xaddeq0 - Mathbox for Thierry Arnoux

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Theorem xaddeq0 32676

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 13076	. . 3 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		simpll 766	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐴 ∈ ℝ)
3	2	rexrd 11224	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐴 ∈ ℝ^)
4		xnegneg 13174	. . . . . . 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 13260	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 +_𝑒 𝐵) = 0) → -_𝑒𝐴 ∈ ℝ^)
7		xaddlid 13202	. . . . . . . . 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 13200	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
11	3, 9, 10	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
12	11	oveq1d 7402	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ((𝐴 +_𝑒 𝐵) +_𝑒 -_𝑒𝐴) = ((𝐵 +_𝑒 𝐴) +_𝑒 -_𝑒𝐴))
13		simpr 484	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (𝐴 +_𝑒 𝐵) = 0)
14	13	oveq1d 7402	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ((𝐴 +_𝑒 𝐵) +_𝑒 -_𝑒𝐴) = (0 +_𝑒 -_𝑒𝐴))
15		xpncan 13211	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) → ((𝐵 +_𝑒 𝐴) +_𝑒 -_𝑒𝐴) = 𝐵)
16	15	ancoms 458	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → ((𝐵 +_𝑒 𝐴) +_𝑒 -_𝑒𝐴) = 𝐵)
17	16	adantr 480	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ((𝐵 +_𝑒 𝐴) +_𝑒 -_𝑒𝐴) = 𝐵)
18	12, 14, 17	3eqtr3d 2772	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (0 +_𝑒 -_𝑒𝐴) = 𝐵)
19	8, 18	eqtr3d 2766	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → -_𝑒𝐴 = 𝐵)
20		xnegeq 13167	. . . . . . 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 2766	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐴 = -_𝑒𝐵)
23	22	ex 412	. . . 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 7402	. . . . . . . . . . . . 13 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (𝐴 +_𝑒 𝐵) = (+∞ +_𝑒 𝐵))
27		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (𝐴 +_𝑒 𝐵) = 0)
28	26, 27	eqtr3d 2766	. . . . . . . . . . . 12 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (+∞ +_𝑒 𝐵) = 0)
29		0re 11176	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
30		renepnf 11222	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℝ → 0 ≠ +∞)
31	29, 30	mp1i 13	. . . . . . . . . . . 12 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 0 ≠ +∞)
32	28, 31	eqnetrd 2992	. . . . . . . . . . 11 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (+∞ +_𝑒 𝐵) ≠ +∞)
33	32	neneqd 2930	. . . . . . . . . 10 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ¬ (+∞ +_𝑒 𝐵) = +∞)
34		xaddpnf2 13187	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (+∞ +_𝑒 𝐵) = +∞)
35	34	stoic1a 1772	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ ¬ (+∞ +_𝑒 𝐵) = +∞) → ¬ 𝐵 ≠ -∞)
36	25, 33, 35	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ¬ 𝐵 ≠ -∞)
37		nne 2929	. . . . . . . . 9 ⊢ (¬ 𝐵 ≠ -∞ ↔ 𝐵 = -∞)
38	36, 37	sylib 218	. . . . . . . 8 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐵 = -∞)
39		xnegeq 13167	. . . . . . . 8 ⊢ (𝐵 = -∞ → -_𝑒𝐵 = -_𝑒-∞)
40	38, 39	syl 17	. . . . . . 7 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → -_𝑒𝐵 = -_𝑒-∞)
41		xnegmnf 13170	. . . . . . 7 ⊢ -_𝑒-∞ = +∞
42	40, 41	eqtr2di 2781	. . . . . 6 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → +∞ = -_𝑒𝐵)
43	24, 42	eqtrd 2764	. . . . 5 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐴 = -_𝑒𝐵)
44	43	ex 412	. . . 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 7402	. . . . . . . . . . . . 13 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (𝐴 +_𝑒 𝐵) = (-∞ +_𝑒 𝐵))
