Theorem xrmaxadd 11304

Description: Distributing addition over maximum. (Contributed by Jim Kingdon, 11-May-2023.)

Assertion

Ref	Expression
xrmaxadd	|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e sup ( { B , C } , RR* , < ) ) )

Proof of Theorem xrmaxadd

Step	Hyp	Ref	Expression
1		simpr 110	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A e. RR ) -> A e. RR )
2		simpl2 1003	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A e. RR ) -> B e. RR* )
3		simpl3 1004	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A e. RR ) -> C e. RR* )
4		xrmaxaddlem 11303	. . 3 |- ( ( A e. RR /\ B e. RR* /\ C e. RR* ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e sup ( { B , C } , RR* , < ) ) )
5	1, 2, 3, 4	syl3anc 1249	. 2 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A e. RR ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e sup ( { B , C } , RR* , < ) ) )
6		simpllr 534	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> A = +oo )
7		simpr 110	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> C = -oo )
8	6, 7	oveq12d 5915	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> ( A +e C ) = ( +oo +e -oo ) )
9		simp1 999	. . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> A e. RR* )
10	9	ad3antrrr 492	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> A e. RR* )
11		simp2 1000	. . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> B e. RR* )
12	11	ad3antrrr 492	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> B e. RR* )
13	10, 12	xaddcld 9916	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> ( A +e B ) e. RR* )
14		simp3 1001	. . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> C e. RR* )
15	14	ad3antrrr 492	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> C e. RR* )
16	10, 15	xaddcld 9916	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> ( A +e C ) e. RR* )
17	13, 16	jca 306	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> ( ( A +e B ) e. RR* /\ ( A +e C ) e. RR* ) )
18		simplr 528	. . . . . . . . . . 11 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) -> A = +oo )
19		simpr 110	. . . . . . . . . . 11 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) -> B = -oo )
20	18, 19	oveq12d 5915	. . . . . . . . . 10 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) -> ( A +e B ) = ( +oo +e -oo ) )
21		pnfaddmnf 9882	. . . . . . . . . 10 |- ( +oo +e -oo ) = 0
22	20, 21	eqtrdi 2238	. . . . . . . . 9 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) -> ( A +e B ) = 0 )
23	22	adantr 276	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> ( A +e B ) = 0 )
24	8, 21	eqtrdi 2238	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> ( A +e C ) = 0 )
25	23, 24	eqtr4d 2225	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> ( A +e B ) = ( A +e C ) )
26	16	xrleidd 9833	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> ( A +e C ) <_ ( A +e C ) )
27	25, 26	eqbrtrd 4040	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> ( A +e B ) <_ ( A +e C ) )
28		xrmaxleim 11287	. . . . . 6 |- ( ( ( A +e B ) e. RR* /\ ( A +e C ) e. RR* ) -> ( ( A +e B ) <_ ( A +e C ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e C ) ) )
29	17, 27, 28	sylc 62	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e C ) )
30	12, 15	jca 306	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> ( B e. RR* /\ C e. RR* ) )
31		simplr 528	. . . . . . . . . 10 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> B = -oo )
32	31, 7	eqtr4d 2225	. . . . . . . . 9 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> B = C )
33	15	xrleidd 9833	. . . . . . . . 9 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> C <_ C )
34	32, 33	eqbrtrd 4040	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> B <_ C )
35		xrmaxleim 11287	. . . . . . . 8 |- ( ( B e. RR* /\ C e. RR* ) -> ( B <_ C -> sup ( { B , C } , RR* , < ) = C ) )
36	30, 34, 35	sylc 62	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> sup ( { B , C } , RR* , < ) = C )
37	36, 7	eqtrd 2222	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> sup ( { B , C } , RR* , < ) = -oo )
