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Theorem iooinsup 11280
Description: Intersection of two open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 22-May-2023.)
Assertion
Ref Expression
iooinsup  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )
)  ->  ( ( A (,) B )  i^i  ( C (,) D
) )  =  ( sup ( { A ,  C } ,  RR* ,  <  ) (,)inf ( { B ,  D } ,  RR* ,  <  )
) )

Proof of Theorem iooinsup
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 inrab 3407 . . 3  |-  ( { z  e.  RR*  |  ( A  <  z  /\  z  <  B ) }  i^i  { z  e. 
RR*  |  ( C  <  z  /\  z  < 
D ) } )  =  { z  e. 
RR*  |  ( ( A  <  z  /\  z  <  B )  /\  ( C  <  z  /\  z  <  D ) ) }
2 iooval 9906 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  =  { z  e.  RR*  |  ( A  <  z  /\  z  <  B ) } )
3 iooval 9906 . . . 4  |-  ( ( C  e.  RR*  /\  D  e.  RR* )  ->  ( C (,) D )  =  { z  e.  RR*  |  ( C  <  z  /\  z  <  D ) } )
42, 3ineqan12d 3338 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )
)  ->  ( ( A (,) B )  i^i  ( C (,) D
) )  =  ( { z  e.  RR*  |  ( A  <  z  /\  z  <  B ) }  i^i  { z  e.  RR*  |  ( C  <  z  /\  z  <  D ) } ) )
5 xrmaxltsup 11261 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  C  e.  RR*  /\  z  e. 
RR* )  ->  ( sup ( { A ,  C } ,  RR* ,  <  )  <  z  <->  ( A  <  z  /\  C  < 
z ) ) )
65ad4ant124 1216 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  C  e.  RR* )  /\  ( B  e.  RR*  /\  D  e. 
RR* ) )  /\  z  e.  RR* )  -> 
( sup ( { A ,  C } ,  RR* ,  <  )  <  z  <->  ( A  < 
z  /\  C  <  z ) ) )
7 xrltmininf 11273 . . . . . . . . . 10  |-  ( ( z  e.  RR*  /\  B  e.  RR*  /\  D  e. 
RR* )  ->  (
z  < inf ( { B ,  D } ,  RR* ,  <  )  <->  ( z  <  B  /\  z  <  D ) ) )
873expb 1204 . . . . . . . . 9  |-  ( ( z  e.  RR*  /\  ( B  e.  RR*  /\  D  e.  RR* ) )  -> 
( z  < inf ( { B ,  D } ,  RR* ,  <  )  <->  ( z  <  B  /\  z  <  D ) ) )
98ancoms 268 . . . . . . . 8  |-  ( ( ( B  e.  RR*  /\  D  e.  RR* )  /\  z  e.  RR* )  ->  ( z  < inf ( { B ,  D } ,  RR* ,  <  )  <->  ( z  <  B  /\  z  <  D ) ) )
109adantll 476 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  C  e.  RR* )  /\  ( B  e.  RR*  /\  D  e. 
RR* ) )  /\  z  e.  RR* )  -> 
( z  < inf ( { B ,  D } ,  RR* ,  <  )  <->  ( z  <  B  /\  z  <  D ) ) )
116, 10anbi12d 473 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  C  e.  RR* )  /\  ( B  e.  RR*  /\  D  e. 
RR* ) )  /\  z  e.  RR* )  -> 
( ( sup ( { A ,  C } ,  RR* ,  <  )  <  z  /\  z  < inf ( { B ,  D } ,  RR* ,  <  ) )  <->  ( ( A  <  z  /\  C  <  z )  /\  (
z  <  B  /\  z  <  D ) ) ) )
12 an4 586 . . . . . 6  |-  ( ( ( A  <  z  /\  z  <  B )  /\  ( C  < 
z  /\  z  <  D ) )  <->  ( ( A  <  z  /\  C  <  z )  /\  (
z  <  B  /\  z  <  D ) ) )
1311, 12bitr4di 198 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  C  e.  RR* )  /\  ( B  e.  RR*  /\  D  e. 
