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   RRcr 8736   0cc0 8737   RR*cxr 8866    < clt 8867    <_ cle 8868   (,)cioo 10656   vol *covol 18822
This theorem is referenced by:  uniioombllem2  18938
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-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  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-cda 7794  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-fl 10925  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-rlim 11963  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  df-vol 18825
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