Theorem xrmaxiflemab 11390

Description: Lemma for xrmaxif 11394. A variation of xrmaxleim 11387- that is, if we know which of two real numbers is larger, we know the maximum of the two. (Contributed by Jim Kingdon, 26-Apr-2023.)

Hypotheses

Ref	Expression
xrmaxiflemab.a	|- ( ph -> A e. RR* )
xrmaxiflemab.b	|- ( ph -> B e. RR* )
xrmaxiflemab.ab	|- ( ph -> A < B )

Assertion

Ref	Expression
xrmaxiflemab	|- ( ph -> if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) ) = B )

Proof of Theorem xrmaxiflemab

Step	Hyp	Ref	Expression
1		simpr 110	. . . 4 |- ( ( ph /\ B = +oo ) -> B = +oo )
2	1	iftrued 3564	. . 3 |- ( ( ph /\ B = +oo ) -> if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) ) = +oo )
3	2, 1	eqtr4d 2229	. 2 |- ( ( ph /\ B = +oo ) -> if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) ) = B )
4		simpr 110	. . . 4 |- ( ( ph /\ -. B = +oo ) -> -. B = +oo )
5	4	iffalsed 3567	. . 3 |- ( ( ph /\ -. 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 , < ) ) ) ) )
6		xrmaxiflemab.ab	. . . . . . 7 |- ( ph -> A < B )
7	6	ad2antrr 488	. . . . . 6 |- ( ( ( ph /\ -. B = +oo ) /\ B = -oo ) -> A < B )
8		simpr 110	. . . . . 6 |- ( ( ( ph /\ -. B = +oo ) /\ B = -oo ) -> B = -oo )
9	7, 8	breqtrd 4055	. . . . 5 |- ( ( ( ph /\ -. B = +oo ) /\ B = -oo ) -> A < -oo )
10		xrmaxiflemab.a	. . . . . . 7 |- ( ph -> A e. RR* )
11		nltmnf 9854	. . . . . . 7 |- ( A e. RR* -> -. A < -oo )
12	10, 11	syl 14	. . . . . 6 |- ( ph -> -. A < -oo )
13	12	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ -. B = +oo ) /\ B = -oo ) -> -. A < -oo )
14	9, 13	pm2.21dd 621	. . . 4 |- ( ( ( ph /\ -. B = +oo ) /\ B = -oo ) -> if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) = B )
15		simpr 110	. . . . . 6 |- ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) -> -. B = -oo )
16	15	iffalsed 3567	. . . . 5 |- ( ( ( ph /\ -. 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 , < ) ) ) )
17		simpr 110	. . . . . . . 8 |- ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ A = +oo ) -> A = +oo )
18	6	ad3antrrr 492	. . . . . . . 8 |- ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ A = +oo ) -> A < B )
19	17, 18	eqbrtrrd 4053	. . . . . . 7 |- ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ A = +oo ) -> +oo < B )
20		xrmaxiflemab.b	. . . . . . . . 9 |- ( ph -> B e. RR* )
21		pnfnlt 9853	. . . . . . . . 9 |- ( B e. RR* -> -. +oo < B )
22	20, 21	syl 14	. . . . . . . 8 |- ( ph -> -. +oo < B )
23	22	ad3antrrr 492	. . . . . . 7 |- ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ A = +oo ) -> -. +oo < B )
24	19, 23	pm2.21dd 621	. . . . . 6 |- ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ A = +oo ) -> if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) = B )
25		simpr 110	. . . . . . . 8 |- ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) -> -. A = +oo )
26	25	iffalsed 3567	. . . . . . 7 |- ( ( ( ( ph /\ -. 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 , < ) ) )
27		simpr 110	. . . . . . . . 9 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ A = -oo ) -> A = -oo )
28	27	iftrued 3564	. . . . . . . 8 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ A = -oo ) -> if ( A = -oo , B , sup ( { A , B } , RR , < ) ) = B )
29		simpr 110	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> -. A = -oo )
30	29	iffalsed 3567	. . . . . . . . 9 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> if ( A = -oo , B , sup ( { A , B } , RR , < ) ) = sup ( { A , B } , RR , < ) )
31	25	adantr 276	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> -. A = +oo )
32		elxr 9842	. . . . . . . . . . . . . 14 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
33	10, 32	sylib 122	. . . . . . . . . . . . 13 |- ( ph -> ( A e. RR \/ A = +oo \/ A = -oo ) )
34	33	ad4antr 494	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> ( A e. RR \/ A = +oo \/ A = -oo ) )
35	31, 29, 34	ecase23d 1361	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> A e. RR )
