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Theorem ioorfOLD 22523
Description: Define a function from open intervals to their endpoints. (Contributed by Mario Carneiro, 26-Mar-2015.) Obsolete version of ioorf 22518 as of 13-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
ioorfOLD.1  |-  F  =  ( x  e.  ran  (,)  |->  if ( x  =  (/) ,  <. 0 ,  0
>. ,  <. sup (
x ,  RR* ,  `'  <  ) ,  sup (
x ,  RR* ,  <  )
>. ) )
Assertion
Ref Expression
ioorfOLD  |-  F : ran  (,) --> (  <_  i^i  ( RR*  X.  RR* )
)

Proof of Theorem ioorfOLD
Dummy variables  a 
b  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ioorfOLD.1 . 2  |-  F  =  ( x  e.  ran  (,)  |->  if ( x  =  (/) ,  <. 0 ,  0
>. ,  <. sup (
x ,  RR* ,  `'  <  ) ,  sup (
x ,  RR* ,  <  )
>. ) )
2 ioof 11729 . . . 4  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
3 ffn 5726 . . . 4  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
4 ovelrn 6442 . . . 4  |-  ( (,) 
Fn  ( RR*  X.  RR* )  ->  ( x  e. 
ran  (,)  <->  E. a  e.  RR*  E. b  e.  RR*  x  =  ( a (,) b ) ) )
52, 3, 4mp2b 10 . . 3  |-  ( x  e.  ran  (,)  <->  E. a  e.  RR*  E. b  e. 
RR*  x  =  ( a (,) b ) )
6 0le0 10696 . . . . . . . . 9  |-  0  <_  0
7 df-br 4402 . . . . . . . . 9  |-  ( 0  <_  0  <->  <. 0 ,  0 >.  e.  <_  )
86, 7mpbi 212 . . . . . . . 8  |-  <. 0 ,  0 >.  e.  <_
9 0xr 9684 . . . . . . . . 9  |-  0  e.  RR*
10 opelxpi 4865 . . . . . . . . 9  |-  ( ( 0  e.  RR*  /\  0  e.  RR* )  ->  <. 0 ,  0 >.  e.  (
RR*  X.  RR* ) )
119, 9, 10mp2an 677 . . . . . . . 8  |-  <. 0 ,  0 >.  e.  (
RR*  X.  RR* )
12 elin 3616 . . . . . . . 8  |-  ( <.
0 ,  0 >.  e.  (  <_  i^i  ( RR*  X.  RR* ) )  <->  ( <. 0 ,  0 >.  e. 
<_  /\  <. 0 ,  0
>.  e.  ( RR*  X.  RR* ) ) )
138, 11, 12mpbir2an 930 . . . . . . 7  |-  <. 0 ,  0 >.  e.  (  <_  i^i  ( RR*  X. 
RR* ) )
1413a1i 11 . . . . . 6  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  x  =  (/) )  ->  <. 0 ,  0 >.  e.  (  <_  i^i  ( RR*  X. 
RR* ) ) )
15 simplr 761 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  x  =  ( a (,) b ) )
1615supeq1d 7957 . . . . . . . . 9  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  sup ( x ,  RR* ,  `'  <  )  =  sup ( ( a (,) b ) ,  RR* ,  `'  <  ) )
17 simplll 767 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> 
a  e.  RR* )
18 simpllr 768 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> 
b  e.  RR* )
19 simpr 463 . . . . . . . . . . . 12  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  -.  x  =  (/) )
2019neqned 2630 . . . . . . . . . . 11  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  x  =/=  (/) )
2115, 20eqnetrrd 2691 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> 
( a (,) b
)  =/=  (/) )
22 df-ioo 11636 . . . . . . . . . . 11  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
23 idd 25 . . . . . . . . . . 11  |-  ( ( w  e.  RR*  /\  b  e.  RR* )  ->  (
w  <  b  ->  w  <  b ) )
24 xrltle 11445 . . . . . . . . . . 11  |-  ( ( w  e.  RR*  /\  b  e.  RR* )  ->  (
w  <  b  ->  w  <_  b ) )
25 idd 25 . . . . . . . . . . 11  |-  ( ( a  e.  RR*  /\  w  e.  RR* )  ->  (
a  <  w  ->  a  <  w ) )
26 xrltle 11445 . . . . . . . . . . 11  |-  ( ( a  e.  RR*  /\  w  e.  RR* )  ->  (
a  <  w  ->  a  <_  w ) )
2722, 23, 24, 25, 26ixxlbOLD 11655 . . . . . . . . . 10  |-  ( ( a  e.  RR*  /\  b  e.  RR*  /\  ( a (,) b )  =/=  (/) )  ->  sup (
( a (,) b
) ,  RR* ,  `'  <  )  =  a )
2817, 18, 21, 27syl3anc 1267 . . . . . . . . 9  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  sup ( ( a (,) b ) ,  RR* ,  `'  <  )  =  a )
2916, 28eqtrd 2484 . . . . . . . 8  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  sup ( x ,  RR* ,  `'  <  )  =  a )
3015supeq1d 7957 . . . . . . . . 9  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  sup ( x ,  RR* ,  <  )  =  sup ( ( a (,) b ) ,  RR* ,  <  ) )
