Theorem xrmaxadd 11224

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 109	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A e. RR ) -> A e. RR )
2		simpl2 996	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A e. RR ) -> B e. RR* )
3		simpl3 997	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A e. RR ) -> C e. RR* )
4		xrmaxaddlem 11223	. . 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 1233	. 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 529	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> A = +oo )
7		simpr 109	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> C = -oo )
8	6, 7	oveq12d 5871	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> ( A +e C ) = ( +oo +e -oo ) )
9		simp1 992	. . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> A e. RR* )
10	9	ad3antrrr 489	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> A e. RR* )
11		simp2 993	. . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> B e. RR* )
12	11	ad3antrrr 489	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> B e. RR* )
13	10, 12	xaddcld 9841	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> ( A +e B ) e. RR* )
14		simp3 994	. . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> C e. RR* )
15	14	ad3antrrr 489	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> C e. RR* )
16	10, 15	xaddcld 9841	. . . . . . 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 304	. . . . . 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 525	. . . . . . . . . . 11 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) -> A = +oo )
19		simpr 109	. . . . . . . . . . 11 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) -> B = -oo )
20	18, 19	oveq12d 5871	. . . . . . . . . 10 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) -> ( A +e B ) = ( +oo +e -oo ) )
21		pnfaddmnf 9807	. . . . . . . . . 10 |- ( +oo +e -oo ) = 0
22	20, 21	eqtrdi 2219	. . . . . . . . 9 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) -> ( A +e B ) = 0 )
23	22	adantr 274	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> ( A +e B ) = 0 )
24	8, 21	eqtrdi 2219	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> ( A +e C ) = 0 )
25	23, 24	eqtr4d 2206	. . . . . . 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 9758	. . . . . . 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 4011	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> ( A +e B ) <_ ( A +e C ) )
28		xrmaxleim 11207	. . . . . 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 304	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> ( B e. RR* /\ C e. RR* ) )
31		simplr 525	. . . . . . . . . 10 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> B = -oo )
32	31, 7	eqtr4d 2206	. . . . . . . . 9 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> B = C )
33	15	xrleidd 9758	. . . . . . . . 9 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> C <_ C )
34	32, 33	eqbrtrd 4011	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C = -oo ) -> B <_ C )
35		xrmaxleim 11207	. . . . . . . 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 2203	. . . . . 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 5871	. . . . 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 2213	. . . 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 529	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> A = +oo )
41	40	oveq1d 5868	. . . . . 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 489	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> C e. RR* )
43		xaddpnf2 9804	. . . . . . 7 |- ( ( C e. RR* /\ C =/= -oo ) -> ( +oo +e C ) = +oo )
44	42, 43	sylancom 418	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> ( +oo +e C ) = +oo )
45	41, 44	eqtrd 2203	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> ( A +e C ) = +oo )
46	9, 11	xaddcld 9841	. . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A +e B ) e. RR* )
47	46	ad3antrrr 489	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> ( A +e B ) e. RR* )
48		pnfge 9746	. . . . . . . 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 4017	. . . . . 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 9841	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A +e C ) e. RR* )
52	51	ad3antrrr 489	. . . . . . 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 409	. . . . . 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 5868	. . . . . 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 489	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> B e. RR* )
57		xrmaxcl 11215	. . . . . . . 8 |- ( ( B e. RR* /\ C e. RR* ) -> sup ( { B , C } , RR* , < ) e. RR* )
58	56, 42, 57	syl2anc 409	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> sup ( { B , C } , RR* , < ) e. RR* )
59		simpr 109	. . . . . . . . . . 11 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> C =/= -oo )
60		nmnfgt 9775	. . . . . . . . . . . 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 166	. . . . . . . . . 10 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> -oo < C )
63	62	olcd 729	. . . . . . . . 9 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> ( -oo < B \/ -oo < C ) )
