xaddeq0 - Mathbox for Thierry Arnoux

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Theorem xaddeq0 31997

Description: Two extended reals which add up to zero are each other's negatives. (Contributed by Thierry Arnoux, 13-Jun-2017.)

**Assertion**
Ref	Expression
xaddeq0	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_𝑒 𝐵) = 0 ↔ 𝐴 = -_𝑒𝐵))

**Proof of Theorem xaddeq0**
Step	Hyp	Ref	Expression
1		elxr 13096	. . 3 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		simpll 766	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐴 ∈ ℝ)
3	2	rexrd 11264	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐴 ∈ ℝ^)
4		xnegneg 13193	. . . . . . 7 ⊢ (𝐴 ∈ ℝ^* → -_𝑒-_𝑒𝐴 = 𝐴)
5	3, 4	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → -_𝑒-_𝑒𝐴 = 𝐴)
6	3	xnegcld 13279	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 +_𝑒 𝐵) = 0) → -_𝑒𝐴 ∈ ℝ^)
7		xaddlid 13221	. . . . . . . . 9 ⊢ (-_𝑒𝐴 ∈ ℝ^* → (0 +_𝑒 -_𝑒𝐴) = -_𝑒𝐴)
8	6, 7	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (0 +_𝑒 -_𝑒𝐴) = -_𝑒𝐴)
9		simplr 768	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐵 ∈ ℝ^)
10		xaddcom 13219	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
11	3, 9, 10	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
12	11	oveq1d 7424	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ((𝐴 +_𝑒 𝐵) +_𝑒 -_𝑒𝐴) = ((𝐵 +_𝑒 𝐴) +_𝑒 -_𝑒𝐴))
13		simpr 486	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (𝐴 +_𝑒 𝐵) = 0)
14	13	oveq1d 7424	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ((𝐴 +_𝑒 𝐵) +_𝑒 -_𝑒𝐴) = (0 +_𝑒 -_𝑒𝐴))
15		xpncan 13230	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) → ((𝐵 +_𝑒 𝐴) +_𝑒 -_𝑒𝐴) = 𝐵)
16	15	ancoms 460	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → ((𝐵 +_𝑒 𝐴) +_𝑒 -_𝑒𝐴) = 𝐵)
17	16	adantr 482	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ((𝐵 +_𝑒 𝐴) +_𝑒 -_𝑒𝐴) = 𝐵)
18	12, 14, 17	3eqtr3d 2781	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (0 +_𝑒 -_𝑒𝐴) = 𝐵)
19	8, 18	eqtr3d 2775	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → -_𝑒𝐴 = 𝐵)
20		xnegeq 13186	. . . . . . 7 ⊢ (-_𝑒𝐴 = 𝐵 → -_𝑒-_𝑒𝐴 = -_𝑒𝐵)
21	19, 20	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → -_𝑒-_𝑒𝐴 = -_𝑒𝐵)
22	5, 21	eqtr3d 2775	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐴 = -_𝑒𝐵)
23	22	ex 414	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_𝑒 𝐵) = 0 → 𝐴 = -_𝑒𝐵))
24		simpll 766	. . . . . 6 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐴 = +∞)
25		simplr 768	. . . . . . . . . 10 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐵 ∈ ℝ^)
26	24	oveq1d 7424	. . . . . . . . . . . . 13 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (𝐴 +_𝑒 𝐵) = (+∞ +_𝑒 𝐵))
27		simpr 486	. . . . . . . . . . . . 13 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (𝐴 +_𝑒 𝐵) = 0)
28	26, 27	eqtr3d 2775	. . . . . . . . . . . 12 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (+∞ +_𝑒 𝐵) = 0)
29		0re 11216	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
30		renepnf 11262	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℝ → 0 ≠ +∞)
31	29, 30	mp1i 13	. . . . . . . . . . . 12 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 0 ≠ +∞)
32	28, 31	eqnetrd 3009	. . . . . . . . . . 11 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (+∞ +_𝑒 𝐵) ≠ +∞)
33	32	neneqd 2946	. . . . . . . . . 10 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ¬ (+∞ +_𝑒 𝐵) = +∞)
34		xaddpnf2 13206	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (+∞ +_𝑒 𝐵) = +∞)
35	34	stoic1a 1775	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ ¬ (+∞ +_𝑒 𝐵) = +∞) → ¬ 𝐵 ≠ -∞)
36	25, 33, 35	syl2anc 585	. . . . . . . . 9 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ¬ 𝐵 ≠ -∞)
37		nne 2945	. . . . . . . . 9 ⊢ (¬ 𝐵 ≠ -∞ ↔ 𝐵 = -∞)
38	36, 37	sylib 217	. . . . . . . 8 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐵 = -∞)
39		xnegeq 13186	. . . . . . . 8 ⊢ (𝐵 = -∞ → -_𝑒𝐵 = -_𝑒-∞)
40	38, 39	syl 17	. . . . . . 7 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → -_𝑒𝐵 = -_𝑒-∞)
41		xnegmnf 13189	. . . . . . 7 ⊢ -_𝑒-∞ = +∞
