Theorem xrmaxiflemab 11957

Description: Lemma for xrmaxif 11961. A variation of xrmaxleim 11954- 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 3633	. . 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 2270	. 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 3636	. . 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 4140	. . . . 5 |- ( ( ( ph /\ -. B = +oo ) /\ B = -oo ) -> A < -oo )
10		xrmaxiflemab.a	. . . . . . 7 |- ( ph -> A e. RR* )
11		nltmnf 10140	. . . . . . 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 625	. . . 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 3636	. . . . 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 4138	. . . . . . 7 |- ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ A = +oo ) -> +oo < B )
20		xrmaxiflemab.b	. . . . . . . . 9 |- ( ph -> B e. RR* )
21		pnfnlt 10139	. . . . . . . . 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 625	. . . . . 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 3636	. . . . . . 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 3633	. . . . . . . 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 3636	. . . . . . . . 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 10128	. . . . . . . . . . . . . 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 1387	. . . . . . . . . . 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 10128	. . . . . . . . . . . . . 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 1387	. . . . . . . . . . 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 8408	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> A <_ B )
45		maxleim 11915	. . . . . . . . . 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 2267	. . . . . . . 8 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> if ( A = -oo , B , sup ( { A , B } , RR , < ) ) = B )
48		xrmnfdc 10195	. . . . . . . . . 10 |- ( A e. RR* -> DECID A = -oo )
49		exmiddc 844	. . . . . . . . . 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 806	. . . . . . 7 |- ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) -> if ( A = -oo , B , sup ( { A , B } , RR , < ) ) = B )
53	26, 52	eqtrd 2267	. . . . . 6 |- ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) -> if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) = B )
54		xrpnfdc 10194	. . . . . . . 8 |- ( A e. RR* -> DECID A = +oo )
55		exmiddc 844	. . . . . . . 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 806	. . . . 5 |- ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) -> if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) = B )
59	16, 58	eqtrd 2267	. . . 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 10195	. . . . . 6 |- ( B e. RR* -> DECID B = -oo )
61		exmiddc 844	. . . . . 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 806	. . 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 2267	. 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 10194	. . 3 |- ( B e. RR* -> DECID B = +oo )
67		exmiddc 844	. . 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 806	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 716  DECID wdc 842   \/ w3o 1004   = wceq 1398   e. wcel 2205   ifcif 3624   {cpr 3695   class class class wbr 4114   supcsup 7286   RRcr 8142   +oocpnf 8321   -oocmnf 8322   RR*cxr 8323   < clt 8324   <_ cle 8325

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-apti 8258

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-xp 4760  df-cnv 4762  df-iota 5317  df-riota 6011  df-sup 7288  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330

This theorem is referenced by:  xrmaxiflemlub  11958