Theorem xrmaxifle 11672

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 9946	. . . 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 3586	. . 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 4087	. 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 9957	. . . . . 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 3586	. . . . 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 4087	. . . 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 3586	. . . . . . 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 4087	. . . . . 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 9949	. . . . . . . . . 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 3586	. . . . . . . . 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 4091	. . . . . . . 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 528	. . . . . . . . . . 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 9933	. . . . . . . . . . . . 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 1363	. . . . . . . . . 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 9933	. . . . . . . . . . . . 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 1363	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> B e. RR )
34		maxle1 11637	. . . . . . . . . 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 3589	. . . . . . . . 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 4087	. . . . . . . 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 10000	. . . . . . . . . 10 |- ( A e. RR* -> DECID A = -oo )
39		exmiddc 838	. . . . . . . . . 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 800	. . . . . . 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 3589	. . . . . . 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 4087	. . . . . 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 9999	. . . . . . . 8 |- ( A e. RR* -> DECID A = +oo )
47		exmiddc 838	. . . . . . . 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 800	. . . . 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 3589	. . . . 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 4087	. . . 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 10000	. . . . . 6 |- ( B e. RR* -> DECID B = -oo )
54		exmiddc 838	. . . . . 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 800	. . 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 3589	. . 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 4087	. 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 9999	. . . 4 |- ( B e. RR* -> DECID B = +oo )
61		exmiddc 838	. . . 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 800	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 710  DECID wdc 836   \/ w3o 980   = wceq 1373   e. wcel 2178   ifcif 3579   {cpr 3644   class class class wbr 4059   supcsup 7110   RRcr 7959   +oocpnf 8139   -oocmnf 8140   RR*cxr 8141   < clt 8142   <_ cle 8143

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-sup 7112  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-rp 9811  df-seqfrec 10630  df-exp 10721  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425

This theorem is referenced by:  xrmaxiflemval  11676  xrmax1sup  11679