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Theorem xnegdi 12911

Description: Extended real version of negdi 11208. (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 12781	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 12781	. . . 4 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3		recn 10892	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
4		recn 10892	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ)
5		negdi 11208	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 + 𝐵) = (-𝐴 + -𝐵))
6	3, 4, 5	syl2an 595	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -(𝐴 + 𝐵) = (-𝐴 + -𝐵))
7		readdcl 10885	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
8		rexneg 12874	. . . . . . . 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 11214	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
11		renegcl 11214	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → -𝐵 ∈ ℝ)
12		rexadd 12895	. . . . . . . 8 ⊢ ((-𝐴 ∈ ℝ ∧ -𝐵 ∈ ℝ) → (-𝐴 +_𝑒 -𝐵) = (-𝐴 + -𝐵))
13	10, 11, 12	syl2an 595	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐴 +_𝑒 -𝐵) = (-𝐴 + -𝐵))
14	6, 9, 13	3eqtr4d 2788	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -_𝑒(𝐴 + 𝐵) = (-𝐴 +_𝑒 -𝐵))
15		rexadd 12895	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +_𝑒 𝐵) = (𝐴 + 𝐵))
16		xnegeq 12870	. . . . . . 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 12874	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 = -𝐴)
19		rexneg 12874	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → -_𝑒𝐵 = -𝐵)
20	18, 19	oveqan12d 7274	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-_𝑒𝐴 +_𝑒 -_𝑒𝐵) = (-𝐴 +_𝑒 -𝐵))
21	14, 17, 20	3eqtr4d 2788	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
22		xnegpnf 12872	. . . . . 6 ⊢ -_𝑒+∞ = -∞
23		oveq2 7263	. . . . . . . 8 ⊢ (𝐵 = +∞ → (𝐴 +_𝑒 𝐵) = (𝐴 +_𝑒 +∞))
24		rexr 10952	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
25		renemnf 10955	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
26		xaddpnf1 12889	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → (𝐴 +_𝑒 +∞) = +∞)
27	24, 25, 26	syl2anc 583	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 +∞) = +∞)
28	23, 27	sylan9eqr 2801	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 +_𝑒 𝐵) = +∞)
29		xnegeq 12870	. . . . . . 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 12870	. . . . . . . . 9 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -_𝑒+∞)
32	31, 22	eqtrdi 2795	. . . . . . . 8 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -∞)
33	32	oveq2d 7271	. . . . . . 7 ⊢ (𝐵 = +∞ → (-_𝑒𝐴 +_𝑒 -_𝑒𝐵) = (-_𝑒𝐴 +_𝑒 -∞))
34	18, 10	eqeltrd 2839	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 ∈ ℝ)
35		rexr 10952	. . . . . . . . 9 ⊢ (-_𝑒𝐴 ∈ ℝ → -_𝑒𝐴 ∈ ℝ^*)
36		renepnf 10954	. . . . . . . . 9 ⊢ (-_𝑒𝐴 ∈ ℝ → -_𝑒𝐴 ≠ +∞)
37		xaddmnf1 12891	. . . . . . . . 9 ⊢ ((-_𝑒𝐴 ∈ ℝ^* ∧ -_𝑒𝐴 ≠ +∞) → (-_𝑒𝐴 +_𝑒 -∞) = -∞)
38	35, 36, 37	syl2anc 583	. . . . . . . 8 ⊢ (-_𝑒𝐴 ∈ ℝ → (-_𝑒𝐴 +_𝑒 -∞) = -∞)
39	34, 38	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (-_𝑒𝐴 +_𝑒 -∞) = -∞)
40	33, 39	sylan9eqr 2801	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (-_𝑒𝐴 +_𝑒 -_𝑒𝐵) = -∞)
41	22, 30, 40	3eqtr4a 2805	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
42		xnegmnf 12873	. . . . . 6 ⊢ -_𝑒-∞ = +∞
43		oveq2 7263	. . . . . . . 8 ⊢ (𝐵 = -∞ → (𝐴 +_𝑒 𝐵) = (𝐴 +_𝑒 -∞))
44		renepnf 10954	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
45		xaddmnf1 12891	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) → (𝐴 +_𝑒 -∞) = -∞)
46	24, 44, 45	syl2anc 583	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 -∞) = -∞)
47	43, 46	sylan9eqr 2801	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 +_𝑒 𝐵) = -∞)