38	6, 37	oveq12d 5915	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> ( A +e sup ( { B , C } , RR* , < ) ) = ( +oo +e -oo ) )
39	8, 29, 38	3eqtr4d 2232	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e sup ( { B , C } , RR* , < ) ) )
40		simpllr 534	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> A = +oo )
41	40	oveq1d 5912	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> ( A +e C ) = ( +oo +e C ) )
42	14	ad3antrrr 492	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> C e. RR* )
43		xaddpnf2 9879	. . . . . . 7 |- ( ( C e. RR* /\ C =/= -oo ) -> ( +oo +e C ) = +oo )
44	42, 43	sylancom 420	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> ( +oo +e C ) = +oo )
45	41, 44	eqtrd 2222	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> ( A +e C ) = +oo )
46	9, 11	xaddcld 9916	. . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A +e B ) e. RR* )
47	46	ad3antrrr 492	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> ( A +e B ) e. RR* )
48		pnfge 9821	. . . . . . . 8 |- ( ( A +e B ) e. RR* -> ( A +e B ) <_ +oo )
49	47, 48	syl 14	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> ( A +e B ) <_ +oo )
50	49, 45	breqtrrd 4046	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> ( A +e B ) <_ ( A +e C ) )
51	9, 14	xaddcld 9916	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A +e C ) e. RR* )
52	51	ad3antrrr 492	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> ( A +e C ) e. RR* )
53	47, 52, 28	syl2anc 411	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> ( ( A +e B ) <_ ( A +e C ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e C ) ) )
54	50, 53	mpd 13	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e C ) )
55	40	oveq1d 5912	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> ( A +e sup ( { B , C } , RR* , < ) ) = ( +oo +e sup ( { B , C } , RR* , < ) ) )
56	11	ad3antrrr 492	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> B e. RR* )
57		xrmaxcl 11295	. . . . . . . 8 |- ( ( B e. RR* /\ C e. RR* ) -> sup ( { B , C } , RR* , < ) e. RR* )
58	56, 42, 57	syl2anc 411	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> sup ( { B , C } , RR* , < ) e. RR* )
59		simpr 110	. . . . . . . . . . 11 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> C =/= -oo )
60		nmnfgt 9850	. . . . . . . . . . . 12 |- ( C e. RR* -> ( -oo < C <-> C =/= -oo ) )
61	42, 60	syl 14	. . . . . . . . . . 11 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> ( -oo < C <-> C =/= -oo ) )
62	59, 61	mpbird 167	. . . . . . . . . 10 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> -oo < C )
63	62	olcd 735	. . . . . . . . 9 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> ( -oo < B \/ -oo < C ) )
64		mnfxr 8045	. . . . . . . . . . 11 |- -oo e. RR*
65	64	a1i 9	. . . . . . . . . 10 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> -oo e. RR* )
66		xrltmaxsup 11300	. . . . . . . . . 10 |- ( ( B e. RR* /\ C e. RR* /\ -oo e. RR* ) -> ( -oo < sup ( { B , C } , RR* , < ) <-> ( -oo < B \/ -oo < C ) ) )
67	56, 42, 65, 66	syl3anc 1249	. . . . . . . . 9 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> ( -oo < sup ( { B , C } , RR* , < ) <-> ( -oo < B \/ -oo < C ) ) )
68	63, 67	mpbird 167	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> -oo < sup ( { B , C } , RR* , < ) )
69		nmnfgt 9850	. . . . . . . . 9 |- ( sup ( { B , C } , RR* , < ) e. RR* -> ( -oo < sup ( { B , C } , RR* , < ) <-> sup ( { B , C } , RR* , < ) =/= -oo ) )
70	58, 69	syl 14	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> ( -oo < sup ( { B , C } , RR* , < ) <-> sup ( { B , C } , RR* , < ) =/= -oo ) )
71	68, 70	mpbid 147	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> sup ( { B , C } , RR* , < ) =/= -oo )
72		xaddpnf2 9879	. . . . . . 7 |- ( ( sup ( { B , C } , RR* , < ) e. RR* /\ sup ( { B , C } , RR* , < ) =/= -oo ) -> ( +oo +e sup ( { B , C } , RR* , < ) ) = +oo )
73	58, 71, 72	syl2anc 411	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> ( +oo +e sup ( { B , C } , RR* , < ) ) = +oo )
