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Theorem xrmaxiflemab 11286
Description: Lemma for xrmaxif 11290. A variation of xrmaxleim 11283- that is, if we know which of two real numbers is larger, we know the maximum of the two. (Contributed by Jim Kingdon, 26-Apr-2023.)
Hypotheses
Ref Expression
xrmaxiflemab.a  |-  ( ph  ->  A  e.  RR* )
xrmaxiflemab.b  |-  ( ph  ->  B  e.  RR* )
xrmaxiflemab.ab  |-  ( ph  ->  A  <  B )
Assertion
Ref Expression
xrmaxiflemab  |-  ( ph  ->  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) )  =  B )

Proof of Theorem xrmaxiflemab
StepHypRef Expression
1 simpr 110 . . . 4  |-  ( (
ph  /\  B  = +oo )  ->  B  = +oo )
21iftrued 3556 . . 3  |-  ( (
ph  /\  B  = +oo )  ->  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) )  = +oo )
32, 1eqtr4d 2225 . 2  |-  ( (
ph  /\  B  = +oo )  ->  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) ) )  =  B )
4 simpr 110 . . . 4  |-  ( (
ph  /\  -.  B  = +oo )  ->  -.  B  = +oo )
54iffalsed 3559 . . 3  |-  ( (
ph  /\  -.  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 ,  <  ) ) ) ) )
6 xrmaxiflemab.ab . . . . . . 7  |-  ( ph  ->  A  <  B )
76ad2antrr 488 . . . . . 6  |-  ( ( ( ph  /\  -.  B  = +oo )  /\  B  = -oo )  ->  A  <  B
)
8 simpr 110 . . . . . 6  |-  ( ( ( ph  /\  -.  B  = +oo )  /\  B  = -oo )  ->  B  = -oo )
97, 8breqtrd 4044 . . . . 5  |-  ( ( ( ph  /\  -.  B  = +oo )  /\  B  = -oo )  ->  A  < -oo )
10 xrmaxiflemab.a . . . . . . 7  |-  ( ph  ->  A  e.  RR* )
11 nltmnf 9817 . . . . . . 7  |-  ( A  e.  RR*  ->  -.  A  < -oo )
1210, 11syl 14 . . . . . 6  |-  ( ph  ->  -.  A  < -oo )
1312ad2antrr 488 . . . . 5  |-  ( ( ( ph  /\  -.  B  = +oo )  /\  B  = -oo )  ->  -.  A  < -oo )
149, 13pm2.21dd 621 . . . 4  |-  ( ( ( ph  /\  -.  B  = +oo )  /\  B  = -oo )  ->  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) )  =  B )
15 simpr 110 . . . . . 6  |-  ( ( ( ph  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  -.  B  = -oo )
1615iffalsed 3559 . . . . 5  |-  ( ( ( ph  /\  -.  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 ,  <  )
) ) )
17 simpr 110 . . . . . . . 8  |-  ( ( ( ( ph  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  A  = +oo )  ->  A  = +oo )
186ad3antrrr 492 . . . . . . . 8  |-  ( ( ( ( ph  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  A  = +oo )  ->  A  <  B )
1917, 18eqbrtrrd 4042 . . . . . . 7  |-  ( ( ( ( ph  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  A  = +oo )  -> +oo  <  B )
20 xrmaxiflemab.b . . . . . . . . 9  |-  ( ph  ->  B  e.  RR* )
21 pnfnlt 9816 . . . . . . . . 9  |-  ( B  e.  RR*  ->  -. +oo  <  B )
2220, 21syl 14 . . . . . . . 8  |-  ( ph  ->  -. +oo  <  B
)
2322ad3antrrr 492 . . . . . . 7  |-  ( ( ( ( ph  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  A  = +oo )  ->  -. +oo 
<  B )
2419, 23pm2.21dd 621 . . . . . 6  |-  ( ( ( ( ph  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  A  = +oo )  ->  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) )  =  B )
25 simpr 110 . . . . . . . 8  |-  ( ( ( ( ph  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  ->  -.  A  = +oo )
2625iffalsed 3559 . . . . . . 7  |-  ( ( ( ( ph  /\  -.  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 ,  <  )
) )
27 simpr 110 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  A  = -oo )  ->  A  = -oo )
2827iftrued 3556 . . . . . . . 8  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  A  = -oo )  ->  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) )  =  B )
29 simpr 110 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  A  = -oo )
3029iffalsed 3559 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) )  =  sup ( { A ,  B } ,  RR ,  <  ) )
3125adantr 276 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  A  = +oo )
32 elxr 9805 . . . . . . . . . . . . . 14  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
3310, 32sylib 122 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
3433ad4antr 494 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
3531, 29, 34ecase23d 1361 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  A  e.  RR )