3122, 23, 24, 25, 26ixxub 11653 . . . . . . . . . 10  |-  ( ( a  e.  RR*  /\  b  e.  RR*  /\  ( a (,) b )  =/=  (/) )  ->  sup (
( a (,) b
) ,  RR* ,  <  )  =  b )
3217, 18, 21, 31syl3anc 1267 . . . . . . . . 9  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  sup ( ( a (,) b ) ,  RR* ,  <  )  =  b )
3330, 32eqtrd 2484 . . . . . . . 8  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  sup ( x ,  RR* ,  <  )  =  b )
3429, 33opeq12d 4173 . . . . . . 7  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  <. sup ( x , 
RR* ,  `'  <  ) ,  sup ( x ,  RR* ,  <  ) >.  =  <. a ,  b
>. )
35 ioon0 11659 . . . . . . . . . . . 12  |-  ( ( a  e.  RR*  /\  b  e.  RR* )  ->  (
( a (,) b
)  =/=  (/)  <->  a  <  b ) )
3635ad2antrr 731 . . . . . . . . . . 11  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> 
( ( a (,) b )  =/=  (/)  <->  a  <  b ) )
3721, 36mpbid 214 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> 
a  <  b )
38 xrltle 11445 . . . . . . . . . . 11  |-  ( ( a  e.  RR*  /\  b  e.  RR* )  ->  (
a  <  b  ->  a  <_  b ) )
3938ad2antrr 731 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> 
( a  <  b  ->  a  <_  b )
)
4037, 39mpd 15 . . . . . . . . 9  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  -> 
a  <_  b )
41 df-br 4402 . . . . . . . . 9  |-  ( a  <_  b  <->  <. a ,  b >.  e.  <_  )
4240, 41sylib 200 . . . . . . . 8  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  <. a ,  b >.  e.  <_  )
43 opelxpi 4865 . . . . . . . . 9  |-  ( ( a  e.  RR*  /\  b  e.  RR* )  ->  <. a ,  b >.  e.  (
RR*  X.  RR* ) )
4443ad2antrr 731 . . . . . . . 8  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  <. a ,  b >.  e.  ( RR*  X.  RR* )
)
4542, 44elind 3617 . . . . . . 7  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  <. a ,  b >.  e.  (  <_  i^i  ( RR*  X.  RR* ) ) )
4634, 45eqeltrd 2528 . . . . . 6  |-  ( ( ( ( a  e. 
RR*  /\  b  e.  RR* )  /\  x  =  ( a (,) b
) )  /\  -.  x  =  (/) )  ->  <. sup ( x , 
RR* ,  `'  <  ) ,  sup ( x ,  RR* ,  <  ) >.  e.  (  <_  i^i  ( RR*  X.  RR* )
) )
4714, 46ifclda 3912 . . . . 5  |-  ( ( ( a  e.  RR*  /\  b  e.  RR* )  /\  x  =  (
a (,) b ) )  ->  if (
x  =  (/) ,  <. 0 ,  0 >. , 
<. sup ( x , 
RR* ,  `'  <  ) ,  sup ( x ,  RR* ,  <  ) >. )  e.  (  <_  i^i  ( RR*  X.  RR* )
) )
4847ex 436 . . . 4  |-  ( ( a  e.  RR*  /\  b  e.  RR* )  ->  (
x  =  ( a (,) b )  ->  if ( x  =  (/) , 
<. 0 ,  0
>. ,  <. sup (
x ,  RR* ,  `'  <  ) ,  sup (
x ,  RR* ,  <  )
>. )  e.  (  <_  i^i  ( RR*  X.  RR* ) ) ) )
4948rexlimivv 2883 . . 3  |-  ( E. a  e.  RR*  E. b  e.  RR*  x  =  ( a (,) b )  ->  if ( x  =  (/) ,  <. 0 ,  0 >. ,  <. sup ( x ,  RR* ,  `'  <  ) ,  sup ( x ,  RR* ,  <  ) >. )  e.  (  <_  i^i  ( RR*  X.  RR* ) ) )
505, 49sylbi 199 . 2  |-  ( x  e.  ran  (,)  ->  if ( x  =  (/) , 
<. 0 ,  0
>. ,  <. sup (
x ,  RR* ,  `'  <  ) ,  sup (
x ,  RR* ,  <  )
>. )  e.  (  <_  i^i  ( RR*  X.  RR* ) ) )
511, 50fmpti 6043 1  |-  F : ran  (,) --> (  <_  i^i  ( RR*  X.  RR* )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1443    e. wcel 1886    =/= wne 2621   E.wrex 2737    i^i cin 3402   (/)c0 3730   ifcif 3880   ~Pcpw 3950   <.cop 3973   class class class wbr 4401    |-> cmpt 4460    X. cxp 4831   `'ccnv 4832   ran crn 4834    Fn wfn 5576   -->wf 5577  (class class class)co 6288   supcsup 7951   RRcr 9535   0cc0 9536   RR*cxr 9671    < clt 9672    <_ cle 9673   (,)cioo 11632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-sup 7953  df-inf 7954  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-n0 10867  df-z 10935  df-uz 11157  df-q 11262  df-ioo 11636
This theorem is referenced by:  ioorclOLD  22527  uniioombllem2OLD  22534
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