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Theorem xnegdi 13252

Description: Extended real version of negdi 11489. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
xnegdi	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))

**Proof of Theorem xnegdi**
Step	Hyp	Ref	Expression
1		elxr 13119	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 13119	. . . 4 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3		recn 11164	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
4		recn 11164	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ)
5		negdi 11489	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 + 𝐵) = (-𝐴 + -𝐵))
6	3, 4, 5	syl2an 605	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -(𝐴 + 𝐵) = (-𝐴 + -𝐵))
7		readdcl 11157	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
8		rexneg 13215	. . . . . . . 8 ⊢ ((𝐴 + 𝐵) ∈ ℝ → -_𝑒(𝐴 + 𝐵) = -(𝐴 + 𝐵))
9	7, 8	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -_𝑒(𝐴 + 𝐵) = -(𝐴 + 𝐵))
10		renegcl 11495	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
11		renegcl 11495	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → -𝐵 ∈ ℝ)
12		rexadd 13236	. . . . . . . 8 ⊢ ((-𝐴 ∈ ℝ ∧ -𝐵 ∈ ℝ) → (-𝐴 +_𝑒 -𝐵) = (-𝐴 + -𝐵))
13	10, 11, 12	syl2an 605	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐴 +_𝑒 -𝐵) = (-𝐴 + -𝐵))
14	6, 9, 13	3eqtr4d 2808	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -_𝑒(𝐴 + 𝐵) = (-𝐴 +_𝑒 -𝐵))
15		rexadd 13236	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +_𝑒 𝐵) = (𝐴 + 𝐵))
16		xnegeq 13211	. . . . . . 7 ⊢ ((𝐴 +_𝑒 𝐵) = (𝐴 + 𝐵) → -_𝑒(𝐴 +_𝑒 𝐵) = -_𝑒(𝐴 + 𝐵))
17	15, 16	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -_𝑒(𝐴 +_𝑒 𝐵) = -_𝑒(𝐴 + 𝐵))
18		rexneg 13215	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 = -𝐴)
19		rexneg 13215	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → -_𝑒𝐵 = -𝐵)
20	18, 19	oveqan12d 7416	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-_𝑒𝐴 +_𝑒 -_𝑒𝐵) = (-𝐴 +_𝑒 -𝐵))
21	14, 17, 20	3eqtr4d 2808	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
22		xnegpnf 13213	. . . . . 6 ⊢ -_𝑒+∞ = -∞
23		oveq2 7405	. . . . . . . 8 ⊢ (𝐵 = +∞ → (𝐴 +_𝑒 𝐵) = (𝐴 +_𝑒 +∞))
24		rexr 11229	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
25		renemnf 11232	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
26		xaddpnf1 13230	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → (𝐴 +_𝑒 +∞) = +∞)
27	24, 25, 26	syl2anc 593	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 +∞) = +∞)
28	23, 27	sylan9eqr 2820	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 +_𝑒 𝐵) = +∞)
29		xnegeq 13211	. . . . . . 7 ⊢ ((𝐴 +_𝑒 𝐵) = +∞ → -_𝑒(𝐴 +_𝑒 𝐵) = -_𝑒+∞)
30	28, 29	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → -_𝑒(𝐴 +_𝑒 𝐵) = -_𝑒+∞)
31		xnegeq 13211	. . . . . . . . 9 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -_𝑒+∞)
32	31, 22	eqtrdi 2814	. . . . . . . 8 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -∞)
33	32	oveq2d 7413	. . . . . . 7 ⊢ (𝐵 = +∞ → (-_𝑒𝐴 +_𝑒 -_𝑒𝐵) = (-_𝑒𝐴 +_𝑒 -∞))
34	18, 10	eqeltrd 2863	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 ∈ ℝ)
35		rexr 11229	. . . . . . . . 9 ⊢ (-_𝑒𝐴 ∈ ℝ → -_𝑒𝐴 ∈ ℝ^*)