42	40, 41	eqtr2di 2790	. . . . . 6 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → +∞ = -_𝑒𝐵)
43	24, 42	eqtrd 2773	. . . . 5 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐴 = -_𝑒𝐵)
44	43	ex 414	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_𝑒 𝐵) = 0 → 𝐴 = -_𝑒𝐵))
45		simpll 766	. . . . . 6 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐴 = -∞)
46		simplr 768	. . . . . . . . . 10 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐵 ∈ ℝ^)
47	45	oveq1d 7424	. . . . . . . . . . . . 13 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (𝐴 +_𝑒 𝐵) = (-∞ +_𝑒 𝐵))
48		simpr 486	. . . . . . . . . . . . 13 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (𝐴 +_𝑒 𝐵) = 0)
49	47, 48	eqtr3d 2775	. . . . . . . . . . . 12 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (-∞ +_𝑒 𝐵) = 0)
50		renemnf 11263	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℝ → 0 ≠ -∞)
51	29, 50	mp1i 13	. . . . . . . . . . . 12 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 0 ≠ -∞)
52	49, 51	eqnetrd 3009	. . . . . . . . . . 11 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (-∞ +_𝑒 𝐵) ≠ -∞)
53	52	neneqd 2946	. . . . . . . . . 10 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ¬ (-∞ +_𝑒 𝐵) = -∞)
54		xaddmnf2 13208	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (-∞ +_𝑒 𝐵) = -∞)
55	54	stoic1a 1775	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ ¬ (-∞ +_𝑒 𝐵) = -∞) → ¬ 𝐵 ≠ +∞)
56	46, 53, 55	syl2anc 585	. . . . . . . . 9 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ¬ 𝐵 ≠ +∞)
57		nne 2945	. . . . . . . . 9 ⊢ (¬ 𝐵 ≠ +∞ ↔ 𝐵 = +∞)
58	56, 57	sylib 217	. . . . . . . 8 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐵 = +∞)
59		xnegeq 13186	. . . . . . . 8 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -_𝑒+∞)
60	58, 59	syl 17	. . . . . . 7 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → -_𝑒𝐵 = -_𝑒+∞)
61		xnegpnf 13188	. . . . . . 7 ⊢ -_𝑒+∞ = -∞
62	60, 61	eqtr2di 2790	. . . . . 6 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → -∞ = -_𝑒𝐵)
63	45, 62	eqtrd 2773	. . . . 5 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐴 = -_𝑒𝐵)
64	63	ex 414	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_𝑒 𝐵) = 0 → 𝐴 = -_𝑒𝐵))
65	23, 44, 64	3jaoian 1430	. . 3 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_𝑒 𝐵) = 0 → 𝐴 = -_𝑒𝐵))
66	1, 65	sylanb 582	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_𝑒 𝐵) = 0 → 𝐴 = -_𝑒𝐵))
67		simpr 486	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 = -_𝑒𝐵) → 𝐴 = -_𝑒𝐵)
68	67	oveq1d 7424	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 = -_𝑒𝐵) → (𝐴 +_𝑒 𝐵) = (-_𝑒𝐵 +_𝑒 𝐵))
69		xnegcl 13192	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → -_𝑒𝐵 ∈ ℝ^*)
70	69	ad2antlr 726	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 = -_𝑒𝐵) → -_𝑒𝐵 ∈ ℝ^)
71		simplr 768	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 = -_𝑒𝐵) → 𝐵 ∈ ℝ^)
72		xaddcom 13219	. . . . 5 ⊢ ((-_𝑒𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐵 +_𝑒 𝐵) = (𝐵 +_𝑒 -_𝑒𝐵))
73	70, 71, 72	syl2anc 585	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 = -_𝑒𝐵) → (-_𝑒𝐵 +_𝑒 𝐵) = (𝐵 +_𝑒 -_𝑒𝐵))
74		xnegid 13217	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → (𝐵 +_𝑒 -_𝑒𝐵) = 0)
75	74	ad2antlr 726	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 = -_𝑒𝐵) → (𝐵 +_𝑒 -_𝑒𝐵) = 0)
76	68, 73, 75	3eqtrd 2777	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 = -_𝑒𝐵) → (𝐴 +_𝑒 𝐵) = 0)
77	76	ex 414	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 = -_𝑒𝐵 → (𝐴 +_𝑒 𝐵) = 0))
78	66, 77	impbid 211	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_𝑒 𝐵) = 0 ↔ 𝐴 = -_𝑒𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ w3o 1087 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 (class class class)co 7409 ℝcr 11109 0cc0 11110 +∞cpnf 11245 -∞cmnf 11246 ℝ^*cxr 11247 -_𝑒cxne 13089 +_𝑒 cxad 13090

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-sub 11446 df-neg 11447 df-xneg 13092 df-xadd 13093

This theorem is referenced by: xrsinvgval 32209

W3C validator