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Theorem xnegdi 13291

Description: Extended real version of negdi 11567. (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 13159	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 13159	. . . 4 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3		recn 11246	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
4		recn 11246	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ)
5		negdi 11567	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 + 𝐵) = (-𝐴 + -𝐵))
6	3, 4, 5	syl2an 596	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -(𝐴 + 𝐵) = (-𝐴 + -𝐵))
7		readdcl 11239	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
8		rexneg 13254	. . . . . . . 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 11573	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
11		renegcl 11573	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → -𝐵 ∈ ℝ)
12		rexadd 13275	. . . . . . . 8 ⊢ ((-𝐴 ∈ ℝ ∧ -𝐵 ∈ ℝ) → (-𝐴 +_𝑒 -𝐵) = (-𝐴 + -𝐵))
13	10, 11, 12	syl2an 596	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐴 +_𝑒 -𝐵) = (-𝐴 + -𝐵))
14	6, 9, 13	3eqtr4d 2786	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -_𝑒(𝐴 + 𝐵) = (-𝐴 +_𝑒 -𝐵))
15		rexadd 13275	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +_𝑒 𝐵) = (𝐴 + 𝐵))
16		xnegeq 13250	. . . . . . 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 13254	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 = -𝐴)
19		rexneg 13254	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → -_𝑒𝐵 = -𝐵)
20	18, 19	oveqan12d 7451	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-_𝑒𝐴 +_𝑒 -_𝑒𝐵) = (-𝐴 +_𝑒 -𝐵))
21	14, 17, 20	3eqtr4d 2786	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
22		xnegpnf 13252	. . . . . 6 ⊢ -_𝑒+∞ = -∞
23		oveq2 7440	. . . . . . . 8 ⊢ (𝐵 = +∞ → (𝐴 +_𝑒 𝐵) = (𝐴 +_𝑒 +∞))
24		rexr 11308	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
25		renemnf 11311	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
26		xaddpnf1 13269	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → (𝐴 +_𝑒 +∞) = +∞)
27	24, 25, 26	syl2anc 584	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 +∞) = +∞)
28	23, 27	sylan9eqr 2798	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 +_𝑒 𝐵) = +∞)
29		xnegeq 13250	. . . . . . 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 13250	. . . . . . . . 9 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -_𝑒+∞)
32	31, 22	eqtrdi 2792	. . . . . . . 8 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -∞)
33	32	oveq2d 7448	. . . . . . 7 ⊢ (𝐵 = +∞ → (-_𝑒𝐴 +_𝑒 -_𝑒𝐵) = (-_𝑒𝐴 +_𝑒 -∞))
34	18, 10	eqeltrd 2840	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 ∈ ℝ)
35		rexr 11308	. . . . . . . . 9 ⊢ (-_𝑒𝐴 ∈ ℝ → -_𝑒𝐴 ∈ ℝ^*)
36		renepnf 11310	. . . . . . . . 9 ⊢ (-_𝑒𝐴 ∈ ℝ → -_𝑒𝐴 ≠ +∞)
37		xaddmnf1 13271	. . . . . . . . 9 ⊢ ((-_𝑒𝐴 ∈ ℝ^* ∧ -_𝑒𝐴 ≠ +∞) → (-_𝑒𝐴 +_𝑒 -∞) = -∞)
38	35, 36, 37	syl2anc 584	. . . . . . . 8 ⊢ (-_𝑒𝐴 ∈ ℝ → (-_𝑒𝐴 +_𝑒 -∞) = -∞)
39	34, 38	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (-_𝑒𝐴 +_𝑒 -∞) = -∞)
40	33, 39	sylan9eqr 2798	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (-_𝑒𝐴 +_𝑒 -_𝑒𝐵) = -∞)
41	22, 30, 40	3eqtr4a 2802	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
42		xnegmnf 13253	. . . . . 6 ⊢ -_𝑒-∞ = +∞
43		oveq2 7440	. . . . . . . 8 ⊢ (𝐵 = -∞ → (𝐴 +_𝑒 𝐵) = (𝐴 +_𝑒 -∞))
44		renepnf 11310	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