64		mnfxr 7976	. . . . . . . . . . 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 11220	. . . . . . . . . 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 1233	. . . . . . . . 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 166	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> -oo < sup ( { B , C } , RR* , < ) )
69		nmnfgt 9775	. . . . . . . . 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 146	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) /\ C =/= -oo ) -> sup ( { B , C } , RR* , < ) =/= -oo )
72		xaddpnf2 9804	. . . . . . 7 |- ( ( sup ( { B , C } , RR* , < ) e. RR* /\ sup ( { B , C } , RR* , < ) =/= -oo ) -> ( +oo +e sup ( { B , C } , RR* , < ) ) = +oo )
73	58, 71, 72	syl2anc 409	. . . . . 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 2203	. . . . 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 2213	. . . 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 9800	. . . . . . 7 |- ( C e. RR* -> DECID C = -oo )
77	76	3ad2ant3 1015	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> DECID C = -oo )
78	77	ad2antrr 485	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) -> DECID C = -oo )
79		dcne 2351	. . . . 5 |- (DECID C = -oo <-> ( C = -oo \/ C =/= -oo ) )
80	78, 79	sylib 121	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B = -oo ) -> ( C = -oo \/ C =/= -oo ) )
81	39, 75, 80	mpjaodan 793	. . 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 485	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> B e. RR* )
83	14	ad2antrr 485	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> C e. RR* )
84	82, 83, 57	syl2anc 409	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> sup ( { B , C } , RR* , < ) e. RR* )
85		simpr 109	. . . . . . . . 9 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> B =/= -oo )
86		nmnfgt 9775	. . . . . . . . . 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 166	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> -oo < B )
89	88	orcd 728	. . . . . . 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 1233	. . . . . . 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 166	. . . . . 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 146	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> sup ( { B , C } , RR* , < ) =/= -oo )
95	84, 94, 72	syl2anc 409	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> ( +oo +e sup ( { B , C } , RR* , < ) ) = +oo )
96		simplr 525	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> A = +oo )
97	96	oveq1d 5868	. . . 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 3659	. . . . . 6 |- { ( A +e B ) , ( A +e C ) } = { ( A +e C ) , ( A +e B ) }
99	98	supeq1i 6965	. . . . 5 |- sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = sup ( { ( A +e C ) , ( A +e B ) } , RR* , < )
100	51	ad2antrr 485	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> ( A +e C ) e. RR* )
101	46	ad2antrr 485	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> ( A +e B ) e. RR* )
102	100, 101	jca 304	. . . . . . 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 9746	. . . . . . . . 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 5868	. . . . . . . . 9 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> ( A +e B ) = ( +oo +e B ) )
106		xaddpnf2 9804	. . . . . . . . . 10 |- ( ( B e. RR* /\ B =/= -oo ) -> ( +oo +e B ) = +oo )
107	82, 106	sylancom 418	. . . . . . . . 9 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> ( +oo +e B ) = +oo )
108	105, 107	eqtrd 2203	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> ( A +e B ) = +oo )
109	104, 108	breqtrrd 4017	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) /\ B =/= -oo ) -> ( A +e C ) <_ ( A +e B ) )
110		xrmaxleim 11207	. . . . . . 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 2203	. . . . 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 2215	. . . 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 2214	. . 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 9800	. . . . . 6 |- ( B e. RR* -> DECID B = -oo )
116		dcne 2351	. . . . . 6 |- (DECID B = -oo <-> ( B = -oo \/ B =/= -oo ) )
117	115, 116	sylib 121	. . . . 5 |- ( B e. RR* -> ( B = -oo \/ B =/= -oo ) )
118	117	3ad2ant2 1014	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( B = -oo \/ B =/= -oo ) )
119	118	adantr 274	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = +oo ) -> ( B = -oo \/ B =/= -oo ) )
120	81, 114, 119	mpjaodan 793	. 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 529	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C = +oo ) -> A = -oo )
122		simpr 109	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C = +oo ) -> C = +oo )
123	121, 122	oveq12d 5871	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C = +oo ) -> ( A +e C ) = ( -oo +e +oo ) )
124		mnfaddpnf 9808	. . . . . 6 |- ( -oo +e +oo ) = 0
125	123, 124	eqtrdi 2219	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C = +oo ) -> ( A +e C ) = 0 )
126	46	ad3antrrr 489	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C = +oo ) -> ( A +e B ) e. RR* )
127	51	ad3antrrr 489	. . . . . . 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 304	. . . . . 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 8967	. . . . . . . 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 525	. . . . . . . . . 10 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> A = -oo )
