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Theorem iooinsup 11287
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 3409 . . 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 9910 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  =  { z  e.  RR*  |  ( A  <  z  /\  z  <  B ) } )
3 iooval 9910 . . . 4  |-  ( ( C  e.  RR*  /\  D  e.  RR* )  ->  ( C (,) D )  =  { z  e.  RR*  |  ( C  <  z  /\  z  <  D ) } )
42, 3ineqan12d 3340 . . 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 11268 . . . . . . . 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 11280 . . . . . . . . . 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 2727 . . . 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 11262 . . . 4  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  sup ( { A ,  C } ,  RR* ,  <  )  e.  RR* )
18 xrmincl 11276 . . . 4  |-  ( ( B  e.  RR*  /\  D  e.  RR* )  -> inf ( { B ,  D } ,  RR* ,  <  )  e.  RR* )
19 iooval 9910 . . . 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 3130   {cpr 3595   class class class wbr 4005  (class class class)co 5877   supcsup 6983  infcinf 6984   RR*cxr 7993    < clt 7994   (,)cioo 9890
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 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933
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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-sup 6985  df-inf 6986  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-rp 9656  df-xneg 9774  df-ioo 9894  df-seqfrec 10448  df-exp 10522  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010
This theorem is referenced by:  qtopbasss  14060  tgioo  14085
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