36		renepnf 11231	. . . . . . . . 9 ⊢ (-_𝑒𝐴 ∈ ℝ → -_𝑒𝐴 ≠ +∞)
37		xaddmnf1 13232	. . . . . . . . 9 ⊢ ((-_𝑒𝐴 ∈ ℝ^* ∧ -_𝑒𝐴 ≠ +∞) → (-_𝑒𝐴 +_𝑒 -∞) = -∞)
38	35, 36, 37	syl2anc 593	. . . . . . . 8 ⊢ (-_𝑒𝐴 ∈ ℝ → (-_𝑒𝐴 +_𝑒 -∞) = -∞)
39	34, 38	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (-_𝑒𝐴 +_𝑒 -∞) = -∞)
40	33, 39	sylan9eqr 2820	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (-_𝑒𝐴 +_𝑒 -_𝑒𝐵) = -∞)
41	22, 30, 40	3eqtr4a 2824	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
42		xnegmnf 13214	. . . . . 6 ⊢ -_𝑒-∞ = +∞
43		oveq2 7405	. . . . . . . 8 ⊢ (𝐵 = -∞ → (𝐴 +_𝑒 𝐵) = (𝐴 +_𝑒 -∞))
44		renepnf 11231	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
45		xaddmnf1 13232	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) → (𝐴 +_𝑒 -∞) = -∞)
46	24, 44, 45	syl2anc 593	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 -∞) = -∞)
47	43, 46	sylan9eqr 2820	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 +_𝑒 𝐵) = -∞)
48		xnegeq 13211	. . . . . . 7 ⊢ ((𝐴 +_𝑒 𝐵) = -∞ → -_𝑒(𝐴 +_𝑒 𝐵) = -_𝑒-∞)
49	47, 48	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → -_𝑒(𝐴 +_𝑒 𝐵) = -_𝑒-∞)
50		xnegeq 13211	. . . . . . . . 9 ⊢ (𝐵 = -∞ → -_𝑒𝐵 = -_𝑒-∞)
51	50, 42	eqtrdi 2814	. . . . . . . 8 ⊢ (𝐵 = -∞ → -_𝑒𝐵 = +∞)
52	51	oveq2d 7413	. . . . . . 7 ⊢ (𝐵 = -∞ → (-_𝑒𝐴 +_𝑒 -_𝑒𝐵) = (-_𝑒𝐴 +_𝑒 +∞))
53		renemnf 11232	. . . . . . . . 9 ⊢ (-_𝑒𝐴 ∈ ℝ → -_𝑒𝐴 ≠ -∞)
54		xaddpnf1 13230	. . . . . . . . 9 ⊢ ((-_𝑒𝐴 ∈ ℝ^* ∧ -_𝑒𝐴 ≠ -∞) → (-_𝑒𝐴 +_𝑒 +∞) = +∞)
55	35, 53, 54	syl2anc 593	. . . . . . . 8 ⊢ (-_𝑒𝐴 ∈ ℝ → (-_𝑒𝐴 +_𝑒 +∞) = +∞)
56	34, 55	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (-_𝑒𝐴 +_𝑒 +∞) = +∞)
57	52, 56	sylan9eqr 2820	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (-_𝑒𝐴 +_𝑒 -_𝑒𝐵) = +∞)
58	42, 49, 57	3eqtr4a 2824	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
59	21, 41, 58	3jaodan 1452	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
60	2, 59	sylan2b 603	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
61		xneg0 13216	. . . . . . 7 ⊢ -_𝑒0 = 0
62		simpr 488	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → 𝐵 = -∞)
63	62	oveq2d 7413	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (+∞ +_𝑒 𝐵) = (+∞ +_𝑒 -∞))
64		pnfaddmnf 13234	. . . . . . . . 9 ⊢ (+∞ +_𝑒 -∞) = 0
65	63, 64	eqtrdi 2814	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (+∞ +_𝑒 𝐵) = 0)
66		xnegeq 13211	. . . . . . . 8 ⊢ ((+∞ +_𝑒 𝐵) = 0 → -_𝑒(+∞ +_𝑒 𝐵) = -_𝑒0)
67	65, 66	syl 17	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → -_𝑒(+∞ +_𝑒 𝐵) = -_𝑒0)
68	51	adantl 485	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → -_𝑒𝐵 = +∞)
69	68	oveq2d 7413	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (-∞ +_𝑒 -_𝑒𝐵) = (-∞ +_𝑒 +∞))
70		mnfaddpnf 13235	. . . . . . . 8 ⊢ (-∞ +_𝑒 +∞) = 0
71	69, 70	eqtrdi 2814	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (-∞ +_𝑒 -_𝑒𝐵) = 0)
72	61, 67, 71	3eqtr4a 2824	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → -_𝑒(+∞ +_𝑒 𝐵) = (-∞ +_𝑒 -_𝑒𝐵))
73		xaddpnf2 13231	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (+∞ +_𝑒 𝐵) = +∞)
74		xnegeq 13211	. . . . . . . 8 ⊢ ((+∞ +_𝑒 𝐵) = +∞ → -_𝑒(+∞ +_𝑒 𝐵) = -_𝑒+∞)
75	73, 74	syl 17	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → -_𝑒(+∞ +_𝑒 𝐵) = -_𝑒+∞)
