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Theorem xrmaxiflemcl 11992
Description: Lemma for xrmaxif 11998. 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 8371 . . 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 492 . . 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 498 . . . . 5 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ A = -oo ) -> B e. RR* )
8 simplr 533 . . . . . . . 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 10160 . . . . . . . . . 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 498 . . . . . . . 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 1391 . . . . . . 7 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> A e. RR )
14 simp-4r 548 . . . . . . . 8 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> -. B = +oo )
15 simpllr 540 . . . . . . . 8 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> -. B = -oo )
16 elxr 10160 . . . . . . . . . 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 498 . . . . . . . 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 1391 . . . . . . 7 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> B e. RR )
20 maxcl 11957 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> sup ( { A , B } , RR , < ) e. RR )
2113, 19, 20syl2anc 415 . . . . . 6 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> sup ( { A , B } , RR , < ) e. RR )
2221rexrd 8368 . . . . 5 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> sup ( { A , B } , RR , < ) e. RR* )
23 xrmnfdc 10227 . . . . . 6 |- ( A e. RR* -> DECID A = -oo )
2423ad4antr 498 . . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) -> DECID A = -oo )
257, 22, 24ifcldadc 3670 . . . 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 10226 . . . . . 6 |- ( A e. RR* -> DECID A = +oo )
273, 26syl 14 . . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> DECID A = +oo )
2827ad2antrr 492 . . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) -> DECID A = +oo )
295, 25, 28ifcldadc 3670 . . 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 10227 . . . 4 |- ( B e. RR* -> DECID B = -oo )
3130ad2antlr 493 . . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) -> DECID B = -oo )
324, 29, 31ifcldadc 3670 . 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 10226 . . 3 |- ( B e. RR* -> DECID B = +oo )
346, 33syl 14 . 2 |- ( ( A e. RR* /\ B e. RR* ) -> DECID B = +oo )
352, 32, 34ifcldadc 3670 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 846   \/ w3o 1008   = wceq 1402   e. wcel 2209   ifcif 3638   {cpr 3709   supcsup 7315   RRcr 8171   +oocpnf 8350   -oocmnf 8351   RR*cxr 8352   < clt 8353
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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8263  ax-resscn 8264  ax-1cn 8265  ax-1re 8266  ax-icn 8267  ax-addcl 8268  ax-addrcl 8269  ax-mulcl 8270  ax-mulrcl 8271  ax-addcom 8272  ax-mulcom 8273  ax-addass 8274  ax-mulass 8275  ax-distr 8276  ax-i2m1 8277  ax-0lt1 8278  ax-1rid 8279  ax-0id 8280  ax-rnegex 8281  ax-precex 8282  ax-cnre 8283  ax-pre-ltirr 8284  ax-pre-ltwlin 8285  ax-pre-lttrn 8286  ax-pre-apti 8287  ax-pre-ltadd 8288  ax-pre-mulgt0 8289  ax-pre-mulext 8290  ax-arch 8291  ax-caucvg 8292
This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6031  df-ov 6081  df-oprab 6082  df-mpo 6083  df-1st 6367  df-2nd 6368  df-recs 6569  df-frec 6655  df-sup 7317  df-pnf 8355  df-mnf 8356  df-xr 8357  df-ltxr 8358  df-le 8359  df-sub 8492  df-neg 8493  df-reap 8896  df-ap 8903  df-div 8996  df-inn 9287  df-2 9345  df-3 9346  df-4 9347  df-n0 9546  df-z 9627  df-uz 9904  df-rp 10037  df-seqfrec 10866  df-exp 10957  df-cj 11588  df-re 11589  df-im 11590  df-rsqrt 11745  df-abs 11746
This theorem is referenced by:  xrmaxiflemlub  11995  xrmaxiflemval  11997  xrmaxcl  11999
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