48		xnegeq 12870	. . . . . . 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 12870	. . . . . . . . 9 ⊢ (𝐵 = -∞ → -_𝑒𝐵 = -_𝑒-∞)
51	50, 42	eqtrdi 2795	. . . . . . . 8 ⊢ (𝐵 = -∞ → -_𝑒𝐵 = +∞)
52	51	oveq2d 7271	. . . . . . 7 ⊢ (𝐵 = -∞ → (-_𝑒𝐴 +_𝑒 -_𝑒𝐵) = (-_𝑒𝐴 +_𝑒 +∞))
53		renemnf 10955	. . . . . . . . 9 ⊢ (-_𝑒𝐴 ∈ ℝ → -_𝑒𝐴 ≠ -∞)
54		xaddpnf1 12889	. . . . . . . . 9 ⊢ ((-_𝑒𝐴 ∈ ℝ^* ∧ -_𝑒𝐴 ≠ -∞) → (-_𝑒𝐴 +_𝑒 +∞) = +∞)
55	35, 53, 54	syl2anc 583	. . . . . . . 8 ⊢ (-_𝑒𝐴 ∈ ℝ → (-_𝑒𝐴 +_𝑒 +∞) = +∞)
56	34, 55	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (-_𝑒𝐴 +_𝑒 +∞) = +∞)
57	52, 56	sylan9eqr 2801	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (-_𝑒𝐴 +_𝑒 -_𝑒𝐵) = +∞)
58	42, 49, 57	3eqtr4a 2805	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
59	21, 41, 58	3jaodan 1428	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
60	2, 59	sylan2b 593	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
61		xneg0 12875	. . . . . . 7 ⊢ -_𝑒0 = 0
62		simpr 484	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → 𝐵 = -∞)
63	62	oveq2d 7271	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (+∞ +_𝑒 𝐵) = (+∞ +_𝑒 -∞))
64		pnfaddmnf 12893	. . . . . . . . 9 ⊢ (+∞ +_𝑒 -∞) = 0
65	63, 64	eqtrdi 2795	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (+∞ +_𝑒 𝐵) = 0)
66		xnegeq 12870	. . . . . . . 8 ⊢ ((+∞ +_𝑒 𝐵) = 0 → -_𝑒(+∞ +_𝑒 𝐵) = -_𝑒0)
67	65, 66	syl 17	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → -_𝑒(+∞ +_𝑒 𝐵) = -_𝑒0)
68	51	adantl 481	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → -_𝑒𝐵 = +∞)
69	68	oveq2d 7271	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (-∞ +_𝑒 -_𝑒𝐵) = (-∞ +_𝑒 +∞))
70		mnfaddpnf 12894	. . . . . . . 8 ⊢ (-∞ +_𝑒 +∞) = 0
71	69, 70	eqtrdi 2795	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (-∞ +_𝑒 -_𝑒𝐵) = 0)
72	61, 67, 71	3eqtr4a 2805	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → -_𝑒(+∞ +_𝑒 𝐵) = (-∞ +_𝑒 -_𝑒𝐵))
73		xaddpnf2 12890	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (+∞ +_𝑒 𝐵) = +∞)
74		xnegeq 12870	. . . . . . . 8 ⊢ ((+∞ +_𝑒 𝐵) = +∞ → -_𝑒(+∞ +_𝑒 𝐵) = -_𝑒+∞)
75	73, 74	syl 17	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → -_𝑒(+∞ +_𝑒 𝐵) = -_𝑒+∞)
76		xnegcl 12876	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ^* → -_𝑒𝐵 ∈ ℝ^*)
77		xnegeq 12870	. . . . . . . . . . . 12 ⊢ (-_𝑒𝐵 = +∞ → -_𝑒-_𝑒𝐵 = -_𝑒+∞)
78	77, 22	eqtrdi 2795	. . . . . . . . . . 11 ⊢ (-_𝑒𝐵 = +∞ → -_𝑒-_𝑒𝐵 = -∞)
79		xnegneg 12877	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℝ^* → -_𝑒-_𝑒𝐵 = 𝐵)
80	79	eqeq1d 2740	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ^* → (-_𝑒-_𝑒𝐵 = -∞ ↔ 𝐵 = -∞))
81	78, 80	syl5ib 243	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ^* → (-_𝑒𝐵 = +∞ → 𝐵 = -∞))
82	81	necon3d 2963	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ^* → (𝐵 ≠ -∞ → -_𝑒𝐵 ≠ +∞))
83	82	imp 406	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → -_𝑒𝐵 ≠ +∞)
84		xaddmnf2 12892	. . . . . . . 8 ⊢ ((-_𝑒𝐵 ∈ ℝ^* ∧ -_𝑒𝐵 ≠ +∞) → (-∞ +_𝑒 -_𝑒𝐵) = -∞)
85	76, 83, 84	syl2an2r 681	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (-∞ +_𝑒 -_𝑒𝐵) = -∞)
86	22, 75, 85	3eqtr4a 2805	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → -_𝑒(+∞ +_𝑒 𝐵) = (-∞ +_𝑒 -_𝑒𝐵))
87	72, 86	pm2.61dane 3031	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → -_𝑒(+∞ +_𝑒 𝐵) = (-∞ +_𝑒 -_𝑒𝐵))
88	87	adantl 481	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒(+∞ +_𝑒 𝐵) = (-∞ +_𝑒 -_𝑒𝐵))
89		simpl 482	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → 𝐴 = +∞)
90	89	oveq1d 7270	. . . . 5 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (+∞ +_𝑒 𝐵))
91		xnegeq 12870	. . . . 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 12870	. . . . . . 7 ⊢ (𝐴 = +∞ → -_𝑒𝐴 = -_𝑒+∞)
94	93	adantr 480	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒𝐴 = -_𝑒+∞)