74	55, 73	eqtrd 2222	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> ( A +e sup ( { B , C } , RR* , < ) ) = +oo )
75	45, 54, 74	3eqtr4d 2232	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e sup ( { B , C } , RR* , < ) ) )
76		xrmnfdc 9875	. . . . . . 7 |- ( C e. RR* -> DECID C = -oo )
77	76	3ad2ant3 1022	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> DECID C = -oo )
78	77	ad2antrr 488	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) -> DECID C = -oo )
79		dcne 2371	. . . . 5 |- (DECID C = -oo <-> ( C = -oo \/ C =/= -oo ) )
80	78, 79	sylib 122	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) -> ( C = -oo \/ C =/= -oo ) )
81	39, 75, 80	mpjaodan 799	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e sup ( { B , C } , RR* , < ) ) )
82	11	ad2antrr 488	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> B e. RR* )
83	14	ad2antrr 488	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> C e. RR* )
84	82, 83, 57	syl2anc 411	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> sup ( { B , C } , RR* , < ) e. RR* )
85		simpr 110	. . . . . . . . 9 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> B =/= -oo )
86		nmnfgt 9850	. . . . . . . . . 10 |- ( B e. RR* -> ( -oo < B <-> B =/= -oo ) )
87	82, 86	syl 14	. . . . . . . . 9 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> ( -oo < B <-> B =/= -oo ) )
88	85, 87	mpbird 167	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> -oo < B )
89	88	orcd 734	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> ( -oo < B \/ -oo < C ) )
90	64	a1i 9	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> -oo e. RR* )
91	82, 83, 90, 66	syl3anc 1249	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> ( -oo < sup ( { B , C } , RR* , < ) <-> ( -oo < B \/ -oo < C ) ) )
92	89, 91	mpbird 167	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> -oo < sup ( { B , C } , RR* , < ) )
93	84, 69	syl 14	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> ( -oo < sup ( { B , C } , RR* , < ) <-> sup ( { B , C } , RR* , < ) =/= -oo ) )
94	92, 93	mpbid 147	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> sup ( { B , C } , RR* , < ) =/= -oo )
95	84, 94, 72	syl2anc 411	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> ( +oo +e sup ( { B , C } , RR* , < ) ) = +oo )
96		simplr 528	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> A = +oo )
97	96	oveq1d 5912	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> ( A +e sup ( { B , C } , RR* , < ) ) = ( +oo +e sup ( { B , C } , RR* , < ) ) )
98		prcom 3683	. . . . . 6 |- { ( A +e B ) , ( A +e C ) } = { ( A +e C ) , ( A +e B ) }
99	98	supeq1i 7018	. . . . 5 |- sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = sup ( { ( A +e C ) , ( A +e B ) } , RR* , < )
100	51	ad2antrr 488	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> ( A +e C ) e. RR* )
101	46	ad2antrr 488	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> ( A +e B ) e. RR* )
102	100, 101	jca 306	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> ( ( A +e C ) e. RR* /\ ( A +e B ) e. RR* ) )
103		pnfge 9821	. . . . . . . . 9 |- ( ( A +e C ) e. RR* -> ( A +e C ) <_ +oo )
104	100, 103	syl 14	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> ( A +e C ) <_ +oo )
105	96	oveq1d 5912	. . . . . . . . 9 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> ( A +e B ) = ( +oo +e B ) )
106		xaddpnf2 9879	. . . . . . . . . 10 |- ( ( B e. RR* /\ B =/= -oo ) -> ( +oo +e B ) = +oo )
107	82, 106	sylancom 420	. . . . . . . . 9 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> ( +oo +e B ) = +oo )
108	105, 107	eqtrd 2222	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> ( A +e B ) = +oo )
109	104, 108	breqtrrd 4046	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> ( A +e C ) <_ ( A +e B ) )
110		xrmaxleim 11287	. . . . . . 7 |- ( ( ( A +e C ) e. RR* /\ ( A +e B ) e. RR* ) -> ( ( A +e C ) <_ ( A +e B ) -> sup ( { ( A +e C ) , ( A +e B ) } , RR* , < ) = ( A +e B ) ) )
111	102, 109, 110	sylc 62	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> sup ( { ( A +e C ) , ( A +e B ) } , RR* , < ) = ( A +e B ) )