RR* ) )  /\  z  e.  RR* )  -> 
( ( sup ( { A ,  C } ,  RR* ,  <  )  <  z  /\  z  < inf ( { B ,  D } ,  RR* ,  <  ) )  <->  ( ( A  <  z  /\  z  <  B )  /\  ( C  <  z  /\  z  <  D ) ) ) )
1413rabbidva 2725 . . . 4  |-  ( ( ( A  e.  RR*  /\  C  e.  RR* )  /\  ( B  e.  RR*  /\  D  e.  RR* )
)  ->  { z  e.  RR*  |  ( sup ( { A ,  C } ,  RR* ,  <  )  <  z  /\  z  < inf ( { B ,  D } ,  RR* ,  <  ) ) }  =  {
z  e.  RR*  |  ( ( A  <  z  /\  z  <  B )  /\  ( C  < 
z  /\  z  <  D ) ) } )
1514an4s 588 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )
)  ->  { z  e.  RR*  |  ( sup ( { A ,  C } ,  RR* ,  <  )  <  z  /\  z  < inf ( { B ,  D } ,  RR* ,  <  ) ) }  =  {
z  e.  RR*  |  ( ( A  <  z  /\  z  <  B )  /\  ( C  < 
z  /\  z  <  D ) ) } )
161, 4, 153eqtr4a 2236 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )
)  ->  ( ( A (,) B )  i^i  ( C (,) D
) )  =  {
z  e.  RR*  |  ( sup ( { A ,  C } ,  RR* ,  <  )  <  z  /\  z  < inf ( { B ,  D } ,  RR* ,  <  )
) } )
17 xrmaxcl 11255 . . . 4  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  sup ( { A ,  C } ,  RR* ,  <  )  e.  RR* )
18 xrmincl 11269 . . . 4  |-  ( ( B  e.  RR*  /\  D  e.  RR* )  -> inf ( { B ,  D } ,  RR* ,  <  )  e.  RR* )
19 iooval 9906 . . . 4  |-  ( ( sup ( { A ,  C } ,  RR* ,  <  )  e.  RR*  /\ inf ( { B ,  D } ,  RR* ,  <  )  e.  RR* )  ->  ( sup ( { A ,  C } ,  RR* ,  <  ) (,)inf ( { B ,  D } ,  RR* ,  <  ) )  =  { z  e.  RR*  |  ( sup ( { A ,  C } ,  RR* ,  <  )  <  z  /\  z  < inf ( { B ,  D } ,  RR* ,  <  ) ) } )
2017, 18, 19syl2an 289 . . 3  |-  ( ( ( A  e.  RR*  /\  C  e.  RR* )  /\  ( B  e.  RR*  /\  D  e.  RR* )
)  ->  ( sup ( { A ,  C } ,  RR* ,  <  ) (,)inf ( { B ,  D } ,  RR* ,  <  ) )  =  { z  e.  RR*  |  ( sup ( { A ,  C } ,  RR* ,  <  )  <  z  /\  z  < inf ( { B ,  D } ,  RR* ,  <  ) ) } )
2120an4s 588 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )
)  ->  ( sup ( { A ,  C } ,  RR* ,  <  ) (,)inf ( { B ,  D } ,  RR* ,  <  ) )  =  { z  e.  RR*  |  ( sup ( { A ,  C } ,  RR* ,  <  )  <  z  /\  z  < inf ( { B ,  D } ,  RR* ,  <  ) ) } )
2216, 21eqtr4d 2213 1  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )
)  ->  ( ( A (,) B )  i^i  ( C (,) D
) )  =  ( sup ( { A ,  C } ,  RR* ,  <  ) (,)inf ( { B ,  D } ,  RR* ,  <  )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   {crab 2459    i^i cin 3128   {cpr 3593   class class class wbr 4003  (class class class)co 5874   supcsup 6980  infcinf 6981   RR*cxr 7989    < clt 7990   (,)cioo 9886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-sup 6982  df-inf 6983  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-reap 8530  df-ap 8537  df-div 8628  df-inn 8918  df-2 8976  df-3 8977  df-4 8978  df-n0 9175  df-z 9252  df-uz 9527  df-rp 9652  df-xneg 9770  df-ioo 9890  df-seqfrec 10443  df-exp 10517  df-cj 10846  df-re 10847  df-im 10848  df-rsqrt 11002  df-abs 11003
This theorem is referenced by:  qtopbasss  13952  tgioo  13977
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