95	94, 22	eqtrdi 2795	. . . . 5 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒𝐴 = -∞)
96	95	oveq1d 7270	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 +_𝑒 -_𝑒𝐵) = (-∞ +_𝑒 -_𝑒𝐵))
97	88, 92, 96	3eqtr4d 2788	. . 3 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
98		simpr 484	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → 𝐵 = +∞)
99	98	oveq2d 7271	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (-∞ +_𝑒 𝐵) = (-∞ +_𝑒 +∞))
100	99, 70	eqtrdi 2795	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (-∞ +_𝑒 𝐵) = 0)
101		xnegeq 12870	. . . . . . . 8 ⊢ ((-∞ +_𝑒 𝐵) = 0 → -_𝑒(-∞ +_𝑒 𝐵) = -_𝑒0)
102	100, 101	syl 17	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → -_𝑒(-∞ +_𝑒 𝐵) = -_𝑒0)
103	32	adantl 481	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → -_𝑒𝐵 = -∞)
104	103	oveq2d 7271	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (+∞ +_𝑒 -_𝑒𝐵) = (+∞ +_𝑒 -∞))
105	104, 64	eqtrdi 2795	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (+∞ +_𝑒 -_𝑒𝐵) = 0)
106	61, 102, 105	3eqtr4a 2805	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → -_𝑒(-∞ +_𝑒 𝐵) = (+∞ +_𝑒 -_𝑒𝐵))
107		xaddmnf2 12892	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (-∞ +_𝑒 𝐵) = -∞)
108		xnegeq 12870	. . . . . . . 8 ⊢ ((-∞ +_𝑒 𝐵) = -∞ → -_𝑒(-∞ +_𝑒 𝐵) = -_𝑒-∞)
109	107, 108	syl 17	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → -_𝑒(-∞ +_𝑒 𝐵) = -_𝑒-∞)
110		xnegeq 12870	. . . . . . . . . . . 12 ⊢ (-_𝑒𝐵 = -∞ → -_𝑒-_𝑒𝐵 = -_𝑒-∞)
111	110, 42	eqtrdi 2795	. . . . . . . . . . 11 ⊢ (-_𝑒𝐵 = -∞ → -_𝑒-_𝑒𝐵 = +∞)
112	79	eqeq1d 2740	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ^* → (-_𝑒-_𝑒𝐵 = +∞ ↔ 𝐵 = +∞))
113	111, 112	syl5ib 243	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ^* → (-_𝑒𝐵 = -∞ → 𝐵 = +∞))
114	113	necon3d 2963	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ^* → (𝐵 ≠ +∞ → -_𝑒𝐵 ≠ -∞))
115	114	imp 406	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → -_𝑒𝐵 ≠ -∞)
116		xaddpnf2 12890	. . . . . . . 8 ⊢ ((-_𝑒𝐵 ∈ ℝ^* ∧ -_𝑒𝐵 ≠ -∞) → (+∞ +_𝑒 -_𝑒𝐵) = +∞)
117	76, 115, 116	syl2an2r 681	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (+∞ +_𝑒 -_𝑒𝐵) = +∞)
118	42, 109, 117	3eqtr4a 2805	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → -_𝑒(-∞ +_𝑒 𝐵) = (+∞ +_𝑒 -_𝑒𝐵))
119	106, 118	pm2.61dane 3031	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → -_𝑒(-∞ +_𝑒 𝐵) = (+∞ +_𝑒 -_𝑒𝐵))
120	119	adantl 481	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒(-∞ +_𝑒 𝐵) = (+∞ +_𝑒 -_𝑒𝐵))
121		simpl 482	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → 𝐴 = -∞)
122	121	oveq1d 7270	. . . . 5 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (-∞ +_𝑒 𝐵))
123		xnegeq 12870	. . . . 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 12870	. . . . . . 7 ⊢ (𝐴 = -∞ → -_𝑒𝐴 = -_𝑒-∞)
126	125	adantr 480	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒𝐴 = -_𝑒-∞)
127	126, 42	eqtrdi 2795	. . . . 5 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒𝐴 = +∞)
128	127	oveq1d 7270	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 +_𝑒 -_𝑒𝐵) = (+∞ +_𝑒 -_𝑒𝐵))
129	120, 124, 128	3eqtr4d 2788	. . 3 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
130	60, 97, 129	3jaoian 1427	. 2 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
131	1, 130	sylanb 580	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ w3o 1084 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 (class class class)co 7255 ℂcc 10800 ℝcr 10801 0cc0 10802 + caddc 10805 +∞cpnf 10937 -∞cmnf 10938 ℝ^*cxr 10939 -cneg 11136 -_𝑒cxne 12774 +_𝑒 cxad 12775

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-po 5494 df-so 5495 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-sub 11137 df-neg 11138 df-xneg 12777 df-xadd 12778

This theorem is referenced by: xaddass2 12913 xposdif 12925 xadddi 12958 xrsxmet 23878

W3C validator