Theorem xrmaxiflemab 10908

Description: Lemma for xrmaxif 10912. A variation of xrmaxleim 10905- 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 109	. . . 4 |- ( ( ph /\ B = +oo ) -> B = +oo )
2	1	iftrued 3447	. . 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 2150	. 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 109	. . . 4 |- ( ( ph /\ -. B = +oo ) -> -. B = +oo )
5	4	iffalsed 3450	. . 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 477	. . . . . 6 |- ( ( ( ph /\ -. B = +oo ) /\ B = -oo ) -> A < B )
8		simpr 109	. . . . . 6 |- ( ( ( ph /\ -. B = +oo ) /\ B = -oo ) -> B = -oo )
9	7, 8	breqtrd 3919	. . . . 5 |- ( ( ( ph /\ -. B = +oo ) /\ B = -oo ) -> A < -oo )
10		xrmaxiflemab.a	. . . . . . 7 |- ( ph -> A e. RR* )
11		nltmnf 9467	. . . . . . 7 |- ( A e. RR* -> -. A < -oo )
12	10, 11	syl 14	. . . . . 6 |- ( ph -> -. A < -oo )
13	12	ad2antrr 477	. . . . 5 |- ( ( ( ph /\ -. B = +oo ) /\ B = -oo ) -> -. A < -oo )
14	9, 13	pm2.21dd 592	. . . 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 109	. . . . . 6 |- ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) -> -. B = -oo )
16	15	iffalsed 3450	. . . . 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 109	. . . . . . . 8 |- ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ A = +oo ) -> A = +oo )
18	6	ad3antrrr 481	. . . . . . . 8 |- ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ A = +oo ) -> A < B )
19	17, 18	eqbrtrrd 3917	. . . . . . 7 |- ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ A = +oo ) -> +oo < B )
20		xrmaxiflemab.b	. . . . . . . . 9 |- ( ph -> B e. RR* )
21		pnfnlt 9466	. . . . . . . . 9 |- ( B e. RR* -> -. +oo < B )
22	20, 21	syl 14	. . . . . . . 8 |- ( ph -> -. +oo < B )
23	22	ad3antrrr 481	. . . . . . 7 |- ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ A = +oo ) -> -. +oo < B )
24	19, 23	pm2.21dd 592	. . . . . 6 |- ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ A = +oo ) -> if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) = B )
25		simpr 109	. . . . . . . 8 |- ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) -> -. A = +oo )
26	25	iffalsed 3450	. . . . . . 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 109	. . . . . . . . 9 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ A = -oo ) -> A = -oo )
28	27	iftrued 3447	. . . . . . . 8 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ A = -oo ) -> if ( A = -oo , B , sup ( { A , B } , RR , < ) ) = B )
29		simpr 109	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> -. A = -oo )
30	29	iffalsed 3450	. . . . . . . . 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 272	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> -. A = +oo )
32		elxr 9456	. . . . . . . . . . . . . 14 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
33	10, 32	sylib 121	. . . . . . . . . . . . 13 |- ( ph -> ( A e. RR \/ A = +oo \/ A = -oo ) )
34	33	ad4antr 483	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> ( A e. RR \/ A = +oo \/ A = -oo ) )
35	31, 29, 34	ecase23d 1311	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> A e. RR )
36	4	ad3antrrr 481	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> -. B = +oo )
37	15	ad2antrr 477	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> -. B = -oo )
38		elxr 9456	. . . . . . . . . . . . . 14 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
39	20, 38	sylib 121	. . . . . . . . . . . . 13 |- ( ph -> ( B e. RR \/ B = +oo \/ B = -oo ) )
40	39	ad4antr 483	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> ( B e. RR \/ B = +oo \/ B = -oo ) )
41	36, 37, 40	ecase23d 1311	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> B e. RR )
42	35, 41	jca 302	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> ( A e. RR /\ B e. RR ) )
43	6	ad4antr 483	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> A < B )
44	35, 41, 43	ltled 7804	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> A <_ B )
45		maxleim 10869	. . . . . . . . . 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 2147	. . . . . . . 8 |- ( ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) /\ -. A = -oo ) -> if ( A = -oo , B , sup ( { A , B } , RR , < ) ) = B )
48		xrmnfdc 9519	. . . . . . . . . 10 |- ( A e. RR* -> DECID A = -oo )
49		exmiddc 804	. . . . . . . . . 10 |- (DECID A = -oo -> ( A = -oo \/ -. A = -oo ) )
50	10, 48, 49	3syl 17	. . . . . . . . 9 |- ( ph -> ( A = -oo \/ -. A = -oo ) )
51	50	ad3antrrr 481	. . . . . . . 8 |- ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) -> ( A = -oo \/ -. A = -oo ) )
52	28, 47, 51	mpjaodan 770	. . . . . . 7 |- ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) -> if ( A = -oo , B , sup ( { A , B } , RR , < ) ) = B )
53	26, 52	eqtrd 2147	. . . . . 6 |- ( ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) /\ -. A = +oo ) -> if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) = B )
54		xrpnfdc 9518	. . . . . . . 8 |- ( A e. RR* -> DECID A = +oo )
55		exmiddc 804	. . . . . . . 8 |- (DECID A = +oo -> ( A = +oo \/ -. A = +oo ) )
56	10, 54, 55	3syl 17	. . . . . . 7 |- ( ph -> ( A = +oo \/ -. A = +oo ) )
57	56	ad2antrr 477	. . . . . 6 |- ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) -> ( A = +oo \/ -. A = +oo ) )
58	24, 53, 57	mpjaodan 770	. . . . 5 |- ( ( ( ph /\ -. B = +oo ) /\ -. B = -oo ) -> if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) = B )
59	16, 58	eqtrd 2147	. . . 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 9519	. . . . . 6 |- ( B e. RR* -> DECID B = -oo )
61		exmiddc 804	. . . . . 6 |- (DECID B = -oo -> ( B = -oo \/ -. B = -oo ) )
62	20, 60, 61	3syl 17	. . . . 5 |- ( ph -> ( B = -oo \/ -. B = -oo ) )
63	62	adantr 272	. . . 4 |- ( ( ph /\ -. B = +oo ) -> ( B = -oo \/ -. B = -oo ) )
64	14, 59, 63	mpjaodan 770	. . 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 2147	. 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 9518	. . 3 |- ( B e. RR* -> DECID B = +oo )
67		exmiddc 804	. . 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 770	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 103   \/ wo 680  DECID wdc 802   \/ w3o 944   = wceq 1314   e. wcel 1463   ifcif 3440   {cpr 3494   class class class wbr 3895   supcsup 6821   RRcr 7546   +oocpnf 7721   -oocmnf 7722   RR*cxr 7723   < clt 7724   <_ cle 7725

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-cnex 7636  ax-resscn 7637  ax-pre-ltirr 7657  ax-pre-lttrn 7659  ax-pre-apti 7660

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rmo 2398  df-rab 2399  df-v 2659  df-sbc 2879  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-xp 4505  df-cnv 4507  df-iota 5046  df-riota 5684  df-sup 6823  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730

This theorem is referenced by:  xrmaxiflemlub  10909