132		simpr 109	. . . . . . . . . 10 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> B = +oo )
133	131, 132	oveq12d 5871	. . . . . . . . 9 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> ( A +e B ) = ( -oo +e +oo ) )
134	133, 124	eqtrdi 2219	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> ( A +e B ) = 0 )
135	134	adantr 274	. . . . . . 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 4021	. . . . . 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 3659	. . . . . . . . . . 11 |- { C , B } = { B , C }
139	138	supeq1i 6965	. . . . . . . . . 10 |- sup ( { C , B } , RR* , < ) = sup ( { B , C } , RR* , < )
140	14	ad2antrr 485	. . . . . . . . . . . 12 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> C e. RR* )
141	11	ad2antrr 485	. . . . . . . . . . . 12 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> B e. RR* )
142	140, 141	jca 304	. . . . . . . . . . 11 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> ( C e. RR* /\ B e. RR* ) )
143		pnfge 9746	. . . . . . . . . . . . . 14 |- ( C e. RR* -> C <_ +oo )
144	143	3ad2ant3 1015	. . . . . . . . . . . . 13 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> C <_ +oo )
145	144	ad2antrr 485	. . . . . . . . . . . 12 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> C <_ +oo )
146	145, 132	breqtrrd 4017	. . . . . . . . . . 11 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> C <_ B )
147		xrmaxleim 11207	. . . . . . . . . . 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 2217	. . . . . . . . 9 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> sup ( { B , C } , RR* , < ) = B )
150	149, 132	eqtrd 2203	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> sup ( { B , C } , RR* , < ) = +oo )
151	150	oveq2d 5869	. . . . . . 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 5868	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> ( A +e +oo ) = ( -oo +e +oo ) )
153	152, 124	eqtrdi 2219	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> ( A +e +oo ) = 0 )
154	151, 153	eqtrd 2203	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> ( A +e sup ( { B , C } , RR* , < ) ) = 0 )
155	154	adantr 274	. . . . 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 2213	. . . 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 489	. . . . . . . . 9 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C =/= +oo ) -> ( A +e C ) e. RR* )
158	46	ad3antrrr 489	. . . . . . . . 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 304	. . . . . . . 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 7966	. . . . . . . . . 10 |- 0 e. RR*
161		mnfle 9749	. . . . . . . . . 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 529	. . . . . . . . . . 11 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C =/= +oo ) -> A = -oo )
164	163	oveq1d 5868	. . . . . . . . . 10 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C =/= +oo ) -> ( A +e C ) = ( -oo +e C ) )
165		xaddmnf2 9806	. . . . . . . . . . 11 |- ( ( C e. RR* /\ C =/= +oo ) -> ( -oo +e C ) = -oo )
166	140, 165	sylan 281	. . . . . . . . . 10 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C =/= +oo ) -> ( -oo +e C ) = -oo )
167	164, 166	eqtrd 2203	. . . . . . . . 9 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) /\ C =/= +oo ) -> ( A +e C ) = -oo )
168	134	adantr 274	. . . . . . . . 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 4021	. . . . . . . 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 2203	. . . . . 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 2215	. . . . 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 274	. . . . 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 2206	. . . 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 9799	. . . . . . 7 |- ( C e. RR* -> DECID C = +oo )
176		dcne 2351	. . . . . . 7 |- (DECID C = +oo <-> ( C = +oo \/ C =/= +oo ) )
177	175, 176	sylib 121	. . . . . 6 |- ( C e. RR* -> ( C = +oo \/ C =/= +oo ) )
178	177	3ad2ant3 1015	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( C = +oo \/ C =/= +oo ) )
179	178	ad2antrr 485	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B = +oo ) -> ( C = +oo \/ C =/= +oo ) )
180	156, 174, 179	mpjaodan 793	. . 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 529	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C = +oo ) -> A = -oo )
182		simpr 109	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C = +oo ) -> C = +oo )
183	181, 182	oveq12d 5871	. . . . 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 485	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) -> ( A +e B ) e. RR* )
185	51	ad2antrr 485	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) -> ( A +e C ) e. RR* )
186	184, 185	jca 304	. . . . . . 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 525	. . . . . . . . . 10 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) -> A = -oo )
188	187	oveq1d 5868	. . . . . . . . 9 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) -> ( A +e B ) = ( -oo +e B ) )
189	11	ad2antrr 485	. . . . . . . . . 10 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) -> B e. RR* )
190		xaddmnf2 9806	. . . . . . . . . 10 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = -oo )
191	189, 190	sylancom 418	. . . . . . . . 9 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) -> ( -oo +e B ) = -oo )