112	111, 108	eqtrd 2222	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> sup ( { ( A +e C ) , ( A +e B ) } , RR* , < ) = +oo )
113	99, 112	eqtrid 2234	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = +oo )
114	95, 97, 113	3eqtr4rd 2233	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e sup ( { B , C } , RR* , < ) ) )
115		xrmnfdc 9875	. . . . . 6 |- ( B e. RR* -> DECID B = -oo )
116		dcne 2371	. . . . . 6 |- (DECID B = -oo <-> ( B = -oo \/ B =/= -oo ) )
117	115, 116	sylib 122	. . . . 5 |- ( B e. RR* -> ( B = -oo \/ B =/= -oo ) )
118	117	3ad2ant2 1021	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( B = -oo \/ B =/= -oo ) )
119	118	adantr 276	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) -> ( B = -oo \/ B =/= -oo ) )
120	81, 114, 119	mpjaodan 799	. 2 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e sup ( { B , C } , RR* , < ) ) )
121		simpllr 534	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C = +oo ) -> A = -oo )
122		simpr 110	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C = +oo ) -> C = +oo )
123	121, 122	oveq12d 5915	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C = +oo ) -> ( A +e C ) = ( -oo +e +oo ) )
124		mnfaddpnf 9883	. . . . . 6 |- ( -oo +e +oo ) = 0
125	123, 124	eqtrdi 2238	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C = +oo ) -> ( A +e C ) = 0 )
126	46	ad3antrrr 492	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C = +oo ) -> ( A +e B ) e. RR* )
127	51	ad3antrrr 492	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C = +oo ) -> ( A +e C ) e. RR* )
128	126, 127	jca 306	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C = +oo ) -> ( ( A +e B ) e. RR* /\ ( A +e C ) e. RR* ) )
129		0le0 9039	. . . . . . . 8 |- 0 <_ 0
130	129	a1i 9	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C = +oo ) -> 0 <_ 0 )
131		simplr 528	. . . . . . . . . 10 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> A = -oo )
132		simpr 110	. . . . . . . . . 10 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> B = +oo )
133	131, 132	oveq12d 5915	. . . . . . . . 9 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> ( A +e B ) = ( -oo +e +oo ) )
134	133, 124	eqtrdi 2238	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> ( A +e B ) = 0 )
135	134	adantr 276	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C = +oo ) -> ( A +e B ) = 0 )
136	130, 135, 125	3brtr4d 4050	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C = +oo ) -> ( A +e B ) <_ ( A +e C ) )
137	128, 136, 28	sylc 62	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C = +oo ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e C ) )
138		prcom 3683	. . . . . . . . . . 11 |- { C , B } = { B , C }
139	138	supeq1i 7018	. . . . . . . . . 10 |- sup ( { C , B } , RR* , < ) = sup ( { B , C } , RR* , < )
140	14	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> C e. RR* )
141	11	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> B e. RR* )
142	140, 141	jca 306	. . . . . . . . . . 11 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> ( C e. RR* /\ B e. RR* ) )
143		pnfge 9821	. . . . . . . . . . . . . 14 |- ( C e. RR* -> C <_ +oo )
144	143	3ad2ant3 1022	. . . . . . . . . . . . 13 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> C <_ +oo )
145	144	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> C <_ +oo )
146	145, 132	breqtrrd 4046	. . . . . . . . . . 11 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> C <_ B )
147		xrmaxleim 11287	. . . . . . . . . . 11 |- ( ( C e. RR* /\ B e. RR* ) -> ( C <_ B -> sup ( { C , B } , RR* , < ) = B ) )
148	142, 146, 147	sylc 62	. . . . . . . . . 10 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> sup ( { C , B } , RR* , < ) = B )
149	139, 148	eqtr3id 2236	. . . . . . . . 9 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> sup ( { B , C } , RR* , < ) = B )
150	149, 132	eqtrd 2222	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> sup ( { B , C } , RR* , < ) = +oo )
151	150	oveq2d 5913	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> ( A +e sup ( { B , C } , RR* , < ) ) = ( A +e +oo ) )