36	4	ad3antrrr 492	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> -. B = +oo )
37	15	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> -. B = -oo )
38		elxr 9842	. . . . . . . . . . . . . 14 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
39	20, 38	sylib 122	. . . . . . . . . . . . 13 |- ( ph -> ( B e. RR \/ B = +oo \/ B = -oo ) )
40	39	ad4antr 494	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> ( B e. RR \/ B = +oo \/ B = -oo ) )
41	36, 37, 40	ecase23d 1361	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> B e. RR )
42	35, 41	jca 306	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> ( A e. RR /\ B e. RR ) )
43	6	ad4antr 494	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> A < B )
44	35, 41, 43	ltled 8138	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> A <_ B )
45		maxleim 11349	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> ( A <_ B -> sup ( { A , B } , RR , < ) = B ) )
46	42, 44, 45	sylc 62	. . . . . . . . 9 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> sup ( { A , B } , RR , < ) = B )
47	30, 46	eqtrd 2226	. . . . . . . 8 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> if ( A = -oo , B , sup ( { A , B } , RR , < ) ) = B )
48		xrmnfdc 9909	. . . . . . . . . 10 |- ( A e. RR* -> DECID A = -oo )
49		exmiddc 837	. . . . . . . . . 10 |- (DECID A = -oo -> ( A = -oo \/ -. A = -oo ) )
50	10, 48, 49	3syl 17	. . . . . . . . 9 |- ( ph -> ( A = -oo \/ -. A = -oo ) )
51	50	ad3antrrr 492	. . . . . . . 8 |- ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) -> ( A = -oo \/ -. A = -oo ) )
52	28, 47, 51	mpjaodan 799	. . . . . . 7 |- ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) -> if ( A = -oo , B , sup ( { A , B } , RR , < ) ) = B )
53	26, 52	eqtrd 2226	. . . . . 6 |- ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) -> if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) = B )
54		xrpnfdc 9908	. . . . . . . 8 |- ( A e. RR* -> DECID A = +oo )
55		exmiddc 837	. . . . . . . 8 |- (DECID A = +oo -> ( A = +oo \/ -. A = +oo ) )
56	10, 54, 55	3syl 17	. . . . . . 7 |- ( ph -> ( A = +oo \/ -. A = +oo ) )
57	56	ad2antrr 488	. . . . . 6 |- ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) -> ( A = +oo \/ -. A = +oo ) )
58	24, 53, 57	mpjaodan 799	. . . . 5 |- ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) -> if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) = B )
59	16, 58	eqtrd 2226	. . . 4 |- ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) -> if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) = B )
60		xrmnfdc 9909	. . . . . 6 |- ( B e. RR* -> DECID B = -oo )
61		exmiddc 837	. . . . . 6 |- (DECID B = -oo -> ( B = -oo \/ -. B = -oo ) )
62	20, 60, 61	3syl 17	. . . . 5 |- ( ph -> ( B = -oo \/ -. B = -oo ) )
63	62	adantr 276	. . . 4 |- ( ( ph /\ -. B = +oo ) -> ( B = -oo \/ -. B = -oo ) )
64	14, 59, 63	mpjaodan 799	. . 3 |- ( ( ph /\ -. B = +oo ) -> if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) = B )
65	5, 64	eqtrd 2226	. 2 |- ( ( ph /\ -. B = +oo ) -> if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) ) = B )
66		xrpnfdc 9908	. . 3 |- ( B e. RR* -> DECID B = +oo )
67		exmiddc 837	. . 3 |- (DECID B = +oo -> ( B = +oo \/ -. B = +oo ) )
68	20, 66, 67	3syl 17	. 2 |- ( ph -> ( B = +oo \/ -. B = +oo ) )
69	3, 65, 68	mpjaodan 799	1 |- ( ph -> if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) ) = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709  DECID wdc 835   \/ w3o 979   = wceq 1364   e. wcel 2164   ifcif 3557   {cpr 3619   class class class wbr 4029   supcsup 7041   RRcr 7871   +oocpnf 8051   -oocmnf 8052   RR*cxr 8053   < clt 8054   <_ cle 8055

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-apti 7987

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-xp 4665  df-cnv 4667  df-iota 5215  df-riota 5873  df-sup 7043  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060

This theorem is referenced by:  xrmaxiflemlub  11391