Theorem xltadd1 9803

Description: Extended real version of ltadd1 8318. (Contributed by Mario Carneiro, 23-Aug-2015.) (Revised by Jim Kingdon, 16-Apr-2023.)

Assertion

Ref	Expression
xltadd1	|- ( ( A e. RR* /\ B e. RR* /\ C e. RR ) -> ( A < B <-> ( A +e C ) < ( B +e C ) ) )

Proof of Theorem xltadd1

Step	Hyp	Ref	Expression
1		simplr 520	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B e. RR ) -> A e. RR )
2		simpr 109	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B e. RR ) -> B e. RR )
3		simpll3 1027	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B e. RR ) -> C e. RR )
4		ltadd1 8318	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> ( A + C ) < ( B + C ) ) )
5		simp1 986	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
6		simp3 988	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
7	5, 6	rexaddd 9781	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A +e C ) = ( A + C ) )
8		simp2 987	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
9	8, 6	rexaddd 9781	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B +e C ) = ( B + C ) )
10	7, 9	breq12d 3989	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A +e C ) < ( B +e C ) <-> ( A + C ) < ( B + C ) ) )
11	4, 10	bitr4d 190	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> ( A +e C ) < ( B +e C ) ) )
12	1, 2, 3, 11	syl3anc 1227	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B e. RR ) -> ( A < B <-> ( A +e C ) < ( B +e C ) ) )
13		ltpnf 9707	. . . . . 6 |- ( A e. RR -> A < +oo )
14	13	ad2antlr 481	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = +oo ) -> A < +oo )
15		breq2 3980	. . . . . 6 |- ( B = +oo -> ( A < B <-> A < +oo ) )
16	15	adantl 275	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = +oo ) -> ( A < B <-> A < +oo ) )
17	14, 16	mpbird 166	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = +oo ) -> A < B )
18		simplr 520	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = +oo ) -> A e. RR )
19		simpll3 1027	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = +oo ) -> C e. RR )
20		rexadd 9779	. . . . . . . 8 |- ( ( A e. RR /\ C e. RR ) -> ( A +e C ) = ( A + C ) )
21		readdcl 7870	. . . . . . . 8 |- ( ( A e. RR /\ C e. RR ) -> ( A + C ) e. RR )
22	20, 21	eqeltrd 2241	. . . . . . 7 |- ( ( A e. RR /\ C e. RR ) -> ( A +e C ) e. RR )
23	18, 19, 22	syl2anc 409	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = +oo ) -> ( A +e C ) e. RR )
24		ltpnf 9707	. . . . . 6 |- ( ( A +e C ) e. RR -> ( A +e C ) < +oo )
25	23, 24	syl 14	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = +oo ) -> ( A +e C ) < +oo )
26		oveq1 5843	. . . . . . 7 |- ( B = +oo -> ( B +e C ) = ( +oo +e C ) )
27	26	adantl 275	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = +oo ) -> ( B +e C ) = ( +oo +e C ) )
28		rexr 7935	. . . . . . . 8 |- ( C e. RR -> C e. RR* )
29		renemnf 7938	. . . . . . . 8 |- ( C e. RR -> C =/= -oo )
30		xaddpnf2 9774	. . . . . . . 8 |- ( ( C e. RR* /\ C =/= -oo ) -> ( +oo +e C ) = +oo )
31	28, 29, 30	syl2anc 409	. . . . . . 7 |- ( C e. RR -> ( +oo +e C ) = +oo )
32	19, 31	syl 14	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = +oo ) -> ( +oo +e C ) = +oo )
33	27, 32	eqtrd 2197	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = +oo ) -> ( B +e C ) = +oo )
34	25, 33	breqtrrd 4004	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = +oo ) -> ( A +e C ) < ( B +e C ) )
35	17, 34	2thd 174	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = +oo ) -> ( A < B <-> ( A +e C ) < ( B +e C ) ) )
36		mnfle 9719	. . . . . . . 8 |- ( A e. RR* -> -oo <_ A )
37	36	3ad2ant1 1007	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR ) -> -oo <_ A )
38	37	ad2antrr 480	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = -oo ) -> -oo <_ A )
39		mnfxr 7946	. . . . . . 7 |- -oo e. RR*
40		simpll1 1025	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = -oo ) -> A e. RR* )
41		xrlenlt 7954	. . . . . . 7 |- ( ( -oo e. RR* /\ A e. RR* ) -> ( -oo <_ A <-> -. A < -oo ) )
