xaddeq0 - Mathbox for Thierry Arnoux

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Theorem xaddeq0 32845

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 13058	. . 3 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		simpll 772	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐴 ∈ ℝ)
3	2	rexrd 11186	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐴 ∈ ℝ^)
4		xnegneg 13157	. . . . . . 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 13243	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 +_𝑒 𝐵) = 0) → -_𝑒𝐴 ∈ ℝ^)
7		xaddlid 13185	. . . . . . . . 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 774	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐵 ∈ ℝ^)
10		xaddcom 13183	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
11	3, 9, 10	syl2anc 590	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
12	11	oveq1d 7371	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ((𝐴 +_𝑒 𝐵) +_𝑒 -_𝑒𝐴) = ((𝐵 +_𝑒 𝐴) +_𝑒 -_𝑒𝐴))
13		simpr 485	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (𝐴 +_𝑒 𝐵) = 0)
14	13	oveq1d 7371	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ((𝐴 +_𝑒 𝐵) +_𝑒 -_𝑒𝐴) = (0 +_𝑒 -_𝑒𝐴))
15		xpncan 13194	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) → ((𝐵 +_𝑒 𝐴) +_𝑒 -_𝑒𝐴) = 𝐵)
16	15	ancoms 459	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → ((𝐵 +_𝑒 𝐴) +_𝑒 -_𝑒𝐴) = 𝐵)
17	16	adantr 481	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ((𝐵 +_𝑒 𝐴) +_𝑒 -_𝑒𝐴) = 𝐵)
18	12, 14, 17	3eqtr3d 2782	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (0 +_𝑒 -_𝑒𝐴) = 𝐵)
19	8, 18	eqtr3d 2776	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → -_𝑒𝐴 = 𝐵)
20		xnegeq 13150	. . . . . . 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 2776	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐴 = -_𝑒𝐵)
23	22	ex 413	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_𝑒 𝐵) = 0 → 𝐴 = -_𝑒𝐵))
24		simpll 772	. . . . . 6 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐴 = +∞)
25		simplr 774	. . . . . . . . . 10 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐵 ∈ ℝ^)
26	24	oveq1d 7371	. . . . . . . . . . . . 13 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (𝐴 +_𝑒 𝐵) = (+∞ +_𝑒 𝐵))
27		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (𝐴 +_𝑒 𝐵) = 0)
28	26, 27	eqtr3d 2776	. . . . . . . . . . . 12 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (+∞ +_𝑒 𝐵) = 0)
29		0re 11137	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
30		renepnf 11184	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℝ → 0 ≠ +∞)
31	29, 30	mp1i 13	. . . . . . . . . . . 12 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 0 ≠ +∞)
32	28, 31	eqnetrd 3001	. . . . . . . . . . 11 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (+∞ +_𝑒 𝐵) ≠ +∞)
33	32	neneqd 2939	. . . . . . . . . 10 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ¬ (+∞ +_𝑒 𝐵) = +∞)
34		xaddpnf2 13170	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (+∞ +_𝑒 𝐵) = +∞)
35	34	stoic1a 1779	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ ¬ (+∞ +_𝑒 𝐵) = +∞) → ¬ 𝐵 ≠ -∞)
36	25, 33, 35	syl2anc 590	. . . . . . . . 9 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ¬ 𝐵 ≠ -∞)
37		nne 2938	. . . . . . . . 9 ⊢ (¬ 𝐵 ≠ -∞ ↔ 𝐵 = -∞)
38	36, 37	sylib 219	. . . . . . . 8 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐵 = -∞)
39		xnegeq 13150	. . . . . . . 8 ⊢ (𝐵 = -∞ → -_𝑒𝐵 = -_𝑒-∞)
40	38, 39	syl 17	. . . . . . 7 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → -_𝑒𝐵 = -_𝑒-∞)
41		xnegmnf 13153	. . . . . . 7 ⊢ -_𝑒-∞ = +∞
42	40, 41	eqtr2di 2791	. . . . . 6 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → +∞ = -_𝑒𝐵)
43	24, 42	eqtrd 2774	. . . . 5 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐴 = -_𝑒𝐵)
44	43	ex 413	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_𝑒 𝐵) = 0 → 𝐴 = -_𝑒𝐵))