48		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (𝐴 +_𝑒 𝐵) = 0)
49	47, 48	eqtr3d 2766	. . . . . . . . . . . 12 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (-∞ +_𝑒 𝐵) = 0)
50		renemnf 11223	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℝ → 0 ≠ -∞)
51	29, 50	mp1i 13	. . . . . . . . . . . 12 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 0 ≠ -∞)
52	49, 51	eqnetrd 2992	. . . . . . . . . . 11 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (-∞ +_𝑒 𝐵) ≠ -∞)
53	52	neneqd 2930	. . . . . . . . . 10 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ¬ (-∞ +_𝑒 𝐵) = -∞)
54		xaddmnf2 13189	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (-∞ +_𝑒 𝐵) = -∞)
55	54	stoic1a 1772	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ ¬ (-∞ +_𝑒 𝐵) = -∞) → ¬ 𝐵 ≠ +∞)
56	46, 53, 55	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ¬ 𝐵 ≠ +∞)
57		nne 2929	. . . . . . . . 9 ⊢ (¬ 𝐵 ≠ +∞ ↔ 𝐵 = +∞)
58	56, 57	sylib 218	. . . . . . . 8 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐵 = +∞)
59		xnegeq 13167	. . . . . . . 8 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -_𝑒+∞)
60	58, 59	syl 17	. . . . . . 7 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → -_𝑒𝐵 = -_𝑒+∞)
61		xnegpnf 13169	. . . . . . 7 ⊢ -_𝑒+∞ = -∞
62	60, 61	eqtr2di 2781	. . . . . 6 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → -∞ = -_𝑒𝐵)
63	45, 62	eqtrd 2764	. . . . 5 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐴 = -_𝑒𝐵)
64	63	ex 412	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_𝑒 𝐵) = 0 → 𝐴 = -_𝑒𝐵))
65	23, 44, 64	3jaoian 1432	. . 3 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_𝑒 𝐵) = 0 → 𝐴 = -_𝑒𝐵))
66	1, 65	sylanb 581	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_𝑒 𝐵) = 0 → 𝐴 = -_𝑒𝐵))
67		simpr 484	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 = -_𝑒𝐵) → 𝐴 = -_𝑒𝐵)
68	67	oveq1d 7402	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 = -_𝑒𝐵) → (𝐴 +_𝑒 𝐵) = (-_𝑒𝐵 +_𝑒 𝐵))
69		xnegcl 13173	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → -_𝑒𝐵 ∈ ℝ^*)
70	69	ad2antlr 727	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 = -_𝑒𝐵) → -_𝑒𝐵 ∈ ℝ^)
71		simplr 768	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 = -_𝑒𝐵) → 𝐵 ∈ ℝ^)
72		xaddcom 13200	. . . . 5 ⊢ ((-_𝑒𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐵 +_𝑒 𝐵) = (𝐵 +_𝑒 -_𝑒𝐵))
73	70, 71, 72	syl2anc 584	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 = -_𝑒𝐵) → (-_𝑒𝐵 +_𝑒 𝐵) = (𝐵 +_𝑒 -_𝑒𝐵))
74		xnegid 13198	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → (𝐵 +_𝑒 -_𝑒𝐵) = 0)
75	74	ad2antlr 727	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 = -_𝑒𝐵) → (𝐵 +_𝑒 -_𝑒𝐵) = 0)
76	68, 73, 75	3eqtrd 2768	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 = -_𝑒𝐵) → (𝐴 +_𝑒 𝐵) = 0)
77	76	ex 412	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 = -_𝑒𝐵 → (𝐴 +_𝑒 𝐵) = 0))
78	66, 77	impbid 212	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_𝑒 𝐵) = 0 ↔ 𝐴 = -_𝑒𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ w3o 1085 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 (class class class)co 7387 ℝcr 11067 0cc0 11068 +∞cpnf 11205 -∞cmnf 11206 ℝ^*cxr 11207 -_𝑒cxne 13069 +_𝑒 cxad 13070

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-po 5546 df-so 5547 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-1st 7968 df-2nd 7969 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-sub 11407 df-neg 11408 df-xneg 13072 df-xadd 13073

This theorem is referenced by: xrsinvgval 32946

W3C validator