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Theorem ioorcl 22406
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.) (Revised by AV, 13-Sep-2020.)
Hypothesis
Ref Expression
ioorf.1  |-  F  =  ( x  e.  ran  (,)  |->  if ( x  =  (/) ,  <. 0 ,  0
>. ,  <.inf ( 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 3688 . . 3  |-  (  <_  i^i  ( RR*  X.  RR* )
)  C_  <_
2 ioorf.1 . . . . . 6  |-  F  =  ( x  e.  ran  (,)  |->  if ( x  =  (/) ,  <. 0 ,  0
>. ,  <.inf ( x ,  RR* ,  <  ) ,  sup ( x , 
RR* ,  <  ) >.
) )
32ioorf 22402 . . . . 5  |-  F : ran  (,) --> (  <_  i^i  ( RR*  X.  RR* )
)
43ffvelrni 6036 . . . 4  |-  ( A  e.  ran  (,)  ->  ( F `  A )  e.  (  <_  i^i  ( RR*  X.  RR* )
) )
54adantr 466 . . 3  |-  ( ( A  e.  ran  (,)  /\  ( vol* `  A )  e.  RR )  ->  ( F `  A )  e.  (  <_  i^i  ( RR*  X. 
RR* ) ) )
61, 5sseldi 3468 . 2  |-  ( ( A  e.  ran  (,)  /\  ( vol* `  A )  e.  RR )  ->  ( F `  A )  e.  <_  )
72ioorval 22403 . . . . . 6  |-  ( A  e.  ran  (,)  ->  ( F `  A )  =  if ( A  =  (/) ,  <. 0 ,  0 >. ,  <.inf ( A ,  RR* ,  <  ) ,  sup ( A ,  RR* ,  <  ) >. ) )
87adantr 466 . . . . 5  |-  ( ( A  e.  ran  (,)  /\  ( vol* `  A )  e.  RR )  ->  ( F `  A )  =  if ( A  =  (/) , 
<. 0 ,  0
>. ,  <.inf ( A ,  RR* ,  <  ) ,  sup ( A ,  RR* ,  <  ) >.
) )
9 iftrue 3921 . . . . 5  |-  ( A  =  (/)  ->  if ( A  =  (/) ,  <. 0 ,  0 >. , 
<.inf ( A ,  RR* ,  <  ) ,  sup ( A ,  RR* ,  <  )
>. )  =  <. 0 ,  0 >. )
108, 9sylan9eq 2490 . . . 4  |-  ( ( ( A  e.  ran  (,) 
/\  ( vol* `  A )  e.  RR )  /\  A  =  (/) )  ->  ( F `  A )  =  <. 0 ,  0 >. )
11 0re 9642 . . . . 5  |-  0  e.  RR
12 opelxpi 4886 . . . . 5  |-  ( ( 0  e.  RR  /\  0  e.  RR )  -> 
<. 0 ,  0
>.  e.  ( RR  X.  RR ) )
1311, 11, 12mp2an 676 . . . 4  |-  <. 0 ,  0 >.  e.  ( RR  X.  RR )
1410, 13syl6eqel 2525 . . 3  |-  ( ( ( A  e.  ran  (,) 
/\  ( vol* `  A )  e.  RR )  /\  A  =  (/) )  ->  ( F `  A )  e.  ( RR  X.  RR ) )
15 ioof 11732 . . . . . 6  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
16 ffn 5746 . . . . . 6  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
17 ovelrn 6459 . . . . . 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 22404 . . . . . . . . . 10  |-  ( ( a (,) b )  =/=  (/)  ->  ( F `  ( a (,) b
) )  =  <. a ,  b >. )
2019adantl 467 . . . . . . . . 9  |-  ( ( ( vol* `  ( a (,) b
) )  e.  RR  /\  ( a (,) b
)  =/=  (/) )  -> 
( F `  (
a (,) b ) )  =  <. a ,  b >. )
21 ioorcl2 22401 . . . . . . . . . . 11  |-  ( ( ( a (,) b
)  =/=  (/)  /\  ( vol* `  ( a (,) b ) )  e.  RR )  -> 
( a  e.  RR  /\  b  e.  RR ) )
2221ancoms 454 . . . . . . . . . 10  |-  ( ( ( vol* `  ( a (,) b
) )  e.  RR  /\  ( a (,) b
)  =/=  (/) )  -> 
( a  e.  RR  /\  b  e.  RR ) )