42	39, 40, 41	sylancr 411	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = -oo ) -> ( -oo <_ A <-> -. A < -oo ) )
43	38, 42	mpbid 146	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = -oo ) -> -. A < -oo )
44		breq2 3980	. . . . . 6 |- ( B = -oo -> ( A < B <-> A < -oo ) )
45	44	adantl 275	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = -oo ) -> ( A < B <-> A < -oo ) )
46	43, 45	mtbird 663	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = -oo ) -> -. A < B )
47	28	3ad2ant3 1009	. . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR ) -> C e. RR* )
48	47	ad2antrr 480	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = -oo ) -> C e. RR* )
49		xaddcl 9787	. . . . . . . 8 |- ( ( A e. RR* /\ C e. RR* ) -> ( A +e C ) e. RR* )
50	40, 48, 49	syl2anc 409	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = -oo ) -> ( A +e C ) e. RR* )
51		mnfle 9719	. . . . . . 7 |- ( ( A +e C ) e. RR* -> -oo <_ ( A +e C ) )
52	50, 51	syl 14	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = -oo ) -> -oo <_ ( A +e C ) )
53		xrlenlt 7954	. . . . . . 7 |- ( ( -oo e. RR* /\ ( A +e C ) e. RR* ) -> ( -oo <_ ( A +e C ) <-> -. ( A +e C ) < -oo ) )
54	39, 50, 53	sylancr 411	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = -oo ) -> ( -oo <_ ( A +e C ) <-> -. ( A +e C ) < -oo ) )
55	52, 54	mpbid 146	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = -oo ) -> -. ( A +e C ) < -oo )
56		simpr 109	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = -oo ) -> B = -oo )
57	56	oveq1d 5851	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = -oo ) -> ( B +e C ) = ( -oo +e C ) )
58		renepnf 7937	. . . . . . . . . 10 |- ( C e. RR -> C =/= +oo )
59	58	3ad2ant3 1009	. . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR ) -> C =/= +oo )
60	59	ad2antrr 480	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = -oo ) -> C =/= +oo )
61		xaddmnf2 9776	. . . . . . . 8 |- ( ( C e. RR* /\ C =/= +oo ) -> ( -oo +e C ) = -oo )
62	48, 60, 61	syl2anc 409	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = -oo ) -> ( -oo +e C ) = -oo )
63	57, 62	eqtrd 2197	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = -oo ) -> ( B +e C ) = -oo )
64	63	breq2d 3988	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = -oo ) -> ( ( A +e C ) < ( B +e C ) <-> ( A +e C ) < -oo ) )
65	55, 64	mtbird 663	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = -oo ) -> -. ( A +e C ) < ( B +e C ) )
66	46, 65	2falsed 692	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) /\ B = -oo ) -> ( A < B <-> ( A +e C ) < ( B +e C ) ) )
67		elxr 9703	. . . . . 6 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
68	67	biimpi 119	. . . . 5 |- ( B e. RR* -> ( B e. RR \/ B = +oo \/ B = -oo ) )
69	68	3ad2ant2 1008	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR ) -> ( B e. RR \/ B = +oo \/ B = -oo ) )
70	69	adantr 274	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) -> ( B e. RR \/ B = +oo \/ B = -oo ) )
71	12, 35, 66, 70	mpjao3dan 1296	. 2 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A e. RR ) -> ( A < B <-> ( A +e C ) < ( B +e C ) ) )
72		simpl2 990	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = +oo ) -> B e. RR* )
73		pnfge 9716	. . . . . 6 |- ( B e. RR* -> B <_ +oo )
74	72, 73	syl 14	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = +oo ) -> B <_ +oo )
75		pnfxr 7942	. . . . . . 7 |- +oo e. RR*
76	75	a1i 9	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = +oo ) -> +oo e. RR* )
77		xrlenlt 7954	. . . . . 6 |- ( ( B e. RR* /\ +oo e. RR* ) -> ( B <_ +oo <-> -. +oo < B ) )
78	72, 76, 77	syl2anc 409	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = +oo ) -> ( B <_ +oo <-> -. +oo < B ) )
79	74, 78	mpbid 146	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = +oo ) -> -. +oo < B )
80		simpr 109	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = +oo ) -> A = +oo )
81	80	breq1d 3986	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = +oo ) -> ( A < B <-> +oo < B ) )
82	79, 81	mtbird 663	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = +oo ) -> -. A < B )