152	131	oveq1d 5912	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> ( A +e +oo ) = ( -oo +e +oo ) )
153	152, 124	eqtrdi 2238	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> ( A +e +oo ) = 0 )
154	151, 153	eqtrd 2222	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> ( A +e sup ( { B , C } , RR* , < ) ) = 0 )
155	154	adantr 276	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C = +oo ) -> ( A +e sup ( { B , C } , RR* , < ) ) = 0 )
156	125, 137, 155	3eqtr4d 2232	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C = +oo ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e sup ( { B , C } , RR* , < ) ) )
157	51	ad3antrrr 492	. . . . . . . . 9 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C =/= +oo ) -> ( A +e C ) e. RR* )
158	46	ad3antrrr 492	. . . . . . . . 9 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C =/= +oo ) -> ( A +e B ) e. RR* )
159	157, 158	jca 306	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C =/= +oo ) -> ( ( A +e C ) e. RR* /\ ( A +e B ) e. RR* ) )
160		0xr 8035	. . . . . . . . . 10 |- 0 e. RR*
161		mnfle 9824	. . . . . . . . . 10 |- ( 0 e. RR* -> -oo <_ 0 )
162	160, 161	mp1i 10	. . . . . . . . 9 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C =/= +oo ) -> -oo <_ 0 )
163		simpllr 534	. . . . . . . . . . 11 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C =/= +oo ) -> A = -oo )
164	163	oveq1d 5912	. . . . . . . . . 10 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C =/= +oo ) -> ( A +e C ) = ( -oo +e C ) )
165		xaddmnf2 9881	. . . . . . . . . . 11 |- ( ( C e. RR* /\ C =/= +oo ) -> ( -oo +e C ) = -oo )
166	140, 165	sylan 283	. . . . . . . . . 10 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C =/= +oo ) -> ( -oo +e C ) = -oo )
167	164, 166	eqtrd 2222	. . . . . . . . 9 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C =/= +oo ) -> ( A +e C ) = -oo )
168	134	adantr 276	. . . . . . . . 9 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C =/= +oo ) -> ( A +e B ) = 0 )
169	162, 167, 168	3brtr4d 4050	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C =/= +oo ) -> ( A +e C ) <_ ( A +e B ) )
170	159, 169, 110	sylc 62	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C =/= +oo ) -> sup ( { ( A +e C ) , ( A +e B ) } , RR* , < ) = ( A +e B ) )
171	170, 168	eqtrd 2222	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C =/= +oo ) -> sup ( { ( A +e C ) , ( A +e B ) } , RR* , < ) = 0 )
172	99, 171	eqtrid 2234	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C =/= +oo ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = 0 )
173	154	adantr 276	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C =/= +oo ) -> ( A +e sup ( { B , C } , RR* , < ) ) = 0 )
174	172, 173	eqtr4d 2225	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C =/= +oo ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e sup ( { B , C } , RR* , < ) ) )
175		xrpnfdc 9874	. . . . . . 7 |- ( C e. RR* -> DECID C = +oo )
176		dcne 2371	. . . . . . 7 |- (DECID C = +oo <-> ( C = +oo \/ C =/= +oo ) )
177	175, 176	sylib 122	. . . . . 6 |- ( C e. RR* -> ( C = +oo \/ C =/= +oo ) )
178	177	3ad2ant3 1022	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( C = +oo \/ C =/= +oo ) )
179	178	ad2antrr 488	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> ( C = +oo \/ C =/= +oo ) )
180	156, 174, 179	mpjaodan 799	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e sup ( { B , C } , RR* , < ) ) )
181		simpllr 534	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C = +oo ) -> A = -oo )
182		simpr 110	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C = +oo ) -> C = +oo )
183	181, 182	oveq12d 5915	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C = +oo ) -> ( A +e C ) = ( -oo +e +oo ) )
184	46	ad2antrr 488	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) -> ( A +e B ) e. RR* )
185	51	ad2antrr 488	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) -> ( A +e C ) e. RR* )
186	184, 185	jca 306	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) -> ( ( A +e B ) e. RR* /\ ( A +e C ) e. RR* ) )