76		xnegcl 13217	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ^* → -_𝑒𝐵 ∈ ℝ^*)
77		xnegeq 13211	. . . . . . . . . . . 12 ⊢ (-_𝑒𝐵 = +∞ → -_𝑒-_𝑒𝐵 = -_𝑒+∞)
78	77, 22	eqtrdi 2814	. . . . . . . . . . 11 ⊢ (-_𝑒𝐵 = +∞ → -_𝑒-_𝑒𝐵 = -∞)
79		xnegneg 13218	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℝ^* → -_𝑒-_𝑒𝐵 = 𝐵)
80	79	eqeq1d 2765	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ^* → (-_𝑒-_𝑒𝐵 = -∞ ↔ 𝐵 = -∞))
81	78, 80	imbitrid 246	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ^* → (-_𝑒𝐵 = +∞ → 𝐵 = -∞))
82	81	necon3d 2979	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ^* → (𝐵 ≠ -∞ → -_𝑒𝐵 ≠ +∞))
83	82	imp 410	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → -_𝑒𝐵 ≠ +∞)
84		xaddmnf2 13233	. . . . . . . 8 ⊢ ((-_𝑒𝐵 ∈ ℝ^* ∧ -_𝑒𝐵 ≠ +∞) → (-∞ +_𝑒 -_𝑒𝐵) = -∞)
85	76, 83, 84	syl2an2r 695	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (-∞ +_𝑒 -_𝑒𝐵) = -∞)
86	22, 75, 85	3eqtr4a 2824	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → -_𝑒(+∞ +_𝑒 𝐵) = (-∞ +_𝑒 -_𝑒𝐵))
87	72, 86	pm2.61dane 3045	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → -_𝑒(+∞ +_𝑒 𝐵) = (-∞ +_𝑒 -_𝑒𝐵))
88	87	adantl 485	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒(+∞ +_𝑒 𝐵) = (-∞ +_𝑒 -_𝑒𝐵))
89		simpl 486	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → 𝐴 = +∞)
90	89	oveq1d 7412	. . . . 5 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (+∞ +_𝑒 𝐵))
91		xnegeq 13211	. . . . 5 ⊢ ((𝐴 +_𝑒 𝐵) = (+∞ +_𝑒 𝐵) → -_𝑒(𝐴 +_𝑒 𝐵) = -_𝑒(+∞ +_𝑒 𝐵))
92	90, 91	syl 17	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = -_𝑒(+∞ +_𝑒 𝐵))
93		xnegeq 13211	. . . . . . 7 ⊢ (𝐴 = +∞ → -_𝑒𝐴 = -_𝑒+∞)
94	93	adantr 484	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒𝐴 = -_𝑒+∞)
95	94, 22	eqtrdi 2814	. . . . 5 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒𝐴 = -∞)
96	95	oveq1d 7412	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 +_𝑒 -_𝑒𝐵) = (-∞ +_𝑒 -_𝑒𝐵))
97	88, 92, 96	3eqtr4d 2808	. . 3 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
98		simpr 488	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → 𝐵 = +∞)
99	98	oveq2d 7413	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (-∞ +_𝑒 𝐵) = (-∞ +_𝑒 +∞))
100	99, 70	eqtrdi 2814	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (-∞ +_𝑒 𝐵) = 0)
101		xnegeq 13211	. . . . . . . 8 ⊢ ((-∞ +_𝑒 𝐵) = 0 → -_𝑒(-∞ +_𝑒 𝐵) = -_𝑒0)
102	100, 101	syl 17	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → -_𝑒(-∞ +_𝑒 𝐵) = -_𝑒0)
103	32	adantl 485	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → -_𝑒𝐵 = -∞)
104	103	oveq2d 7413	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (+∞ +_𝑒 -_𝑒𝐵) = (+∞ +_𝑒 -∞))
105	104, 64	eqtrdi 2814	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (+∞ +_𝑒 -_𝑒𝐵) = 0)
106	61, 102, 105	3eqtr4a 2824	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → -_𝑒(-∞ +_𝑒 𝐵) = (+∞ +_𝑒 -_𝑒𝐵))
107		xaddmnf2 13233	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (-∞ +_𝑒 𝐵) = -∞)
108		xnegeq 13211	. . . . . . . 8 ⊢ ((-∞ +_𝑒 𝐵) = -∞ → -_𝑒(-∞ +_𝑒 𝐵) = -_𝑒-∞)
109	107, 108	syl 17	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → -_𝑒(-∞ +_𝑒 𝐵) = -_𝑒-∞)