83	47	adantr 274	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = +oo ) -> C e. RR* )
84		xaddcl 9787	. . . . . . . 8 |- ( ( B e. RR* /\ C e. RR* ) -> ( B +e C ) e. RR* )
85	72, 83, 84	syl2anc 409	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = +oo ) -> ( B +e C ) e. RR* )
86		pnfge 9716	. . . . . . 7 |- ( ( B +e C ) e. RR* -> ( B +e C ) <_ +oo )
87	85, 86	syl 14	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = +oo ) -> ( B +e C ) <_ +oo )
88	29	3ad2ant3 1009	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR ) -> C =/= -oo )
89	88	adantr 274	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = +oo ) -> C =/= -oo )
90	83, 89, 30	syl2anc 409	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = +oo ) -> ( +oo +e C ) = +oo )
91	87, 90	breqtrrd 4004	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = +oo ) -> ( B +e C ) <_ ( +oo +e C ) )
92		xaddcl 9787	. . . . . . 7 |- ( ( +oo e. RR* /\ C e. RR* ) -> ( +oo +e C ) e. RR* )
93	75, 83, 92	sylancr 411	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = +oo ) -> ( +oo +e C ) e. RR* )
94		xrlenlt 7954	. . . . . 6 |- ( ( ( B +e C ) e. RR* /\ ( +oo +e C ) e. RR* ) -> ( ( B +e C ) <_ ( +oo +e C ) <-> -. ( +oo +e C ) < ( B +e C ) ) )
95	85, 93, 94	syl2anc 409	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = +oo ) -> ( ( B +e C ) <_ ( +oo +e C ) <-> -. ( +oo +e C ) < ( B +e C ) ) )
96	91, 95	mpbid 146	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = +oo ) -> -. ( +oo +e C ) < ( B +e C ) )
97	80	oveq1d 5851	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = +oo ) -> ( A +e C ) = ( +oo +e C ) )
98	97	breq1d 3986	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = +oo ) -> ( ( A +e C ) < ( B +e C ) <-> ( +oo +e C ) < ( B +e C ) ) )
99	96, 98	mtbird 663	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = +oo ) -> -. ( A +e C ) < ( B +e C ) )
100	82, 99	2falsed 692	. 2 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = +oo ) -> ( A < B <-> ( A +e C ) < ( B +e C ) ) )
101		simplr 520	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B e. RR ) -> A = -oo )
102		mnflt 9710	. . . . . 6 |- ( B e. RR -> -oo < B )
103	102	adantl 275	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B e. RR ) -> -oo < B )
104	101, 103	eqbrtrd 3998	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B e. RR ) -> A < B )
105	101	oveq1d 5851	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B e. RR ) -> ( A +e C ) = ( -oo +e C ) )
106		simpll3 1027	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B e. RR ) -> C e. RR )
107	106, 28	syl 14	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B e. RR ) -> C e. RR* )
108	106, 58	syl 14	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B e. RR ) -> C =/= +oo )
109	107, 108, 61	syl2anc 409	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B e. RR ) -> ( -oo +e C ) = -oo )
110	105, 109	eqtrd 2197	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B e. RR ) -> ( A +e C ) = -oo )
111		simpr 109	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B e. RR ) -> B e. RR )
112		rexadd 9779	. . . . . . . 8 |- ( ( B e. RR /\ C e. RR ) -> ( B +e C ) = ( B + C ) )
113		readdcl 7870	. . . . . . . 8 |- ( ( B e. RR /\ C e. RR ) -> ( B + C ) e. RR )
114	112, 113	eqeltrd 2241	. . . . . . 7 |- ( ( B e. RR /\ C e. RR ) -> ( B +e C ) e. RR )
115	111, 106, 114	syl2anc 409	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B e. RR ) -> ( B +e C ) e. RR )
116		mnflt 9710	. . . . . 6 |- ( ( B +e C ) e. RR -> -oo < ( B +e C ) )
117	115, 116	syl 14	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B e. RR ) -> -oo < ( B +e C ) )
118	110, 117	eqbrtrd 3998	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B e. RR ) -> ( A +e C ) < ( B +e C ) )
119	104, 118	2thd 174	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B e. RR ) -> ( A < B <-> ( A +e C ) < ( B +e C ) ) )
120		simplr 520	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B = +oo ) -> A = -oo )
121		simpr 109	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B = +oo ) -> B = +oo )
