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Theorem xrmaxiflemab 10908
Description: Lemma for xrmaxif 10912. A variation of xrmaxleim 10905- 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 109 . . . 4  |-  ( (
ph  /\  B  = +oo )  ->  B  = +oo )
21iftrued 3447 . . 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 2150 . 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 109 . . . 4  |-  ( (
ph  /\  -.  B  = +oo )  ->  -.  B  = +oo )
54iffalsed 3450 . . 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 477 . . . . . 6  |-  ( ( ( ph  /\  -.  B  = +oo )  /\  B  = -oo )  ->  A  <  B
)
8 simpr 109 . . . . . 6  |-  ( ( ( ph  /\  -.  B  = +oo )  /\  B  = -oo )  ->  B  = -oo )
97, 8breqtrd 3919 . . . . 5  |-  ( ( ( ph  /\  -.  B  = +oo )  /\  B  = -oo )  ->  A  < -oo )
10 xrmaxiflemab.a . . . . . . 7  |-  ( ph  ->  A  e.  RR* )
11 nltmnf 9467 . . . . . . 7  |-  ( A  e.  RR*  ->  -.  A  < -oo )
1210, 11syl 14 . . . . . 6  |-  ( ph  ->  -.  A  < -oo )
1312ad2antrr 477 . . . . 5  |-  ( ( ( ph  /\  -.  B  = +oo )  /\  B  = -oo )  ->  -.  A  < -oo )
149, 13pm2.21dd 592 . . . 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 109 . . . . . 6  |-  ( ( ( ph  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  -.  B  = -oo )
1615iffalsed 3450 . . . . 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 109 . . . . . . . 8  |-  ( ( ( ( ph  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  A  = +oo )  ->  A  = +oo )
186ad3antrrr 481 . . . . . . . 8  |-  ( ( ( ( ph  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  A  = +oo )  ->  A  <  B )
1917, 18eqbrtrrd 3917 . . . . . . 7  |-  ( ( ( ( ph  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  A  = +oo )  -> +oo  <  B )
20 xrmaxiflemab.b . . . . . . . . 9  |-  ( ph  ->  B  e.  RR* )
21 pnfnlt 9466 . . . . . . . . 9  |-  ( B  e.  RR*  ->  -. +oo  <  B )
2220, 21syl 14 . . . . . . . 8  |-  ( ph  ->  -. +oo  <  B
)
2322ad3antrrr 481 . . . . . . 7  |-  ( ( ( ( ph  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  A  = +oo )  ->  -. +oo 
<  B )
2419, 23pm2.21dd 592 . . . . . 6  |-  ( ( ( ( ph  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  A  = +oo )  ->  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) )  =  B )
25 simpr 109 . . . . . . . 8  |-  ( ( ( ( ph  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  ->  -.  A  = +oo )
2625iffalsed 3450 . . . . . . 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 109 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  A  = -oo )  ->  A  = -oo )
2827iftrued 3447 . . . . . . . 8  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  A  = -oo )  ->  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) )  =  B )
29 simpr 109 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  A  = -oo )
3029iffalsed 3450 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) )  =  sup ( { A ,  B } ,  RR ,  <  ) )
3125adantr 272 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  A  = +oo )
32 elxr 9456 . . . . . . . . . . . . . 14  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
3310, 32sylib 121 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
3433ad4antr 483 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
3531, 29, 34ecase23d 1311 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  A  e.  RR )
364ad3antrrr 481 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  B  = +oo )
3715ad2antrr 477 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  B  = -oo )
38 elxr 9456 . . . . . . . . . . . . . 14  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
3920, 38sylib 121 . . . . . . . . . . . . 13  |-  ( ph  ->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
4039ad4antr 483 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
4136, 37, 40ecase23d 1311 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  B  e.  RR )
4235, 41jca 302 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  ( A  e.  RR  /\  B  e.  RR ) )
436ad4antr 483 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  A  <  B
)
4435, 41, 43ltled 7804 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  A  <_  B
)
45 maxleim 10869 . . . . . . . . . 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 2147 . . . . . . . 8  |-  ( ( ( ( ( ph  /\ 
-.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) )  =  B )
48 xrmnfdc 9519 . . . . . . . . . 10  |-  ( A  e.  RR*  -> DECID  A  = -oo )
49 exmiddc 804 . . . . . . . . . 10  |-  (DECID  A  = -oo  ->  ( A  = -oo  \/  -.  A  = -oo ) )
5010, 48, 493syl 17 . . . . . . . . 9  |-  ( ph  ->  ( A  = -oo  \/  -.  A  = -oo ) )
5150ad3antrrr 481 . . . . . . . 8  |-  ( ( ( ( ph  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  ->  ( A  = -oo  \/  -.  A  = -oo ) )
5228, 47, 51mpjaodan 770 . . . . . . 7  |-  ( ( ( ( ph  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  ->  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) )  =  B )
5326, 52eqtrd 2147 . . . . . 6  |-  ( ( ( ( ph  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  ->  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) )  =  B )
54 xrpnfdc 9518 . . . . . . . 8  |-  ( A  e.  RR*  -> DECID  A  = +oo )
55 exmiddc 804 . . . . . . . 8  |-  (DECID  A  = +oo  ->  ( A  = +oo  \/  -.  A  = +oo ) )
5610, 54, 553syl 17 . . . . . . 7  |-  ( ph  ->  ( A  = +oo  \/  -.  A  = +oo ) )
5756ad2antrr 477 . . . . . 6  |-  ( ( ( ph  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  ( A  = +oo  \/  -.  A  = +oo ) )
5824, 53, 57mpjaodan 770 . . . . 5  |-  ( ( ( ph  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) )  =  B )
5916, 58eqtrd 2147 . . . 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 9519 . . . . . 6  |-  ( B  e.  RR*  -> DECID  B  = -oo )
61 exmiddc 804 . . . . . 6  |-  (DECID  B  = -oo  ->  ( B  = -oo  \/  -.  B  = -oo ) )
6220, 60, 613syl 17 . . . . 5  |-  ( ph  ->  ( B  = -oo  \/  -.  B  = -oo ) )
6362adantr 272 . . . 4  |-  ( (
ph  /\  -.  B  = +oo )  ->  ( B  = -oo  \/  -.  B  = -oo )
)
6414, 59, 63mpjaodan 770 . . 3  |-  ( (
ph  /\  -.  B  = +oo )  ->  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) )  =  B )
655, 64eqtrd 2147 . 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 9518 . . 3  |-  ( B  e.  RR*  -> DECID  B  = +oo )
67 exmiddc 804 . . 3  |-  (DECID  B  = +oo  ->  ( B  = +oo  \/  -.  B  = +oo ) )
6820, 66, 673syl 17 . 2  |-  ( ph  ->  ( B  = +oo  \/  -.  B  = +oo ) )
693, 65, 68mpjaodan 770 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 103    \/ wo 680  DECID wdc 802    \/ w3o 944    = wceq 1314    e. wcel 1463   ifcif 3440   {cpr 3494   class class class wbr 3895   supcsup 6821   RRcr 7546   +oocpnf 7721   -oocmnf 7722   RR*cxr 7723    < clt 7724    <_ cle 7725
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-cnex 7636  ax-resscn 7637  ax-pre-ltirr 7657  ax-pre-lttrn 7659  ax-pre-apti 7660
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rmo 2398  df-rab 2399  df-v 2659  df-sbc 2879  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-xp 4505  df-cnv 4507  df-iota 5046  df-riota 5684  df-sup 6823  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730
This theorem is referenced by:  xrmaxiflemlub  10909
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