45		xaddmnf1 13271	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) → (𝐴 +_𝑒 -∞) = -∞)
46	24, 44, 45	syl2anc 584	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 -∞) = -∞)
47	43, 46	sylan9eqr 2798	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 +_𝑒 𝐵) = -∞)
48		xnegeq 13250	. . . . . . 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 13250	. . . . . . . . 9 ⊢ (𝐵 = -∞ → -_𝑒𝐵 = -_𝑒-∞)
51	50, 42	eqtrdi 2792	. . . . . . . 8 ⊢ (𝐵 = -∞ → -_𝑒𝐵 = +∞)
52	51	oveq2d 7448	. . . . . . 7 ⊢ (𝐵 = -∞ → (-_𝑒𝐴 +_𝑒 -_𝑒𝐵) = (-_𝑒𝐴 +_𝑒 +∞))
53		renemnf 11311	. . . . . . . . 9 ⊢ (-_𝑒𝐴 ∈ ℝ → -_𝑒𝐴 ≠ -∞)
54		xaddpnf1 13269	. . . . . . . . 9 ⊢ ((-_𝑒𝐴 ∈ ℝ^* ∧ -_𝑒𝐴 ≠ -∞) → (-_𝑒𝐴 +_𝑒 +∞) = +∞)
55	35, 53, 54	syl2anc 584	. . . . . . . 8 ⊢ (-_𝑒𝐴 ∈ ℝ → (-_𝑒𝐴 +_𝑒 +∞) = +∞)
56	34, 55	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (-_𝑒𝐴 +_𝑒 +∞) = +∞)
57	52, 56	sylan9eqr 2798	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (-_𝑒𝐴 +_𝑒 -_𝑒𝐵) = +∞)
58	42, 49, 57	3eqtr4a 2802	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
59	21, 41, 58	3jaodan 1432	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
60	2, 59	sylan2b 594	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
61		xneg0 13255	. . . . . . 7 ⊢ -_𝑒0 = 0
62		simpr 484	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → 𝐵 = -∞)
63	62	oveq2d 7448	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (+∞ +_𝑒 𝐵) = (+∞ +_𝑒 -∞))
64		pnfaddmnf 13273	. . . . . . . . 9 ⊢ (+∞ +_𝑒 -∞) = 0
65	63, 64	eqtrdi 2792	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (+∞ +_𝑒 𝐵) = 0)
66		xnegeq 13250	. . . . . . . 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 7448	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (-∞ +_𝑒 -_𝑒𝐵) = (-∞ +_𝑒 +∞))
70		mnfaddpnf 13274	. . . . . . . 8 ⊢ (-∞ +_𝑒 +∞) = 0
71	69, 70	eqtrdi 2792	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (-∞ +_𝑒 -_𝑒𝐵) = 0)
72	61, 67, 71	3eqtr4a 2802	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → -_𝑒(+∞ +_𝑒 𝐵) = (-∞ +_𝑒 -_𝑒𝐵))
73		xaddpnf2 13270	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (+∞ +_𝑒 𝐵) = +∞)
74		xnegeq 13250	. . . . . . . 8 ⊢ ((+∞ +_𝑒 𝐵) = +∞ → -_𝑒(+∞ +_𝑒 𝐵) = -_𝑒+∞)
75	73, 74	syl 17	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → -_𝑒(+∞ +_𝑒 𝐵) = -_𝑒+∞)
76		xnegcl 13256	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ^* → -_𝑒𝐵 ∈ ℝ^*)
77		xnegeq 13250	. . . . . . . . . . . 12 ⊢ (-_𝑒𝐵 = +∞ → -_𝑒-_𝑒𝐵 = -_𝑒+∞)
78	77, 22	eqtrdi 2792	. . . . . . . . . . 11 ⊢ (-_𝑒𝐵 = +∞ → -_𝑒-_𝑒𝐵 = -∞)
79		xnegneg 13257	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℝ^* → -_𝑒-_𝑒𝐵 = 𝐵)
80	79	eqeq1d 2738	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ^* → (-_𝑒-_𝑒𝐵 = -∞ ↔ 𝐵 = -∞))
81	78, 80	imbitrid 244	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ^* → (-_𝑒𝐵 = +∞ → 𝐵 = -∞))
82	81	necon3d 2960	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ^* → (𝐵 ≠ -∞ → -_𝑒𝐵 ≠ +∞))
83	82	imp 406	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → -_𝑒𝐵 ≠ +∞)
84		xaddmnf2 13272	. . . . . . . 8 ⊢ ((-_𝑒𝐵 ∈ ℝ^* ∧ -_𝑒𝐵 ≠ +∞) → (-∞ +_𝑒 -_𝑒𝐵) = -∞)
85	76, 83, 84	syl2an2r 685	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (-∞ +_𝑒 -_𝑒𝐵) = -∞)
86	22, 75, 85	3eqtr4a 2802	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → -_𝑒(+∞ +_𝑒 𝐵) = (-∞ +_𝑒 -_𝑒𝐵))
87	72, 86	pm2.61dane 3028	. . . . 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 7447	. . . . 5 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (+∞ +_𝑒 𝐵))
91		xnegeq 13250	. . . . 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 13250	. . . . . . 7 ⊢ (𝐴 = +∞ → -_𝑒𝐴 = -_𝑒+∞)