122	120, 121	breq12d 3989	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B = +oo ) -> ( A < B <-> -oo < +oo ) )
123		oveq1 5843	. . . . . . 7 |- ( A = -oo -> ( A +e C ) = ( -oo +e C ) )
124	47, 59, 61	syl2anc 409	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR ) -> ( -oo +e C ) = -oo )
125	123, 124	sylan9eqr 2219	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) -> ( A +e C ) = -oo )
126	125	adantr 274	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B = +oo ) -> ( A +e C ) = -oo )
127	26	adantl 275	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B = +oo ) -> ( B +e C ) = ( +oo +e C ) )
128	47, 88, 30	syl2anc 409	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR ) -> ( +oo +e C ) = +oo )
129	128	ad2antrr 480	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B = +oo ) -> ( +oo +e C ) = +oo )
130	127, 129	eqtrd 2197	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B = +oo ) -> ( B +e C ) = +oo )
131	126, 130	breq12d 3989	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B = +oo ) -> ( ( A +e C ) < ( B +e C ) <-> -oo < +oo ) )
132	122, 131	bitr4d 190	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B = +oo ) -> ( A < B <-> ( A +e C ) < ( B +e C ) ) )
133		simplr 520	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B = -oo ) -> A = -oo )
134		simpr 109	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B = -oo ) -> B = -oo )
135	133, 134	breq12d 3989	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B = -oo ) -> ( A < B <-> -oo < -oo ) )
136	124	ad2antrr 480	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B = -oo ) -> ( -oo +e C ) = -oo )
137	123	eqeq1d 2173	. . . . . . 7 |- ( A = -oo -> ( ( A +e C ) = -oo <-> ( -oo +e C ) = -oo ) )
138	137	ad2antlr 481	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B = -oo ) -> ( ( A +e C ) = -oo <-> ( -oo +e C ) = -oo ) )
139	136, 138	mpbird 166	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B = -oo ) -> ( A +e C ) = -oo )
140	134	oveq1d 5851	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B = -oo ) -> ( B +e C ) = ( -oo +e C ) )
141	140, 136	eqtrd 2197	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B = -oo ) -> ( B +e C ) = -oo )
142	139, 141	breq12d 3989	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B = -oo ) -> ( ( A +e C ) < ( B +e C ) <-> -oo < -oo ) )
143	135, 142	bitr4d 190	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) /\ B = -oo ) -> ( A < B <-> ( A +e C ) < ( B +e C ) ) )
144	69	adantr 274	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) -> ( B e. RR \/ B = +oo \/ B = -oo ) )
145	119, 132, 143, 144	mpjao3dan 1296	. 2 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR ) /\ A = -oo ) -> ( A < B <-> ( A +e C ) < ( B +e C ) ) )
146		elxr 9703	. . . 4 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
147	146	biimpi 119	. . 3 |- ( A e. RR* -> ( A e. RR \/ A = +oo \/ A = -oo ) )
148	147	3ad2ant1 1007	. 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR ) -> ( A e. RR \/ A = +oo \/ A = -oo ) )
149	71, 100, 145, 148	mpjao3dan 1296	1 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR ) -> ( A < B <-> ( A +e C ) < ( B +e C ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ w3o 966   /\ w3a 967   = wceq 1342   e. wcel 2135   =/= wne 2334   class class class wbr 3976  (class class class)co 5836   RRcr 7743   + caddc 7747   +oocpnf 7921   -oocmnf 7922   RR*cxr 7923   < clt 7924   <_ cle 7925   +ecxad 9697

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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-cnex 7835  ax-resscn 7836  ax-1cn 7837  ax-1re 7838  ax-icn 7839  ax-addcl 7840  ax-addrcl 7841  ax-mulcl 7842  ax-addcom 7844  ax-addass 7846  ax-i2m1 7849  ax-0id 7852  ax-rnegex 7853  ax-pre-ltadd 7860

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-if 3516  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-fv 5190  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-xadd 9700

This theorem is referenced by:  xltadd2  9804  xlt2add  9807  xrmaxaddlem  11187