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Theorem xrmaxifle 10854
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 9416 . . . 4  |-  ( A  e.  RR*  ->  A  <_ +oo )
21ad2antrr 475 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  ->  A  <_ +oo )
3 simpr 109 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  ->  B  = +oo )
43iftrued 3428 . . 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 3901 . 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 9427 . . . . . 6  |-  ( A  e.  RR*  ->  A  <_  A )
76ad3antrrr 479 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  A  <_  A )
8 simpr 109 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  B  = -oo )
98iftrued 3428 . . . . 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 3901 . . . 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 481 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  A  = +oo )  ->  A  <_ +oo )
12 simpr 109 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  A  = +oo )  ->  A  = +oo )
1312iftrued 3428 . . . . . . 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 3901 . . . . . 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 9419 . . . . . . . . . 10  |-  ( B  e.  RR*  -> -oo  <_  B )
1615ad5antlr 484 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  A  = -oo )  -> -oo  <_  B )
17 simpr 109 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  A  = -oo )  ->  A  = -oo )
1817iftrued 3428 . . . . . . . . 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 3905 . . . . . . . 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 500 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  A  = +oo )
21 simpr 109 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  A  = -oo )
22 elxr 9404 . . . . . . . . . . . . 13  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2322biimpi 119 . . . . . . . . . . . 12  |-  ( A  e.  RR*  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2423ad5antr 483 . . . . . . . . . . 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 1296 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  A  e.  RR )
26 simpr 109 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  ->  -.  B  = +oo )
2726ad3antrrr 479 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  B  = +oo )
28 simpr 109 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  -.  B  = -oo )
2928ad2antrr 475 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  B  = -oo )
30 elxr 9404 . . . . . . . . . . . . 13  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3130biimpi 119 . . . . . . . . . . . 12  |-  ( B  e.  RR*  ->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3231ad5antlr 484 . . . . . . . . . . 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 1296 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  B  e.  RR )
34 maxle1 10823 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  <_  sup ( { A ,  B } ,  RR ,  <  )
)
3525, 33, 34syl2anc 406 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  A  <_  sup ( { A ,  B } ,  RR ,  <  )
)
3621iffalsed 3431 . . . . . . . . 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 3901 . . . . . . . 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 9467 . . . . . . . . . 10  |-  ( A  e.  RR*  -> DECID  A  = -oo )
39 exmiddc 788 . . . . . . . . . 10  |-  (DECID  A  = -oo  ->  ( A  = -oo  \/  -.  A  = -oo ) )
4038, 39syl 14 . . . . . . . . 9  |-  ( A  e.  RR*  ->  ( A  = -oo  \/  -.  A  = -oo )
)
4140ad4antr 481 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  ->  ( A  = -oo  \/  -.  A  = -oo )
)
4219, 37, 41mpjaodan 753 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  ->  A  <_  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) )
43 simpr 109 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  ->  -.  A  = +oo )
4443iffalsed 3431 . . . . . . 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 3901 . . . . . 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 9466 . . . . . . . 8  |-  ( A  e.  RR*  -> DECID  A  = +oo )
47 exmiddc 788 . . . . . . . 8  |-  (DECID  A  = +oo  ->  ( A  = +oo  \/  -.  A  = +oo ) )
4846, 47syl 14 . . . . . . 7  |-  ( A  e.  RR*  ->  ( A  = +oo  \/  -.  A  = +oo )
)
4948ad3antrrr 479 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  ( A  = +oo  \/  -.  A  = +oo ) )
5014, 45, 49mpjaodan 753 . . . . 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 3431 . . . . 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 3901 . . . 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 9467 . . . . . 6  |-  ( B  e.  RR*  -> DECID  B  = -oo )
54 exmiddc 788 . . . . . 6  |-  (DECID  B  = -oo  ->  ( B  = -oo  \/  -.  B  = -oo ) )
5553, 54syl 14 . . . . 5  |-  ( B  e.  RR*  ->  ( B  = -oo  \/  -.  B  = -oo )
)
5655ad2antlr 476 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  ->  ( B  = -oo  \/  -.  B  = -oo ) )
5710, 52, 56mpjaodan 753 . . 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 3431 . . 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 3901 . 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 9466 . . . 4  |-  ( B  e.  RR*  -> DECID  B  = +oo )
61 exmiddc 788 . . . 4  |-  (DECID  B  = +oo  ->  ( B  = +oo  \/  -.  B  = +oo ) )
6260, 61syl 14 . . 3  |-  ( B  e.  RR*  ->  ( B  = +oo  \/  -.  B  = +oo )
)
6362adantl 273 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  = +oo  \/  -.  B  = +oo )
)
645, 59, 63mpjaodan 753 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 103    \/ wo 670  DECID wdc 786    \/ w3o 929    = wceq 1299    e. wcel 1448   ifcif 3421   {cpr 3475   class class class wbr 3875   supcsup 6784   RRcr 7499   +oocpnf 7669   -oocmnf 7670   RR*cxr 7671    < clt 7672    <_ cle 7673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614  ax-caucvg 7615
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-sup 6786  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-3 8638  df-4 8639  df-n0 8830  df-z 8907  df-uz 9177  df-rp 9292  df-seqfrec 10060  df-exp 10134  df-cj 10455  df-re 10456  df-im 10457  df-rsqrt 10610  df-abs 10611
This theorem is referenced by:  xrmaxiflemval  10858  xrmax1sup  10861
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