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Theorem xrmaxiflemcl 11201
Description: Lemma for xrmaxif 11207. Closure. (Contributed by Jim Kingdon, 29-Apr-2023.)
Assertion
Ref Expression
xrmaxiflemcl  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) )  e.  RR* )

Proof of Theorem xrmaxiflemcl
StepHypRef Expression
1 pnfxr 7965 . . 3  |- +oo  e.  RR*
21a1i 9 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  B  = +oo )  -> +oo  e.  RR* )
3 simpl 108 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A  e.  RR* )
43ad2antrr 485 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  B  = -oo )  ->  A  e.  RR* )
51a1i 9 . . . 4  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  A  = +oo )  -> +oo  e.  RR* )
6 simpr 109 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  B  e.  RR* )
76ad4antr 491 . . . . 5  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  A  = -oo )  ->  B  e.  RR* )
8 simplr 525 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  A  = +oo )
9 simpr 109 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  A  = -oo )
10 elxr 9726 . . . . . . . . . 10  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
113, 10sylib 121 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
1211ad4antr 491 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
138, 9, 12ecase23d 1345 . . . . . . 7  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  A  e.  RR )
14 simp-4r 537 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  B  = +oo )
15 simpllr 529 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  -.  B  = -oo )
16 elxr 9726 . . . . . . . . . 10  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
176, 16sylib 121 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
1817ad4antr 491 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
1914, 15, 18ecase23d 1345 . . . . . . 7  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  B  e.  RR )
20 maxcl 11167 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  sup ( { A ,  B } ,  RR ,  <  )  e.  RR )
2113, 19, 20syl2anc 409 . . . . . 6  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  sup ( { A ,  B } ,  RR ,  <  )  e.  RR )
2221rexrd 7962 . . . . 5  |-  ( ( ( ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  /\  -.  A  = -oo )  ->  sup ( { A ,  B } ,  RR ,  <  )  e.  RR* )
23 xrmnfdc 9793 . . . . . 6  |-  ( A  e.  RR*  -> DECID  A  = -oo )
2423ad4antr 491 . . . . 5  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  -> DECID  A  = -oo )
257, 22, 24ifcldadc 3554 . . . 4  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  /\  -.  A  = +oo )  ->  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
)  e.  RR* )
26 xrpnfdc 9792 . . . . . 6  |-  ( A  e.  RR*  -> DECID  A  = +oo )
273, 26syl 14 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  -> DECID  A  = +oo )
2827ad2antrr 485 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  -> DECID  A  = +oo )
295, 25, 28ifcldadc 3554 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  /\  -.  B  = -oo )  ->  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) )  e.  RR* )
30 xrmnfdc 9793 . . . 4  |-  ( B  e.  RR*  -> DECID  B  = -oo )
3130ad2antlr 486 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  -> DECID 
B  = -oo )
324, 29, 31ifcldadc 3554 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  B  = +oo )  ->  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  )
) ) )  e. 
RR* )
33 xrpnfdc 9792 . . 3  |-  ( B  e.  RR*  -> DECID  B  = +oo )
346, 33syl 14 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  -> DECID  B  = +oo )
352, 32, 34ifcldadc 3554 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  if ( B  = +oo , +oo ,  if ( B  = -oo ,  A ,  if ( A  = +oo , +oo ,  if ( A  = -oo ,  B ,  sup ( { A ,  B } ,  RR ,  <  ) ) ) ) )  e.  RR* )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103  DECID wdc 829    \/ w3o 972    = wceq 1348    e. wcel 2141   ifcif 3525   {cpr 3582   supcsup 6957   RRcr 7766   +oocpnf 7944   -oocmnf 7945   RR*cxr 7946    < clt 7947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7858  ax-resscn 7859  ax-1cn 7860  ax-1re 7861  ax-icn 7862  ax-addcl 7863  ax-addrcl 7864  ax-mulcl 7865  ax-mulrcl 7866  ax-addcom 7867  ax-mulcom 7868  ax-addass 7869  ax-mulass 7870  ax-distr 7871  ax-i2m1 7872  ax-0lt1 7873  ax-1rid 7874  ax-0id 7875  ax-rnegex 7876  ax-precex 7877  ax-cnre 7878  ax-pre-ltirr 7879  ax-pre-ltwlin 7880  ax-pre-lttrn 7881  ax-pre-apti 7882  ax-pre-ltadd 7883  ax-pre-mulgt0 7884  ax-pre-mulext 7885  ax-arch 7886  ax-caucvg 7887
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-recs 6282  df-frec 6368  df-sup 6959  df-pnf 7949  df-mnf 7950  df-xr 7951  df-ltxr 7952  df-le 7953  df-sub 8085  df-neg 8086  df-reap 8487  df-ap 8494  df-div 8583  df-inn 8872  df-2 8930  df-3 8931  df-4 8932  df-n0 9129  df-z 9206  df-uz 9481  df-rp 9604  df-seqfrec 10395  df-exp 10469  df-cj 10799  df-re 10800  df-im 10801  df-rsqrt 10955  df-abs 10956
This theorem is referenced by:  xrmaxiflemlub  11204  xrmaxiflemval  11206  xrmaxcl  11208
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