110		xnegeq 13211	. . . . . . . . . . . 12 ⊢ (-_𝑒𝐵 = -∞ → -_𝑒-_𝑒𝐵 = -_𝑒-∞)
111	110, 42	eqtrdi 2814	. . . . . . . . . . 11 ⊢ (-_𝑒𝐵 = -∞ → -_𝑒-_𝑒𝐵 = +∞)
112	79	eqeq1d 2765	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ^* → (-_𝑒-_𝑒𝐵 = +∞ ↔ 𝐵 = +∞))
113	111, 112	imbitrid 246	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ^* → (-_𝑒𝐵 = -∞ → 𝐵 = +∞))
114	113	necon3d 2979	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ^* → (𝐵 ≠ +∞ → -_𝑒𝐵 ≠ -∞))
115	114	imp 410	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → -_𝑒𝐵 ≠ -∞)
116		xaddpnf2 13231	. . . . . . . 8 ⊢ ((-_𝑒𝐵 ∈ ℝ^* ∧ -_𝑒𝐵 ≠ -∞) → (+∞ +_𝑒 -_𝑒𝐵) = +∞)
117	76, 115, 116	syl2an2r 695	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (+∞ +_𝑒 -_𝑒𝐵) = +∞)
118	42, 109, 117	3eqtr4a 2824	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → -_𝑒(-∞ +_𝑒 𝐵) = (+∞ +_𝑒 -_𝑒𝐵))
119	106, 118	pm2.61dane 3045	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → -_𝑒(-∞ +_𝑒 𝐵) = (+∞ +_𝑒 -_𝑒𝐵))
120	119	adantl 485	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒(-∞ +_𝑒 𝐵) = (+∞ +_𝑒 -_𝑒𝐵))
121		simpl 486	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → 𝐴 = -∞)
122	121	oveq1d 7412	. . . . 5 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (-∞ +_𝑒 𝐵))
123		xnegeq 13211	. . . . 5 ⊢ ((𝐴 +_𝑒 𝐵) = (-∞ +_𝑒 𝐵) → -_𝑒(𝐴 +_𝑒 𝐵) = -_𝑒(-∞ +_𝑒 𝐵))
124	122, 123	syl 17	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = -_𝑒(-∞ +_𝑒 𝐵))
125		xnegeq 13211	. . . . . . 7 ⊢ (𝐴 = -∞ → -_𝑒𝐴 = -_𝑒-∞)
126	125	adantr 484	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒𝐴 = -_𝑒-∞)
127	126, 42	eqtrdi 2814	. . . . 5 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒𝐴 = +∞)
128	127	oveq1d 7412	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 +_𝑒 -_𝑒𝐵) = (+∞ +_𝑒 -_𝑒𝐵))
129	120, 124, 128	3eqtr4d 2808	. . 3 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
130	60, 97, 129	3jaoian 1451	. 2 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
131	1, 130	sylanb 590	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∨ w3o 1098 = wceq 1561 ∈ wcel 2143 ≠ wne 2958 (class class class)co 7397 ℂcc 11072 ℝcr 11073 0cc0 11074 + caddc 11077 +∞cpnf 11214 -∞cmnf 11215 ℝ^*cxr 11216 -cneg 11416 -_𝑒cxne 13112 +_𝑒 cxad 13113

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1816 ax-4 1830 ax-5 1931 ax-6 1988 ax-7 2029 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5247 ax-nul 5257 ax-pow 5323 ax-pr 5391 ax-un 7719 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1564 df-fal 1574 df-ex 1801 df-nf 1805 df-sb 2092 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3369 df-rab 3416 df-v 3457 df-sbc 3746 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-br 5102 df-opab 5164 df-mpt 5183 df-id 5543 df-po 5556 df-so 5557 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-iota 6478 df-fun 6524 df-fn 6525 df-f 6526 df-f1 6527 df-fo 6528 df-f1o 6529 df-fv 6530 df-riota 7354 df-ov 7400 df-oprab 7401 df-mpo 7402 df-er 8679 df-en 8929 df-dom 8930 df-sdom 8931 df-pnf 11219 df-mnf 11220 df-xr 11221 df-ltxr 11222 df-sub 11417 df-neg 11418 df-xneg 13115 df-xadd 13116

This theorem is referenced by: xaddass2 13254 xposdif 13266 xadddi 13299 xrsxmet 24871

W3C validator