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Theorem xrmaxifle 11273
Description: An upper bound for  { A ,  B } in the extended reals. (Contributed by Jim Kingdon, 26-Apr-2023.)
Assertion
Ref Expression
xrmaxifle  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A  <_  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )

Proof of Theorem xrmaxifle
StepHypRef Expression
1 pnfge 9808 . . . 4  |-  ( A  e.  RR*  ->  A  <_ +oo )
21ad2antrr 488 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  ->  A  <_ +oo )
3 simpr 110 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  ->  B  = +oo )
43iftrued 3556 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  ->  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) )  = +oo )
52, 4breqtrrd 4046 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  ->  A  <_  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )
6 xrleid 9819 . . . . . 6  |-  ( A  e.  RR*  ->  A  <_  A )
76ad3antrrr 492 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  A  <_  A )
8 simpr 110 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  B  = -oo )
98iftrued 3556 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) )  =  A )
107, 9breqtrrd 4046 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  A  <_  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) )
111ad4antr 494 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  A  = +oo )  ->  A  <_ +oo )
12 simpr 110 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  A  = +oo )  ->  A  = +oo )
1312iftrued 3556 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  A  = +oo )  ->  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) )  = +oo )
1411, 13breqtrrd 4046 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  A  = +oo )  ->  A  <_  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) )
15 mnfle 9811 . . . . . . . . . 10  |-  ( B  e.  RR*  -> -oo  <_  B )
1615ad5antlr 497 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  A  = -oo )  -> -oo  <_  B )
17 simpr 110 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  A  = -oo )  ->  A  = -oo )
1817iftrued 3556 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  A  = -oo )  ->  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
)  =  B )
1916, 17, 183brtr4d 4050 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  A  = -oo )  ->  A  <_  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) )
20 simplr 528 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  A  = +oo )
21 simpr 110 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  A  = -oo )
22 elxr 9795 . . . . . . . . . . . . 13  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2322biimpi 120 . . . . . . . . . . . 12  |-  ( A  e.  RR*  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2423ad5antr 496 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2520, 21, 24ecase23d 1361 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  A  e.  RR )
26 simpr 110 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  ->  -.  B  = +oo )
2726ad3antrrr 492 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  B  = +oo )
28 simpr 110 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  -.  B  = -oo )
2928ad2antrr 488 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  B  = -oo )
30 elxr 9795 . . . . . . . . . . . . 13  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3130biimpi 120 . . . . . . . . . . . 12  |-  ( B  e.  RR*  ->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3231ad5antlr 497 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3327, 29, 32ecase23d 1361 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  B  e.  RR )
34 maxle1 11239 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  <_  sup ( { A ,  B } ,  RR ,  <  )
)
3525, 33, 34syl2anc 411 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  A  <_  sup ( { A ,  B } ,  RR ,  <  )
)
3621iffalsed 3559 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) )  =  sup ( { A ,  B } ,  RR ,  <  ) )
3735, 36breqtrrd 4046 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  A  <_  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) )
38 xrmnfdc 9862 . . . . . . . . . 10  |-  ( A  e.  RR*  -> DECID  A  = -oo )
39 exmiddc 837 . . . . . . . . . 10  |-  (DECID  A  = -oo  ->  ( A  = -oo  \/  -.  A  = -oo ) )
4038, 39syl 14 . . . . . . . . 9  |-  ( A  e.  RR*  ->  ( A  = -oo  \/  -.  A  = -oo )
)
4140ad4antr 494 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  ->  ( A  = -oo  \/  -.  A  = -oo )
)
4219, 37, 41mpjaodan 799 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  ->  A  <_  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) )
43 simpr 110 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  ->  -.  A  = +oo )
4443iffalsed 3559 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  ->  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) )  =  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) )
4542, 44breqtrrd 4046 . . . . . 6  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  ->  A  <_  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) )
46 xrpnfdc 9861 . . . . . . . 8  |-  ( A  e.  RR*  -> DECID  A  = +oo )
47 exmiddc 837 . . . . . . . 8  |-  (DECID  A  = +oo  ->  ( A  = +oo  \/  -.  A  = +oo ) )
4846, 47syl 14 . . . . . . 7  |-  ( A  e.  RR*  ->  ( A  = +oo  \/  -.  A  = +oo )
)
4948ad3antrrr 492 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  ( A  = +oo  \/  -.  A  = +oo ) )
5014, 45, 49mpjaodan 799 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  A  <_  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) )
5128iffalsed 3559 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) )  =  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) )
5250, 51breqtrrd 4046 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  A  <_  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) )
53 xrmnfdc 9862 . . . . . 6  |-  ( B  e.  RR*  -> DECID  B  = -oo )
54 exmiddc 837 . . . . . 6  |-  (DECID  B  = -oo  ->  ( B  = -oo  \/  -.  B  = -oo ) )
5553, 54syl 14 . . . . 5  |-  ( B  e.  RR*  ->  ( B  = -oo  \/  -.  B  = -oo )
)
5655ad2antlr 489 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  ->  ( B  = -oo  \/  -.  B  = -oo ) )
5710, 52, 56mpjaodan 799 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  ->  A  <_  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) )
5826iffalsed 3559 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  ->  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) )  =  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) )
5957, 58breqtrrd 4046 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  ->  A  <_  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )
60 xrpnfdc 9861 . . . 4  |-  ( B  e.  RR*  -> DECID  B  = +oo )
61 exmiddc 837 . . . 4  |-  (DECID  B  = +oo  ->  ( B  = +oo  \/  -.  B  = +oo ) )
6260, 61syl 14 . . 3  |-  ( B  e.  RR*  ->  ( B  = +oo  \/  -.  B  = +oo )
)
6362adantl 277 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  = +oo  \/  -.  B  = +oo )
)
645, 59, 63mpjaodan 799 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A  <_  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    \/ wo 709  DECID wdc 835    \/ w3o 979    = wceq 1364    e. wcel 2160   ifcif 3549   {cpr 3608   class class class wbr 4018   supcsup 7000   RRcr 7829   +oocpnf 8008   -oocmnf 8009   RR*cxr 8010    < clt 8011    <_ cle 8012
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-mulrcl 7929  ax-addcom 7930  ax-mulcom 7931  ax-addass 7932  ax-mulass 7933  ax-distr 7934  ax-i2m1 7935  ax-0lt1 7936  ax-1rid 7937  ax-0id 7938  ax-rnegex 7939  ax-precex 7940  ax-cnre 7941  ax-pre-ltirr 7942  ax-pre-ltwlin 7943  ax-pre-lttrn 7944  ax-pre-apti 7945  ax-pre-ltadd 7946  ax-pre-mulgt0 7947  ax-pre-mulext 7948  ax-arch 7949  ax-caucvg 7950
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-frec 6410  df-sup 7002  df-pnf 8013  df-mnf 8014  df-xr 8015  df-ltxr 8016  df-le 8017  df-sub 8149  df-neg 8150  df-reap 8551  df-ap 8558  df-div 8649  df-inn 8939  df-2 8997  df-3 8998  df-4 8999  df-n0 9196  df-z 9273  df-uz 9548  df-rp 9673  df-seqfrec 10465  df-exp 10539  df-cj 10870  df-re 10871  df-im 10872  df-rsqrt 11026  df-abs 11027
This theorem is referenced by:  xrmaxiflemval  11277  xrmax1sup  11280
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