187		simplr 528	. . . . . . . . . 10 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) -> A = -oo )
188	187	oveq1d 5912	. . . . . . . . 9 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) -> ( A +e B ) = ( -oo +e B ) )
189	11	ad2antrr 488	. . . . . . . . . 10 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) -> B e. RR* )
190		xaddmnf2 9881	. . . . . . . . . 10 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = -oo )
191	189, 190	sylancom 420	. . . . . . . . 9 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) -> ( -oo +e B ) = -oo )
192	188, 191	eqtrd 2222	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) -> ( A +e B ) = -oo )
193		mnfle 9824	. . . . . . . . 9 |- ( ( A +e C ) e. RR* -> -oo <_ ( A +e C ) )
194	185, 193	syl 14	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) -> -oo <_ ( A +e C ) )
195	192, 194	eqbrtrd 4040	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) -> ( A +e B ) <_ ( A +e C ) )
196	186, 195, 28	sylc 62	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e C ) )
197	196	adantr 276	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C = +oo ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e C ) )
198	189	adantr 276	. . . . . . . . 9 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C = +oo ) -> B e. RR* )
199	14	ad3antrrr 492	. . . . . . . . 9 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C = +oo ) -> C e. RR* )
200	198, 199	jca 306	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C = +oo ) -> ( B e. RR* /\ C e. RR* ) )
201		simpr 110	. . . . . . . . . . . 12 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) -> B =/= +oo )
202		npnflt 9847	. . . . . . . . . . . . 13 |- ( B e. RR* -> ( B < +oo <-> B =/= +oo ) )
203	189, 202	syl 14	. . . . . . . . . . . 12 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) -> ( B < +oo <-> B =/= +oo ) )
204	201, 203	mpbird 167	. . . . . . . . . . 11 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) -> B < +oo )
205	204	adantr 276	. . . . . . . . . 10 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C = +oo ) -> B < +oo )
206	205, 182	breqtrrd 4046	. . . . . . . . 9 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C = +oo ) -> B < C )
207	198, 199, 206	xrltled 9831	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C = +oo ) -> B <_ C )
208	200, 207, 35	sylc 62	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C = +oo ) -> sup ( { B , C } , RR* , < ) = C )
209	208, 182	eqtrd 2222	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C = +oo ) -> sup ( { B , C } , RR* , < ) = +oo )
210	181, 209	oveq12d 5915	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C = +oo ) -> ( A +e sup ( { B , C } , RR* , < ) ) = ( -oo +e +oo ) )
211	183, 197, 210	3eqtr4d 2232	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C = +oo ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e sup ( { B , C } , RR* , < ) ) )
212	189	adantr 276	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> B e. RR* )
213	14	ad3antrrr 492	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> C e. RR* )
214	212, 213, 57	syl2anc 411	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> sup ( { B , C } , RR* , < ) e. RR* )
215	204	adantr 276	. . . . . . . . 9 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> B < +oo )
216		simpr 110	. . . . . . . . . 10 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> C =/= +oo )
217		npnflt 9847	. . . . . . . . . . 11 |- ( C e. RR* -> ( C < +oo <-> C =/= +oo ) )
218	213, 217	syl 14	. . . . . . . . . 10 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> ( C < +oo <-> C =/= +oo ) )
219	216, 218	mpbird 167	. . . . . . . . 9 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> C < +oo )
220	215, 219	jca 306	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> ( B < +oo /\ C < +oo ) )
221		pnfxr 8041	. . . . . . . . . 10 |- +oo e. RR*
222	221	a1i 9	. . . . . . . . 9 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> +oo e. RR* )
