Theorem xrmaxifle 11956

Description: An upper bound for { A , B } in the extended reals. (Contributed by Jim Kingdon, 26-Apr-2023.)

Assertion

Ref	Expression
xrmaxifle	|- ( ( A e. RR* /\ B e. RR* ) -> A <_ if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) ) )

Proof of Theorem xrmaxifle

Step	Hyp	Ref	Expression
1		pnfge 10141	. . . 4 |- ( A e. RR* -> A <_ +oo )
2	1	ad2antrr 488	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ B = +oo ) -> A <_ +oo )
3		simpr 110	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* ) /\ B = +oo ) -> B = +oo )
4	3	iftrued 3633	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ B = +oo ) -> if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) ) = +oo )
5	2, 4	breqtrrd 4142	. 2 |- ( ( ( A e. RR* /\ B e. RR* ) /\ B = +oo ) -> A <_ if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) ) )
6		xrleid 10152	. . . . . 6 |- ( A e. RR* -> A <_ A )
7	6	ad3antrrr 492	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ B = -oo ) -> A <_ A )
8		simpr 110	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ B = -oo ) -> B = -oo )
9	8	iftrued 3633	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ B = -oo ) -> if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) = A )
10	7, 9	breqtrrd 4142	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ B = -oo ) -> A <_ if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) )
11	1	ad4antr 494	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ A = +oo ) -> A <_ +oo )
12		simpr 110	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ A = +oo ) -> A = +oo )
13	12	iftrued 3633	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ A = +oo ) -> if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) = +oo )
14	11, 13	breqtrrd 4142	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ A = +oo ) -> A <_ if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) )
15		mnfle 10144	. . . . . . . . . 10 |- ( B e. RR* -> -oo <_ B )
16	15	ad5antlr 497	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ A = -oo ) -> -oo <_ B )
17		simpr 110	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ A = -oo ) -> A = -oo )
18	17	iftrued 3633	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ A = -oo ) -> if ( A = -oo , B , sup ( { A , B } , RR , < ) ) = B )
19	16, 17, 18	3brtr4d 4146	. . . . . . . 8 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ A = -oo ) -> A <_ if ( A = -oo , B , sup ( { A , B } , RR , < ) ) )
20		simplr 529	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> -. A = +oo )
21		simpr 110	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> -. A = -oo )
22		elxr 10128	. . . . . . . . . . . . 13 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
23	22	biimpi 120	. . . . . . . . . . . 12 |- ( A e. RR* -> ( A e. RR \/ A = +oo \/ A = -oo ) )
24	23	ad5antr 496	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> ( A e. RR \/ A = +oo \/ A = -oo ) )
25	20, 21, 24	ecase23d 1387	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> A e. RR )
26		simpr 110	. . . . . . . . . . . 12 |- ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) -> -. B = +oo )
27	26	ad3antrrr 492	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> -. B = +oo )
28		simpr 110	. . . . . . . . . . . 12 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) -> -. B = -oo )
29	28	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> -. B = -oo )
30		elxr 10128	. . . . . . . . . . . . 13 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
31	30	biimpi 120	. . . . . . . . . . . 12 |- ( B e. RR* -> ( B e. RR \/ B = +oo \/ B = -oo ) )
32	31	ad5antlr 497	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> ( B e. RR \/ B = +oo \/ B = -oo ) )
33	27, 29, 32	ecase23d 1387	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> B e. RR )
34		maxle1 11921	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> A <_ sup ( { A , B } , RR , < ) )
35	25, 33, 34	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> A <_ sup ( { A , B } , RR , < ) )
36	21	iffalsed 3636	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> if ( A = -oo , B , sup ( { A , B } , RR , < ) ) = sup ( { A , B } , RR , < ) )
37	35, 36	breqtrrd 4142	. . . . . . . 8 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> A <_ if ( A = -oo , B , sup ( { A , B } , RR , < ) ) )
38		xrmnfdc 10195	. . . . . . . . . 10 |- ( A e. RR* -> DECID A = -oo )
39		exmiddc 844	. . . . . . . . . 10 |- (DECID A = -oo -> ( A = -oo \/ -. A = -oo ) )
40	38, 39	syl 14	. . . . . . . . 9 |- ( A e. RR* -> ( A = -oo \/ -. A = -oo ) )
41	40	ad4antr 494	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) -> ( A = -oo \/ -. A = -oo ) )
42	19, 37, 41	mpjaodan 806	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) -> A <_ if ( A = -oo , B , sup ( { A , B } , RR , < ) ) )
43		simpr 110	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) -> -. A = +oo )
44	43	iffalsed 3636	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. 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 , < ) ) )
45	42, 44	breqtrrd 4142	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) -> A <_ if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) )
46		xrpnfdc 10194	. . . . . . . 8 |- ( A e. RR* -> DECID A = +oo )
47		exmiddc 844	. . . . . . . 8 |- (DECID A = +oo -> ( A = +oo \/ -. A = +oo ) )
48	46, 47	syl 14	. . . . . . 7 |- ( A e. RR* -> ( A = +oo \/ -. A = +oo ) )
49	48	ad3antrrr 492	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) -> ( A = +oo \/ -. A = +oo ) )
50	14, 45, 49	mpjaodan 806	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) -> A <_ if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) )
51	28	iffalsed 3636	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. 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 , < ) ) ) )
52	50, 51	breqtrrd 4142	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) -> A <_ if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) )
53		xrmnfdc 10195	. . . . . 6 |- ( B e. RR* -> DECID B = -oo )
54		exmiddc 844	. . . . . 6 |- (DECID B = -oo -> ( B = -oo \/ -. B = -oo ) )
55	53, 54	syl 14	. . . . 5 |- ( B e. RR* -> ( B = -oo \/ -. B = -oo ) )
56	55	ad2antlr 489	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) -> ( B = -oo \/ -. B = -oo ) )
57	10, 52, 56	mpjaodan 806	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) -> A <_ if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) )
58	26	iffalsed 3636	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ -. 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 , < ) ) ) ) )
59	57, 58	breqtrrd 4142	. 2 |- ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) -> A <_ if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) ) )
60		xrpnfdc 10194	. . . 4 |- ( B e. RR* -> DECID B = +oo )
61		exmiddc 844	. . . 4 |- (DECID B = +oo -> ( B = +oo \/ -. B = +oo ) )
62	60, 61	syl 14	. . 3 |- ( B e. RR* -> ( B = +oo \/ -. B = +oo ) )
63	62	adantl 277	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( B = +oo \/ -. B = +oo ) )
64	5, 59, 63	mpjaodan 806	1 |- ( ( A e. RR* /\ B e. RR* ) -> A <_ if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 716  DECID wdc 842   \/ w3o 1004   = wceq 1398   e. wcel 2205   ifcif 3624   {cpr 3695   class class class wbr 4114   supcsup 7286   RRcr 8142   +oocpnf 8321   -oocmnf 8322   RR*cxr 8323   < clt 8324   <_ cle 8325

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:  xrmaxiflemval  11960  xrmax1sup  11963