MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ioorinvOLD Structured version   Unicode version

Theorem ioorinvOLD 22531
Description: The function  F is an "inverse" of sorts to the open interval function. (Contributed by Mario Carneiro, 26-Mar-2015.) Obsolete version of ioorinv 22526 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
ioorinvOLD  |-  ( A  e.  ran  (,)  ->  ( (,) `  ( F `
 A ) )  =  A )
Distinct variable group:    x, A
Allowed substitution hint:    F( x)

Proof of Theorem ioorinvOLD
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ioof 11739 . . . 4  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
2 ffn 5746 . . . 4  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
3 ovelrn 6459 . . . 4  |-  ( (,) 
Fn  ( RR*  X.  RR* )  ->  ( A  e. 
ran  (,)  <->  E. a  e.  RR*  E. b  e.  RR*  A  =  ( a (,) b ) ) )
41, 2, 3mp2b 10 . . 3  |-  ( A  e.  ran  (,)  <->  E. a  e.  RR*  E. b  e. 
RR*  A  =  (
a (,) b ) )
5 ioorfOLD.1 . . . . . . . . 9  |-  F  =  ( x  e.  ran  (,)  |->  if ( x  =  (/) ,  <. 0 ,  0
>. ,  <. sup (
x ,  RR* ,  `'  <  ) ,  sup (
x ,  RR* ,  <  )
>. ) )
65ioorinv2OLD 22530 . . . . . . . 8  |-  ( ( a (,) b )  =/=  (/)  ->  ( F `  ( a (,) b
) )  =  <. a ,  b >. )
76fveq2d 5885 . . . . . . 7  |-  ( ( a (,) b )  =/=  (/)  ->  ( (,) `  ( F `  (
a (,) b ) ) )  =  ( (,) `  <. a ,  b >. )
)
8 df-ov 6308 . . . . . . 7  |-  ( a (,) b )  =  ( (,) `  <. a ,  b >. )
97, 8syl6eqr 2481 . . . . . 6  |-  ( ( a (,) b )  =/=  (/)  ->  ( (,) `  ( F `  (
a (,) b ) ) )  =  ( a (,) b ) )
10 df-ne 2616 . . . . . . . 8  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
11 neeq1 2701 . . . . . . . 8  |-  ( A  =  ( a (,) b )  ->  ( A  =/=  (/)  <->  ( a (,) b )  =/=  (/) ) )
1210, 11syl5bbr 262 . . . . . . 7  |-  ( A  =  ( a (,) b )  ->  ( -.  A  =  (/)  <->  ( a (,) b )  =/=  (/) ) )
13 fveq2 5881 . . . . . . . . 9  |-  ( A  =  ( a (,) b )  ->  ( F `  A )  =  ( F `  ( a (,) b
) ) )
1413fveq2d 5885 . . . . . . . 8  |-  ( A  =  ( a (,) b )  ->  ( (,) `  ( F `  A ) )  =  ( (,) `  ( F `  ( a (,) b ) ) ) )
15 id 22 . . . . . . . 8  |-  ( A  =  ( a (,) b )  ->  A  =  ( a (,) b ) )
1614, 15eqeq12d 2444 . . . . . . 7  |-  ( A  =  ( a (,) b )  ->  (
( (,) `  ( F `  A )
)  =  A  <->  ( (,) `  ( F `  (
a (,) b ) ) )  =  ( a (,) b ) ) )
1712, 16imbi12d 321 . . . . . 6  |-  ( A  =  ( a (,) b )  ->  (
( -.  A  =  (/)  ->  ( (,) `  ( F `  A )
)  =  A )  <-> 
( ( a (,) b )  =/=  (/)  ->  ( (,) `  ( F `  ( a (,) b
) ) )  =  ( a (,) b
) ) ) )
189, 17mpbiri 236 . . . . 5  |-  ( A  =  ( a (,) b )  ->  ( -.  A  =  (/)  ->  ( (,) `  ( F `  A ) )  =  A ) )
1918a1i 11 . . . 4  |-  ( ( a  e.  RR*  /\  b  e.  RR* )  ->  ( A  =  ( a (,) b )  ->  ( -.  A  =  (/)  ->  ( (,) `  ( F `  A ) )  =  A ) ) )
2019rexlimivv 2919 . . 3  |-  ( E. a  e.  RR*  E. b  e.  RR*  A  =  ( a (,) b )  ->  ( -.  A  =  (/)  ->  ( (,) `  ( F `  A
) )  =  A ) )
214, 20sylbi 198 . 2  |-  ( A  e.  ran  (,)  ->  ( -.  A  =  (/)  ->  ( (,) `  ( F `  A )
)  =  A ) )
22 ioorebas 11743 . . . . . . 7  |-  ( 0 (,) 0 )  e. 
