Theorem xrmaxifle 11222

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 9760	. . . 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 3539	. . 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 4026	. 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 9771	. . . . . 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 3539	. . . . 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 4026	. . . 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 3539	. . . . . . 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 4026	. . . . . 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 9763	. . . . . . . . . 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 3539	. . . . . . . . 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 4030	. . . . . . . 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 9747	. . . . . . . . . . . . 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 1350	. . . . . . . . . 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 9747	. . . . . . . . . . . . 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 1350	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> B e. RR )
34		maxle1 11188	. . . . . . . . . 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 3542	. . . . . . . . 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 4026	. . . . . . . 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 9814	. . . . . . . . . 10 |- ( A e. RR* -> DECID A = -oo )
39		exmiddc 836	. . . . . . . . . 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 798	. . . . . . 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 3542	. . . . . . 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 4026	. . . . . 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 9813	. . . . . . . 8 |- ( A e. RR* -> DECID A = +oo )
47		exmiddc 836	. . . . . . . 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 798	. . . . 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 3542	. . . . 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 4026	. . . 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 9814	. . . . . 6 |- ( B e. RR* -> DECID B = -oo )
54		exmiddc 836	. . . . . 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 798	. . 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 3542	. . 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 4026	. 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 9813	. . . 4 |- ( B e. RR* -> DECID B = +oo )
61		exmiddc 836	. . . 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 798	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 708  DECID wdc 834   \/ w3o 977   = wceq 1353   e. wcel 2146   ifcif 3532   {cpr 3590   class class class wbr 3998   supcsup 6971   RRcr 7785   +oocpnf 7963   -oocmnf 7964   RR*cxr 7965   < clt 7966   <_ cle 7967

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-sup 6973  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8603  df-inn 8893  df-2 8951  df-3 8952  df-4 8953  df-n0 9150  df-z 9227  df-uz 9502  df-rp 9625  df-seqfrec 10416  df-exp 10490  df-cj 10819  df-re 10820  df-im 10821  df-rsqrt 10975  df-abs 10976

This theorem is referenced by:  xrmaxiflemval  11226  xrmax1sup  11229