Theorem xrmaxifle 11209

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 9746	. . . 4 |- ( A e. RR* -> A <_ +oo )
2	1	ad2antrr 485	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ B = +oo ) -> A <_ +oo )
3		simpr 109	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* ) /\ B = +oo ) -> B = +oo )
4	3	iftrued 3533	. . 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 4017	. 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 9757	. . . . . 6 |- ( A e. RR* -> A <_ A )
7	6	ad3antrrr 489	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ B = -oo ) -> A <_ A )
8		simpr 109	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ B = -oo ) -> B = -oo )
9	8	iftrued 3533	. . . . 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 4017	. . . 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 491	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ A = +oo ) -> A <_ +oo )
12		simpr 109	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ A = +oo ) -> A = +oo )
13	12	iftrued 3533	. . . . . . 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 4017	. . . . . 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 9749	. . . . . . . . . 10 |- ( B e. RR* -> -oo <_ B )
16	15	ad5antlr 494	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ A = -oo ) -> -oo <_ B )
17		simpr 109	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ A = -oo ) -> A = -oo )
18	17	iftrued 3533	. . . . . . . . 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 4021	. . . . . . . 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 525	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> -. A = +oo )
21		simpr 109	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> -. A = -oo )
22		elxr 9733	. . . . . . . . . . . . 13 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
23	22	biimpi 119	. . . . . . . . . . . 12 |- ( A e. RR* -> ( A e. RR \/ A = +oo \/ A = -oo ) )
24	23	ad5antr 493	. . . . . . . . . . 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 1345	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> A e. RR )
26		simpr 109	. . . . . . . . . . . 12 |- ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) -> -. B = +oo )
27	26	ad3antrrr 489	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> -. B = +oo )
28		simpr 109	. . . . . . . . . . . 12 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) -> -. B = -oo )
29	28	ad2antrr 485	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> -. B = -oo )
30		elxr 9733	. . . . . . . . . . . . 13 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
31	30	biimpi 119	. . . . . . . . . . . 12 |- ( B e. RR* -> ( B e. RR \/ B = +oo \/ B = -oo ) )
32	31	ad5antlr 494	. . . . . . . . . . 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 1345	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> B e. RR )
34		maxle1 11175	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> A <_ sup ( { A , B } , RR , < ) )
35	25, 33, 34	syl2anc 409	. . . . . . . . 9 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> A <_ sup ( { A , B } , RR , < ) )
36	21	iffalsed 3536	. . . . . . . . 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 4017	. . . . . . . 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 9800	. . . . . . . . . 10 |- ( A e. RR* -> DECID A = -oo )
39		exmiddc 831	. . . . . . . . . 10 |- (DECID A = -oo -> ( A = -oo \/ -. A = -oo ) )
40	38, 39	syl 14	. . . . . . . . 9 |- ( A e. RR* -> ( A = -oo \/ -. A = -oo ) )
41	40	ad4antr 491	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) -> ( A = -oo \/ -. A = -oo ) )
42	19, 37, 41	mpjaodan 793	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) -> A <_ if ( A = -oo , B , sup ( { A , B } , RR , < ) ) )
43		simpr 109	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) -> -. A = +oo )
44	43	iffalsed 3536	. . . . . . 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 4017	. . . . . 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 9799	. . . . . . . 8 |- ( A e. RR* -> DECID A = +oo )
47		exmiddc 831	. . . . . . . 8 |- (DECID A = +oo -> ( A = +oo \/ -. A = +oo ) )
48	46, 47	syl 14	. . . . . . 7 |- ( A e. RR* -> ( A = +oo \/ -. A = +oo ) )
49	48	ad3antrrr 489	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) -> ( A = +oo \/ -. A = +oo ) )
50	14, 45, 49	mpjaodan 793	. . . . 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 3536	. . . . 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 4017	. . . 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 9800	. . . . . 6 |- ( B e. RR* -> DECID B = -oo )
54		exmiddc 831	. . . . . 6 |- (DECID B = -oo -> ( B = -oo \/ -. B = -oo ) )
55	53, 54	syl 14	. . . . 5 |- ( B e. RR* -> ( B = -oo \/ -. B = -oo ) )
56	55	ad2antlr 486	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) -> ( B = -oo \/ -. B = -oo ) )
57	10, 52, 56	mpjaodan 793	. . 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 3536	. . 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 4017	. 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 9799	. . . 4 |- ( B e. RR* -> DECID B = +oo )
61		exmiddc 831	. . . 4 |- (DECID B = +oo -> ( B = +oo \/ -. B = +oo ) )
62	60, 61	syl 14	. . 3 |- ( B e. RR* -> ( B = +oo \/ -. B = +oo ) )
63	62	adantl 275	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( B = +oo \/ -. B = +oo ) )
64	5, 59, 63	mpjaodan 793	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 103   \/ wo 703  DECID wdc 829   \/ w3o 972   = wceq 1348   e. wcel 2141   ifcif 3526   {cpr 3584   class class class wbr 3989   supcsup 6959   RRcr 7773   +oocpnf 7951   -oocmnf 7952   RR*cxr 7953   < clt 7954   <_ cle 7955

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-sup 6961  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-rp 9611  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963

This theorem is referenced by:  xrmaxiflemval  11213  xrmax1sup  11216