192	188, 191	eqtrd 2203	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) -> ( A +e B ) = -oo )
193		mnfle 9749	. . . . . . . . 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 4011	. . . . . . 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 274	. . . . 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 274	. . . . . . . . 9 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C = +oo ) -> B e. RR* )
199	14	ad3antrrr 489	. . . . . . . . 9 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C = +oo ) -> C e. RR* )
200	198, 199	jca 304	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C = +oo ) -> ( B e. RR* /\ C e. RR* ) )
201		simpr 109	. . . . . . . . . . . 12 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) -> B =/= +oo )
202		npnflt 9772	. . . . . . . . . . . . 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 166	. . . . . . . . . . 11 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) -> B < +oo )
205	204	adantr 274	. . . . . . . . . 10 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C = +oo ) -> B < +oo )
206	205, 182	breqtrrd 4017	. . . . . . . . 9 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C = +oo ) -> B < C )
207	198, 199, 206	xrltled 9756	. . . . . . . 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 2203	. . . . . 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 5871	. . . . 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 2213	. . . 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 274	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> B e. RR* )
213	14	ad3antrrr 489	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> C e. RR* )
214	212, 213, 57	syl2anc 409	. . . . . 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 274	. . . . . . . . 9 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> B < +oo )
216		simpr 109	. . . . . . . . . 10 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> C =/= +oo )
217		npnflt 9772	. . . . . . . . . . 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 166	. . . . . . . . 9 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> C < +oo )
220	215, 219	jca 304	. . . . . . . 8 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> ( B < +oo /\ C < +oo ) )
221		pnfxr 7972	. . . . . . . . . 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 11221	. . . . . . . . 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 1233	. . . . . . . 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 166	. . . . . . 7 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> sup ( { B , C } , RR* , < ) < +oo )
226		npnflt 9772	. . . . . . . 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 146	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> sup ( { B , C } , RR* , < ) =/= +oo )
229		xaddmnf2 9806	. . . . . 6 |- ( ( sup ( { B , C } , RR* , < ) e. RR* /\ sup ( { B , C } , RR* , < ) =/= +oo ) -> ( -oo +e sup ( { B , C } , RR* , < ) ) = -oo )
230	214, 228, 229	syl2anc 409	. . . . 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 529	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> A = -oo )
232	231	oveq1d 5868	. . . . 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 274	. . . . . . 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 5868	. . . . . . 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 2203	. . . . . 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 418	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) /\ C =/= +oo ) -> ( -oo +e C ) = -oo )
237	235, 236	eqtrd 2203	. . . . 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 2214	. . . 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 485	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) /\ B =/= +oo ) -> ( C = +oo \/ C =/= +oo ) )
240	211, 238, 239	mpjaodan 793	. . 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 9799	. . . . . 6 |- ( B e. RR* -> DECID B = +oo )
242	241	3ad2ant2 1014	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> DECID B = +oo )
243		dcne 2351	. . . . 5 |- (DECID B = +oo <-> ( B = +oo \/ B =/= +oo ) )
244	242, 243	sylib 121	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( B = +oo \/ B =/= +oo ) )
245	244	adantr 274	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A = -oo ) -> ( B = +oo \/ B =/= +oo ) )
246	180, 240, 245	mpjaodan 793	. 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 9733	. . . 4 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
248	247	biimpi 119	. . 3 |- ( A e. RR* -> ( A e. RR \/ A = +oo \/ A = -oo ) )
249	248	3ad2ant1 1013	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A e. RR \/ A = +oo \/ A = -oo ) )
250	5, 120, 246, 249	mpjao3dan 1302	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 103   <-> wb 104   \/ wo 703  DECID wdc 829   \/ w3o 972   /\ w3a 973   = wceq 1348   e. wcel 2141   =/= wne 2340   {cpr 3584   class class class wbr 3989  (class class class)co 5853   supcsup 6959   RRcr 7773   0cc0 7774   +oocpnf 7951   -oocmnf 7952   RR*cxr 7953   < clt 7954   <_ cle 7955   +ecxad 9727

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-sup 6961  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-rp 9611  df-xneg 9729  df-xadd 9730  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963

This theorem is referenced by:  xrminadd  11238