Theorem xrmaxifle 11806

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 10023	. . . 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 3612	. . 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 4116	. 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 10034	. . . . . 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 3612	. . . . 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 4116	. . . 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 3612	. . . . . . 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 4116	. . . . . 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 10026	. . . . . . . . . 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 3612	. . . . . . . . 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 4120	. . . . . . . 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 10010	. . . . . . . . . . . . 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 1386	. . . . . . . . . 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 10010	. . . . . . . . . . . . 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 1386	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> B e. RR )
34		maxle1 11771	. . . . . . . . . 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 3615	. . . . . . . . 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 4116	. . . . . . . 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 10077	. . . . . . . . . 10 |- ( A e. RR* -> DECID A = -oo )
39		exmiddc 843	. . . . . . . . . 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 805	. . . . . . 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 3615	. . . . . . 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 4116	. . . . . 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 10076	. . . . . . . 8 |- ( A e. RR* -> DECID A = +oo )
47		exmiddc 843	. . . . . . . 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 805	. . . . 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 3615	. . . . 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 4116	. . . 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 10077	. . . . . 6 |- ( B e. RR* -> DECID B = -oo )
54		exmiddc 843	. . . . . 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 805	. . 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 3615	. . 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 4116	. 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 10076	. . . 4 |- ( B e. RR* -> DECID B = +oo )
61		exmiddc 843	. . . 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 805	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 715  DECID wdc 841   \/ w3o 1003   = wceq 1397   e. wcel 2202   ifcif 3605   {cpr 3670   class class class wbr 4088   supcsup 7180   RRcr 8030   +oocpnf 8210   -oocmnf 8211   RR*cxr 8212   < clt 8213   <_ cle 8214

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-sup 7182  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-rp 9888  df-seqfrec 10709  df-exp 10800  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559

This theorem is referenced by:  xrmaxiflemval  11810  xrmax1sup  11813