45		simpll 772	. . . . . 6 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐴 = -∞)
46		simplr 774	. . . . . . . . . 10 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐵 ∈ ℝ^)
47	45	oveq1d 7371	. . . . . . . . . . . . 13 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (𝐴 +_𝑒 𝐵) = (-∞ +_𝑒 𝐵))
48		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (𝐴 +_𝑒 𝐵) = 0)
49	47, 48	eqtr3d 2776	. . . . . . . . . . . 12 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (-∞ +_𝑒 𝐵) = 0)
50		renemnf 11185	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℝ → 0 ≠ -∞)
51	29, 50	mp1i 13	. . . . . . . . . . . 12 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 0 ≠ -∞)
52	49, 51	eqnetrd 3001	. . . . . . . . . . 11 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (-∞ +_𝑒 𝐵) ≠ -∞)
53	52	neneqd 2939	. . . . . . . . . 10 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ¬ (-∞ +_𝑒 𝐵) = -∞)
54		xaddmnf2 13172	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (-∞ +_𝑒 𝐵) = -∞)
55	54	stoic1a 1779	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ ¬ (-∞ +_𝑒 𝐵) = -∞) → ¬ 𝐵 ≠ +∞)
56	46, 53, 55	syl2anc 590	. . . . . . . . 9 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ¬ 𝐵 ≠ +∞)
57		nne 2938	. . . . . . . . 9 ⊢ (¬ 𝐵 ≠ +∞ ↔ 𝐵 = +∞)
58	56, 57	sylib 219	. . . . . . . 8 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐵 = +∞)
59		xnegeq 13150	. . . . . . . 8 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -_𝑒+∞)
60	58, 59	syl 17	. . . . . . 7 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → -_𝑒𝐵 = -_𝑒+∞)
61		xnegpnf 13152	. . . . . . 7 ⊢ -_𝑒+∞ = -∞
62	60, 61	eqtr2di 2791	. . . . . 6 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → -∞ = -_𝑒𝐵)
63	45, 62	eqtrd 2774	. . . . 5 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐴 = -_𝑒𝐵)
64	63	ex 413	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_𝑒 𝐵) = 0 → 𝐴 = -_𝑒𝐵))
65	23, 44, 64	3jaoian 1438	. . 3 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_𝑒 𝐵) = 0 → 𝐴 = -_𝑒𝐵))
66	1, 65	sylanb 587	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_𝑒 𝐵) = 0 → 𝐴 = -_𝑒𝐵))
67		simpr 485	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 = -_𝑒𝐵) → 𝐴 = -_𝑒𝐵)
68	67	oveq1d 7371	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 = -_𝑒𝐵) → (𝐴 +_𝑒 𝐵) = (-_𝑒𝐵 +_𝑒 𝐵))
69		xnegcl 13156	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → -_𝑒𝐵 ∈ ℝ^*)
70	69	ad2antlr 733	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 = -_𝑒𝐵) → -_𝑒𝐵 ∈ ℝ^)
71		simplr 774	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 = -_𝑒𝐵) → 𝐵 ∈ ℝ^)
72		xaddcom 13183	. . . . 5 ⊢ ((-_𝑒𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐵 +_𝑒 𝐵) = (𝐵 +_𝑒 -_𝑒𝐵))
73	70, 71, 72	syl2anc 590	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 = -_𝑒𝐵) → (-_𝑒𝐵 +_𝑒 𝐵) = (𝐵 +_𝑒 -_𝑒𝐵))
74		xnegid 13181	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → (𝐵 +_𝑒 -_𝑒𝐵) = 0)
75	74	ad2antlr 733	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 = -_𝑒𝐵) → (𝐵 +_𝑒 -_𝑒𝐵) = 0)
76	68, 73, 75	3eqtrd 2778	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 = -_𝑒𝐵) → (𝐴 +_𝑒 𝐵) = 0)
77	76	ex 413	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 = -_𝑒𝐵 → (𝐴 +_𝑒 𝐵) = 0))
78	66, 77	impbid 213	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_𝑒 𝐵) = 0 ↔ 𝐴 = -_𝑒𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ w3o 1091 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 (class class class)co 7356 ℝcr 11028 0cc0 11029 +∞cpnf 11167 -∞cmnf 11168 ℝ^*cxr 11169 -_𝑒cxne 13051 +_𝑒 cxad 13052

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-id 5513 df-po 5526 df-so 5527 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-1st 7931 df-2nd 7932 df-er 8633 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-sub 11370 df-neg 11371 df-xneg 13054 df-xadd 13055

This theorem is referenced by: xrsinvgval 33087

W3C validator