ran  (,)
235ioorvalOLD 22529 . . . . . . 7  |-  ( ( 0 (,) 0 )  e.  ran  (,)  ->  ( F `  ( 0 (,) 0 ) )  =  if ( ( 0 (,) 0 )  =  (/) ,  <. 0 ,  0 >. ,  <. sup ( ( 0 (,) 0 ) ,  RR* ,  `'  <  ) ,  sup ( ( 0 (,) 0 ) ,  RR* ,  <  ) >. )
)
2422, 23ax-mp 5 . . . . . 6  |-  ( F `
 ( 0 (,) 0 ) )  =  if ( ( 0 (,) 0 )  =  (/) ,  <. 0 ,  0
>. ,  <. sup (
( 0 (,) 0
) ,  RR* ,  `'  <  ) ,  sup (
( 0 (,) 0
) ,  RR* ,  <  )
>. )
25 iooid 11671 . . . . . . 7  |-  ( 0 (,) 0 )  =  (/)
2625iftruei 3918 . . . . . 6  |-  if ( ( 0 (,) 0
)  =  (/) ,  <. 0 ,  0 >. , 
<. sup ( ( 0 (,) 0 ) , 
RR* ,  `'  <  ) ,  sup ( ( 0 (,) 0 ) ,  RR* ,  <  ) >. )  =  <. 0 ,  0 >.
2724, 26eqtri 2451 . . . . 5  |-  ( F `
 ( 0 (,) 0 ) )  = 
<. 0 ,  0
>.
2827fveq2i 5884 . . . 4  |-  ( (,) `  ( F `  (
0 (,) 0 ) ) )  =  ( (,) `  <. 0 ,  0 >. )
29 df-ov 6308 . . . 4  |-  ( 0 (,) 0 )  =  ( (,) `  <. 0 ,  0 >. )
3028, 29eqtr4i 2454 . . 3  |-  ( (,) `  ( F `  (
0 (,) 0 ) ) )  =  ( 0 (,) 0 )
3125eqeq2i 2440 . . . . . 6  |-  ( A  =  ( 0 (,) 0 )  <->  A  =  (/) )
3231biimpri 209 . . . . 5  |-  ( A  =  (/)  ->  A  =  ( 0 (,) 0
) )
3332fveq2d 5885 . . . 4  |-  ( A  =  (/)  ->  ( F `
 A )  =  ( F `  (
0 (,) 0 ) ) )
3433fveq2d 5885 . . 3  |-  ( A  =  (/)  ->  ( (,) `  ( F `  A
) )  =  ( (,) `  ( F `
 ( 0 (,) 0 ) ) ) )
3530, 34, 323eqtr4a 2489 . 2  |-  ( A  =  (/)  ->  ( (,) `  ( F `  A
) )  =  A )
3621, 35pm2.61d2 163 1  |-  ( A  e.  ran  (,)  ->  ( (,) `  ( F `
 A ) )  =  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2614   E.wrex 2772   (/)c0 3761   ifcif 3911   ~Pcpw 3981   <.cop 4004    |-> cmpt 4482    X. cxp 4851   `'ccnv 4852   ran crn 4854    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305   supcsup 7963   RRcr 9545   0cc0 9546   RR*cxr 9681    < clt 9682   (,)cioo 11642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623  ax-pre-sup 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  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-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-sup 7965  df-inf 7966  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-div 10277  df-nn 10617  df-n0 10877  df-z 10945  df-uz 11167  df-q 11272  df-ioo 11646
This theorem is referenced by:  uniioombllem2OLD  22539
  Copyright terms: Public domain W3C validator