223		xrmaxltsup 11301	. . . . . . . . 9 |- ( ( B e. RR* /\ C e. RR* /\ +oo e. RR* ) -> ( sup ( { B , C } , RR* , < ) < +oo <-> ( B < +oo /\ C < +oo ) ) )
224	212, 213, 222, 223	syl3anc 1249	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> ( sup ( { B , C } , RR* , < ) < +oo <-> ( B < +oo /\ C < +oo ) ) )
225	220, 224	mpbird 167	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> sup ( { B , C } , RR* , < ) < +oo )
226		npnflt 9847	. . . . . . . 8 |- ( sup ( { B , C } , RR* , < ) e. RR* -> ( sup ( { B , C } , RR* , < ) < +oo <-> sup ( { B , C } , RR* , < ) =/= +oo ) )
227	214, 226	syl 14	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> ( sup ( { B , C } , RR* , < ) < +oo <-> sup ( { B , C } , RR* , < ) =/= +oo ) )
228	225, 227	mpbid 147	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> sup ( { B , C } , RR* , < ) =/= +oo )
229		xaddmnf2 9881	. . . . . 6 |- ( ( sup ( { B , C } , RR* , < ) e. RR* /\ sup ( { B , C } , RR* , < ) =/= +oo ) -> ( -oo +e sup ( { B , C } , RR* , < ) ) = -oo )
230	214, 228, 229	syl2anc 411	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> ( -oo +e sup ( { B , C } , RR* , < ) ) = -oo )
231		simpllr 534	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> A = -oo )
232	231	oveq1d 5912	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> ( A +e sup ( { B , C } , RR* , < ) ) = ( -oo +e sup ( { B , C } , RR* , < ) ) )
233	196	adantr 276	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e C ) )
234	231	oveq1d 5912	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> ( A +e C ) = ( -oo +e C ) )
235	233, 234	eqtrd 2222	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( -oo +e C ) )
236	213, 165	sylancom 420	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> ( -oo +e C ) = -oo )
237	235, 236	eqtrd 2222	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = -oo )
238	230, 232, 237	3eqtr4rd 2233	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e sup ( { B , C } , RR* , < ) ) )
239	178	ad2antrr 488	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) -> ( C = +oo \/ C =/= +oo ) )
240	211, 238, 239	mpjaodan 799	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e sup ( { B , C } , RR* , < ) ) )
241		xrpnfdc 9874	. . . . . 6 |- ( B e. RR* -> DECID B = +oo )
242	241	3ad2ant2 1021	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> DECID B = +oo )
243		dcne 2371	. . . . 5 |- (DECID B = +oo <-> ( B = +oo \/ B =/= +oo ) )
244	242, 243	sylib 122	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( B = +oo \/ B =/= +oo ) )
245	244	adantr 276	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) -> ( B = +oo \/ B =/= +oo ) )
246	180, 240, 245	mpjaodan 799	. 2 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e sup ( { B , C } , RR* , < ) ) )
247		elxr 9808	. . . 4 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
248	247	biimpi 120	. . 3 |- ( A e. RR* -> ( A e. RR \/ A = +oo \/ A = -oo ) )
249	248	3ad2ant1 1020	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A e. RR \/ A = +oo \/ A = -oo ) )
250	5, 120, 246, 249	mpjao3dan 1318	1 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e sup ( { B , C } , RR* , < ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  DECID wdc 835   \/ w3o 979   /\ w3a 980   = wceq 1364   e. wcel 2160   =/= wne 2360   {cpr 3608   class class class wbr 4018  (class class class)co 5897   supcsup 7012   RRcr 7841   0cc0 7842   +oocpnf 8020   -oocmnf 8021   RR*cxr 8022   < clt 8023   <_ cle 8024   +ecxad 9802

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 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959  ax-pre-mulext 7960  ax-arch 7961  ax-caucvg 7962

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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-frec 6417  df-sup 7014  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570  df-div 8661  df-inn 8951  df-2 9009  df-3 9010  df-4 9011  df-n0 9208  df-z 9285  df-uz 9560  df-rp 9686  df-xneg 9804  df-xadd 9805  df-seqfrec 10479  df-exp 10554  df-cj 10886  df-re 10887  df-im 10888  df-rsqrt 11042  df-abs 11043

This theorem is referenced by:  xrminadd  11318