23 opelxpi 4886 . . . . . . . . . 10  |-  ( ( a  e.  RR  /\  b  e.  RR )  -> 
<. a ,  b >.  e.  ( RR  X.  RR ) )
2422, 23syl 17 . . . . . . . . 9  |-  ( ( ( vol* `  ( a (,) b
) )  e.  RR  /\  ( a (,) b
)  =/=  (/) )  ->  <. a ,  b >.  e.  ( RR  X.  RR ) )
2520, 24eqeltrd 2517 . . . . . . . 8  |-  ( ( ( vol* `  ( a (,) b
) )  e.  RR  /\  ( a (,) b
)  =/=  (/) )  -> 
( F `  (
a (,) b ) )  e.  ( RR 
X.  RR ) )
26 fveq2 5881 . . . . . . . . . . 11  |-  ( A  =  ( a (,) b )  ->  ( vol* `  A )  =  ( vol* `  ( a (,) b
) ) )
2726eleq1d 2498 . . . . . . . . . 10  |-  ( A  =  ( a (,) b )  ->  (
( vol* `  A )  e.  RR  <->  ( vol* `  (
a (,) b ) )  e.  RR ) )
28 neeq1 2712 . . . . . . . . . 10  |-  ( A  =  ( a (,) b )  ->  ( A  =/=  (/)  <->  ( a (,) b )  =/=  (/) ) )
2927, 28anbi12d 715 . . . . . . . . 9  |-  ( A  =  ( a (,) b )  ->  (
( ( vol* `  A )  e.  RR  /\  A  =/=  (/) )  <->  ( ( vol* `  ( a (,) b ) )  e.  RR  /\  (
a (,) b )  =/=  (/) ) ) )
30 fveq2 5881 . . . . . . . . . 10  |-  ( A  =  ( a (,) b )  ->  ( F `  A )  =  ( F `  ( a (,) b
) ) )
3130eleq1d 2498 . . . . . . . . 9  |-  ( A  =  ( a (,) b )  ->  (
( F `  A
)  e.  ( RR 
X.  RR )  <->  ( F `  ( a (,) b
) )  e.  ( RR  X.  RR ) ) )
3229, 31imbi12d 321 . . . . . . . 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 236 . . . . . . 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 2929 . . . . 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 198 . . . 4  |-  ( A  e.  ran  (,)  ->  ( ( ( vol* `  A )  e.  RR  /\  A  =/=  (/) )  -> 
( F `  A
)  e.  ( RR 
X.  RR ) ) )
3736impl 624 . . 3  |-  ( ( ( A  e.  ran  (,) 
/\  ( vol* `  A )  e.  RR )  /\  A  =/=  (/) )  -> 
( F `  A
)  e.  ( RR 
X.  RR ) )
3814, 37pm2.61dane 2749 . 2  |-  ( ( A  e.  ran  (,)  /\  ( vol* `  A )  e.  RR )  ->  ( F `  A )  e.  ( RR  X.  RR ) )
396, 38elind 3656 1  |-  ( ( A  e.  ran  (,)  /\  ( vol* `  A )  e.  RR )  ->  ( F `  A )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   E.wrex 2783    i^i cin 3441   (/)c0 3767   ifcif 3915   ~Pcpw 3985   <.cop 4008    |-> cmpt 4484    X. cxp 4852   ran crn 4855    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305   supcsup 7960  infcinf 7961   RRcr 9537   0cc0 9538   RR*cxr 9673    < clt 9674    <_ cle 9675   (,)cioo 11635   vol*covol 22294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fi 7931  df-sup 7962  df-inf 7963  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-rlim 13531  df-sum 13731  df-rest 15280  df-topgen 15301  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-top 19852  df-bases 19853  df-topon 19854  df-cmp 20333  df-ovol 22296  df-vol 22297
This theorem is referenced by:  uniioombllem2  22417
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