94	93	adantr 480	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒𝐴 = -_𝑒+∞)
95	94, 22	eqtrdi 2792	. . . . 5 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒𝐴 = -∞)
96	95	oveq1d 7447	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 +_𝑒 -_𝑒𝐵) = (-∞ +_𝑒 -_𝑒𝐵))
97	88, 92, 96	3eqtr4d 2786	. . 3 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
98		simpr 484	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → 𝐵 = +∞)
99	98	oveq2d 7448	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (-∞ +_𝑒 𝐵) = (-∞ +_𝑒 +∞))
100	99, 70	eqtrdi 2792	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (-∞ +_𝑒 𝐵) = 0)
101		xnegeq 13250	. . . . . . . 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 7448	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (+∞ +_𝑒 -_𝑒𝐵) = (+∞ +_𝑒 -∞))
105	104, 64	eqtrdi 2792	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (+∞ +_𝑒 -_𝑒𝐵) = 0)
106	61, 102, 105	3eqtr4a 2802	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → -_𝑒(-∞ +_𝑒 𝐵) = (+∞ +_𝑒 -_𝑒𝐵))
107		xaddmnf2 13272	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (-∞ +_𝑒 𝐵) = -∞)
108		xnegeq 13250	. . . . . . . 8 ⊢ ((-∞ +_𝑒 𝐵) = -∞ → -_𝑒(-∞ +_𝑒 𝐵) = -_𝑒-∞)
109	107, 108	syl 17	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → -_𝑒(-∞ +_𝑒 𝐵) = -_𝑒-∞)
110		xnegeq 13250	. . . . . . . . . . . 12 ⊢ (-_𝑒𝐵 = -∞ → -_𝑒-_𝑒𝐵 = -_𝑒-∞)
111	110, 42	eqtrdi 2792	. . . . . . . . . . 11 ⊢ (-_𝑒𝐵 = -∞ → -_𝑒-_𝑒𝐵 = +∞)
112	79	eqeq1d 2738	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ^* → (-_𝑒-_𝑒𝐵 = +∞ ↔ 𝐵 = +∞))
113	111, 112	imbitrid 244	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ^* → (-_𝑒𝐵 = -∞ → 𝐵 = +∞))
114	113	necon3d 2960	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ^* → (𝐵 ≠ +∞ → -_𝑒𝐵 ≠ -∞))
115	114	imp 406	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → -_𝑒𝐵 ≠ -∞)
116		xaddpnf2 13270	. . . . . . . 8 ⊢ ((-_𝑒𝐵 ∈ ℝ^* ∧ -_𝑒𝐵 ≠ -∞) → (+∞ +_𝑒 -_𝑒𝐵) = +∞)
117	76, 115, 116	syl2an2r 685	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (+∞ +_𝑒 -_𝑒𝐵) = +∞)
118	42, 109, 117	3eqtr4a 2802	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → -_𝑒(-∞ +_𝑒 𝐵) = (+∞ +_𝑒 -_𝑒𝐵))
119	106, 118	pm2.61dane 3028	. . . . 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 7447	. . . . 5 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (-∞ +_𝑒 𝐵))
123		xnegeq 13250	. . . . 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 13250	. . . . . . 7 ⊢ (𝐴 = -∞ → -_𝑒𝐴 = -_𝑒-∞)
126	125	adantr 480	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒𝐴 = -_𝑒-∞)
127	126, 42	eqtrdi 2792	. . . . 5 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒𝐴 = +∞)
128	127	oveq1d 7447	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 +_𝑒 -_𝑒𝐵) = (+∞ +_𝑒 -_𝑒𝐵))
129	120, 124, 128	3eqtr4d 2786	. . 3 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
130	60, 97, 129	3jaoian 1431	. 2 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
131	1, 130	sylanb 581	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 1085 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 (class class class)co 7432 ℂcc 11154 ℝcr 11155 0cc0 11156 + caddc 11159 +∞cpnf 11293 -∞cmnf 11294 ℝ^*cxr 11295 -cneg 11494 -_𝑒cxne 13152 +_𝑒 cxad 13153

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-sub 11495 df-neg 11496 df-xneg 13155 df-xadd 13156

This theorem is referenced by: xaddass2 13293 xposdif 13305 xadddi 13338 xrsxmet 24832

W3C validator