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Theorem ioorcl 18948
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 3402 . . 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 18944 . . . . 5  |-  F : ran  (,) --> (  <_  i^i  ( RR*  X.  RR* )
)
43ffvelrni 5680 . . . 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 3191 . 2  |-  ( ( A  e.  ran  (,)  /\  ( vol * `  A )  e.  RR )  ->  ( F `  A )  e.  <_  )
72ioorval 18945 . . . . . 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 3584 . . . . 5  |-  ( A  =  (/)  ->  if ( A  =  (/) ,  <. 0 ,  0 >. , 
<. sup ( A ,  RR* ,  `'  <  ) ,  sup ( A ,  RR* ,  <  ) >.
)  =  <. 0 ,  0 >. )
108, 9sylan9eq 2348 . . . 4  |-  ( ( ( A  e.  ran  (,) 
/\  ( vol * `  A )  e.  RR )  /\  A  =  (/) )  ->  ( F `  A )  =  <. 0 ,  0 >. )
11 0re 8854 . . . . 5  |-  0  e.  RR
12 opelxpi 4737 . . . . 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 2384 . . 3  |-  ( ( ( A  e.  ran  (,) 
/\  ( vol * `  A )  e.  RR )  /\  A  =  (/) )  ->  ( F `  A )  e.  ( RR  X.  RR ) )
15 ioof 10757 . . . . . 6  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
16 ffn 5405 . . . . . 6  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
17 ovelrn 6012 . . . . . 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 18946 . . . . . . . . . 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 18943 . . . . . . . . . . 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 4737 . . . . . . . . . 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 2370 . . . . . . . 8  |-  ( ( ( vol * `  ( a (,) b
) )  e.  RR  /\  ( a (,) b
)  =/=  (/) )  -> 
( F `  (
a (,) b ) )  e.  ( RR 
X.  RR ) )
26 fveq2 5541 . . . . . . . . . . 11  |-  ( A  =  ( a (,) b )  ->  ( vol * `  A )  =  ( vol * `  ( a (,) b
) ) )
2726eleq1d 2362 . . . . . . . . . 10  |-  ( A  =  ( a (,) b )  ->  (
( vol * `  A )  e.  RR  <->  ( vol * `  (
a (,) b ) )  e.  RR ) )
28 neeq1 2467 . . . . . . . . . 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 5541 . . . . . . . . . 10  |-  ( A  =  ( a (,) b )  ->  ( F `  A )  =  ( F `  ( a (,) b
) ) )
3130eleq1d 2362 . . . . . . . . 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 2685 . . . . 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 2537 . 2  |-  ( ( A  e.  ran  (,)  /\  ( vol * `  A )  e.  RR )  ->  ( F `  A )  e.  ( RR  X.  RR ) )
39 elin 3371 . 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 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557    i^i cin 3164   (/)c0 3468   ifcif 3578   ~Pcpw 3638   <.cop 3656    e. cmpt 4093    X. cxp 4703   `'ccnv 4704   ran crn 4706    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   supcsup 7209   RRcr 8752   0cc0 8753   RR*cxr 8882    < clt 8883    <_ cle 8884   (,)cioo 10672   vol *covol 18838
This theorem is referenced by:  uniioombllem2  18954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-rest 13343  df-topgen 13360  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-top 16652  df-bases 16654  df-topon 16655  df-cmp 17130  df-ovol 18840  df-vol 18841
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