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Theorem xrmaxiflemcl 11186
Description: Lemma for xrmaxif 11192. 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 7951 . . 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 480 . . 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 486 . . . . 5 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ A = -oo ) -> B e. RR* )
8 simplr 520 . . . . . . . 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 9712 . . . . . . . . . 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 486 . . . . . . . 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 1340 . . . . . . 7 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> A e. RR )
14 simp-4r 532 . . . . . . . 8 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> -. B = +oo )
15 simpllr 524 . . . . . . . 8 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> -. B = -oo )
16 elxr 9712 . . . . . . . . . 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 486 . . . . . . . 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 1340 . . . . . . 7 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> B e. RR )
20 maxcl 11152 . . . . . . 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 7948 . . . . 5 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> sup ( { A , B } , RR , < ) e. RR* )
23 xrmnfdc 9779 . . . . . 6 |- ( A e. RR* -> DECID A = -oo )
2423ad4antr 486 . . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) -> DECID A = -oo )
257, 22, 24ifcldadc 3549 . . . 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 9778 . . . . . 6 |- ( A e. RR* -> DECID A = +oo )
273, 26syl 14 . . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> DECID A = +oo )
2827ad2antrr 480 . . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) -> DECID A = +oo )
295, 25, 28ifcldadc 3549 . . 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 9779 . . . 4 |- ( B e. RR* -> DECID B = -oo )
3130ad2antlr 481 . . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) -> DECID B = -oo )
324, 29, 31ifcldadc 3549 . 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 9778 . . 3 |- ( B e. RR* -> DECID B = +oo )
346, 33syl 14 . 2 |- ( ( A e. RR* /\ B e. RR* ) -> DECID B = +oo )
352, 32, 34ifcldadc 3549 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 824   \/ w3o 967   = wceq 1343   e. wcel 2136   ifcif 3520   {cpr 3577   supcsup 6947   RRcr 7752   +oocpnf 7930   -oocmnf 7931   RR*cxr 7932   < clt 7933
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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-sup 6949  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-rp 9590  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941
This theorem is referenced by:  xrmaxiflemlub  11189  xrmaxiflemval  11191  xrmaxcl  11193
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