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Theorem ioorcl 18934
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 3391 . . 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 18930 . . . . 5  |-  F : ran  (,) --> (  <_  i^i  ( RR*  X.  RR* )
)
43ffvelrni 5666 . . . 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 3180 . 2  |-  ( ( A  e.  ran  (,)  /\  ( vol * `  A )  e.  RR )  ->  ( F `  A )  e.  <_  )
72ioorval 18931 . . . . . 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 3573 . . . . 5  |-  ( A  =  (/)  ->  if ( A  =  (/) ,  <. 0 ,  0 >. , 
<. sup ( A ,  RR* ,  `'  <  ) ,  sup ( A ,  RR* ,  <  ) >.
)  =  <. 0 ,  0 >. )
108, 9sylan9eq 2337 . . . 4  |-  ( ( ( A  e.  ran  (,) 
/\  ( vol * `  A )  e.  RR )  /\  A  =  (/) )  ->  ( F `  A )  =  <. 0 ,  0 >. )
11 0re 8840 . . . . 5  |-  0  e.  RR
12 opelxpi 4723 . . . . 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 2373 . . 3  |-  ( ( ( A  e.  ran  (,) 
/\  ( vol * `  A )  e.  RR )  /\  A  =  (/) )  ->  ( F `  A )  e.  ( RR  X.  RR ) )
15 ioof 10743 . . . . . 6  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
16 ffn 5391 . . . . . 6  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
17 ovelrn 5998 . . . . . 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 18932 . . . . . . . . . 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 18929 . . . . . . . . . . 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 4723 . . . . . . . . . 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 2359 . . . . . . . 8  |-  ( ( ( vol * `  ( a (,) b
) )  e.  RR  /\  ( a (,) b
)  =/=  (/) )  -> 
( F `  (
a (,) b ) )  e.  ( RR 
X.  RR ) )
26 fveq2 5527 . . . . . . . . . . 11  |-  ( A  =  ( a (,) b )  ->  ( vol * `  A )  =  ( vol * `  ( a (,) b
) ) )
2726eleq1d 2351 . . . . . . . . . 10  |-  ( A  =  ( a (,) b )  ->  (
( vol * `  A )  e.  RR  <->  ( vol * `  (
a (,) b ) )  e.  RR ) )
28 neeq1 2456 . . . . . . . . . 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 5527 . . . . . . . . . 10  |-  ( A  =  ( a (,) b )  ->  ( F `  A )  =  ( F `  ( a (,) b
) ) )
3130eleq1d 2351 . . . . . . . . 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 2674 . . . . 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 2526 . 2  |-  ( ( A  e.  ran  (,)  /\  ( vol * `  A )  e.  RR )  ->  ( F `  A )  e.  ( RR  X.  RR ) )
39 elin 3360 . 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 1625    e. wcel 1686    =/= wne 2448   E.wrex 2546    i^i cin 3153   (/)c0 3457   ifcif 3567   ~Pcpw 3627   <.cop 3645    e. cmpt 4079    X. cxp 4689   `'ccnv 4690   ran crn 4692    Fn wfn 5252   -->wf 5253   ` cfv 5257  (class class class)co 5860   supcsup 7195   RRcr 8738   0cc0 8739   RR*cxr 8868    < clt 8869    <_ cle 8870   (,)cioo 10658   vol *covol 18824
This theorem is referenced by:  uniioombllem2  18940
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-er 6662  df-map 6776  df-pm 6777  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-fi 7167  df-sup 7196  df-oi 7227  df-card 7574  df-cda 7796  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-n0 9968  df-z 10027  df-uz 10233  df-q 10319  df-rp 10357  df-xneg 10454  df-xadd 10455  df-xmul 10456  df-ioo 10662  df-ico 10664  df-icc 10665  df-fz 10785  df-fzo 10873  df-fl 10927  df-seq 11049  df-exp 11107  df-hash 11340  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-clim 11964  df-rlim 11965  df-sum 12161  df-rest 13329  df-topgen 13346  df-xmet 16375  df-met 16376  df-bl 16377  df-mopn 16378  df-top 16638  df-bases 16640  df-topon 16641  df-cmp 17116  df-ovol 18826  df-vol 18827
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