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Theorem xrmaxiflemcl 11955
Description: Lemma for xrmaxif 11961. 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 8342 . . 3 |- +oo e. RR*
21a1i 9 . 2 |- ( ( ( A e. RR* /\ B e. RR* ) /\ B = +oo ) -> +oo e. RR* )
3 simpl 109 . . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> A e. RR* )
43ad2antrr 488 . . 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 110 . . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> B e. RR* )
76ad4antr 494 . . . . 5 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ A = -oo ) -> B e. RR* )
8 simplr 529 . . . . . . . 8 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> -. A = +oo )
9 simpr 110 . . . . . . . 8 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> -. A = -oo )
10 elxr 10128 . . . . . . . . . 10 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
113, 10sylib 122 . . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* ) -> ( A e. RR \/ A = +oo \/ A = -oo ) )
1211ad4antr 494 . . . . . . . 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 1387 . . . . . . 7 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> A e. RR )
14 simp-4r 544 . . . . . . . 8 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> -. B = +oo )
15 simpllr 536 . . . . . . . 8 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> -. B = -oo )
16 elxr 10128 . . . . . . . . . 10 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
176, 16sylib 122 . . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* ) -> ( B e. RR \/ B = +oo \/ B = -oo ) )
1817ad4antr 494 . . . . . . . 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 1387 . . . . . . 7 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> B e. RR )
20 maxcl 11920 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> sup ( { A , B } , RR , < ) e. RR )
2113, 19, 20syl2anc 411 . . . . . 6 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> sup ( { A , B } , RR , < ) e. RR )
2221rexrd 8339 . . . . 5 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> sup ( { A , B } , RR , < ) e. RR* )
23 xrmnfdc 10195 . . . . . 6 |- ( A e. RR* -> DECID A = -oo )
2423ad4antr 494 . . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) -> DECID A = -oo )
257, 22, 24ifcldadc 3656 . . . 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 10194 . . . . . 6 |- ( A e. RR* -> DECID A = +oo )
273, 26syl 14 . . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> DECID A = +oo )
2827ad2antrr 488 . . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) -> DECID A = +oo )
295, 25, 28ifcldadc 3656 . . 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 10195 . . . 4 |- ( B e. RR* -> DECID B = -oo )
3130ad2antlr 489 . . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) -> DECID B = -oo )
324, 29, 31ifcldadc 3656 . 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 10194 . . 3 |- ( B e. RR* -> DECID B = +oo )
346, 33syl 14 . 2 |- ( ( A e. RR* /\ B e. RR* ) -> DECID B = +oo )
352, 32, 34ifcldadc 3656 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 104  DECID wdc 842   \/ w3o 1004   = wceq 1398   e. wcel 2205   ifcif 3624   {cpr 3695   supcsup 7286   RRcr 8142   +oocpnf 8321   -oocmnf 8322   RR*cxr 8323   < clt 8324
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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263
This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-sup 7288  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-rp 10005  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709
This theorem is referenced by:  xrmaxiflemlub  11958  xrmaxiflemval  11960  xrmaxcl  11962
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