364ad3antrrr 492 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  B  = +oo )
3715ad2antrr 488 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  B  = -oo )
38 elxr 9805 . . . . . . . . . . . . . 14  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3920, 38sylib 122 . . . . . . . . . . . . 13  |-  ( ph  ->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
4039ad4antr 494 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
4136, 37, 40ecase23d 1361 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  B  e.  RR )
4235, 41jca 306 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  ( A  e.  RR  /\  B  e.  RR ) )
436ad4antr 494 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  A  <  B
)
4435, 41, 43ltled 8105 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  A  <_  B
)
45 maxleim 11245 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  ->  sup ( { A ,  B } ,  RR ,  <  )  =  B ) )
4642, 44, 45sylc 62 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  sup ( { A ,  B } ,  RR ,  <  )  =  B )
4730, 46eqtrd 2222 . . . . . . . 8  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) )  =  B )
48 xrmnfdc 9872 . . . . . . . . . 10  |-  ( A  e.  RR*  -> DECID  A  = -oo )
49 exmiddc 837 . . . . . . . . . 10  |-  (DECID  A  = -oo  ->  ( A  = -oo  \/  -.  A  = -oo ) )
5010, 48, 493syl 17 . . . . . . . . 9  |-  ( ph  ->  ( A  = -oo  \/  -.  A  = -oo ) )
5150ad3antrrr 492 . . . . . . . 8  |-  ( ( ( ( ph  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  ->  ( A  = -oo  \/  -.  A  = -oo ) )
5228, 47, 51mpjaodan 799 . . . . . . 7  |-  ( ( ( ( ph  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  ->  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) )  =  B )
5326, 52eqtrd 2222 . . . . . 6  |-  ( ( ( ( ph  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  ->  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) )  =  B )
54 xrpnfdc 9871 . . . . . . . 8  |-  ( A  e.  RR*  -> DECID  A  = +oo )
55 exmiddc 837 . . . . . . . 8  |-  (DECID  A  = +oo  ->  ( A  = +oo  \/  -.  A  = +oo ) )
5610, 54, 553syl 17 . . . . . . 7  |-  ( ph  ->  ( A  = +oo  \/  -.  A  = +oo ) )
5756ad2antrr 488 . . . . . 6  |-  ( ( ( ph  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  ( A  = +oo  \/  -.  A  = +oo ) )
5824, 53, 57mpjaodan 799 . . . . 5  |-  ( ( ( ph  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) )  =  B )
5916, 58eqtrd 2222 . . . 4  |-  ( ( ( ph  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) )  =  B )
60 xrmnfdc 9872 . . . . . 6  |-  ( B  e.  RR*  -> DECID  B  = -oo )
61 exmiddc 837 . . . . . 6  |-  (DECID  B  = -oo  ->  ( B  = -oo  \/  -.  B  = -oo ) )
6220, 60, 613syl 17 . . . . 5  |-  ( ph  ->  ( B  = -oo  \/  -.  B  = -oo ) )
6362adantr 276 . . . 4  |-  ( (
ph  /\  -.  B  = +oo )  ->  ( B  = -oo  \/  -.  B  = -oo )
)
6414, 59, 63mpjaodan 799 . . 3  |-  ( (
ph  /\  -.  B  = +oo )  ->  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) )  =  B )
655, 64eqtrd 2222 . 2  |-  ( (
ph  /\  -.  B  = +oo )  ->  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) )  =  B )
66 xrpnfdc 9871 . . 3  |-  ( B  e.  RR*  -> DECID  B  = +oo )
67 exmiddc 837 . . 3  |-  (DECID  B  = +oo  ->  ( B  = +oo  \/  -.  B  = +oo ) )
6820, 66, 673syl 17 . 2  |-  ( ph  ->  ( B  = +oo  \/  -.  B  = +oo ) )
693, 65, 68mpjaodan 799 1  |-  ( ph  ->  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) )  =  B )
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 7010   RRcr 7839   +oocpnf 8018   -oocmnf 8019   RR*cxr 8020    < clt 8021    <_ cle 8022
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-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-pre-ltirr 7952  ax-pre-lttrn 7954  ax-pre-apti 7955
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-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4650  df-cnv 4652  df-iota 5196  df-riota 5851  df-sup 7012  df-pnf 8023  df-mnf 8024  df-xr 8025  df-ltxr 8026  df-le 8027
This theorem is referenced by:  xrmaxiflemlub  11287
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