xaddeq0 - Mathbox for Thierry Arnoux

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Theorem xaddeq0 32833

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 13030	. . 3 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		simpll 766	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐴 ∈ ℝ)
3	2	rexrd 11182	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐴 ∈ ℝ^)
4		xnegneg 13129	. . . . . . 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 13215	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 +_𝑒 𝐵) = 0) → -_𝑒𝐴 ∈ ℝ^)
7		xaddlid 13157	. . . . . . . . 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 13155	. . . . . . . . . . 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 7373	. . . . . . . . 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 7373	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ((𝐴 +_𝑒 𝐵) +_𝑒 -_𝑒𝐴) = (0 +_𝑒 -_𝑒𝐴))
15		xpncan 13166	. . . . . . . . . . 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 2779	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (0 +_𝑒 -_𝑒𝐴) = 𝐵)
19	8, 18	eqtr3d 2773	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → -_𝑒𝐴 = 𝐵)
20		xnegeq 13122	. . . . . . 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 2773	. . . . 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 7373	. . . . . . . . . . . . 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 2773	. . . . . . . . . . . 12 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (+∞ +_𝑒 𝐵) = 0)
29		0re 11134	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
30		renepnf 11180	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℝ → 0 ≠ +∞)
31	29, 30	mp1i 13	. . . . . . . . . . . 12 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 0 ≠ +∞)
32	28, 31	eqnetrd 2999	. . . . . . . . . . 11 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (+∞ +_𝑒 𝐵) ≠ +∞)
33	32	neneqd 2937	. . . . . . . . . 10 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ¬ (+∞ +_𝑒 𝐵) = +∞)
34		xaddpnf2 13142	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (+∞ +_𝑒 𝐵) = +∞)
35	34	stoic1a 1773	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ ¬ (+∞ +_𝑒 𝐵) = +∞) → ¬ 𝐵 ≠ -∞)
36	25, 33, 35	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ¬ 𝐵 ≠ -∞)
37		nne 2936	. . . . . . . . 9 ⊢ (¬ 𝐵 ≠ -∞ ↔ 𝐵 = -∞)
38	36, 37	sylib 218	. . . . . . . 8 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐵 = -∞)
39		xnegeq 13122	. . . . . . . 8 ⊢ (𝐵 = -∞ → -_𝑒𝐵 = -_𝑒-∞)
40	38, 39	syl 17	. . . . . . 7 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → -_𝑒𝐵 = -_𝑒-∞)
41		xnegmnf 13125	. . . . . . 7 ⊢ -_𝑒-∞ = +∞
42	40, 41	eqtr2di 2788	. . . . . 6 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → +∞ = -_𝑒𝐵)
43	24, 42	eqtrd 2771	. . . . 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 7373	. . . . . . . . . . . . 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 2773	. . . . . . . . . . . 12 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (-∞ +_𝑒 𝐵) = 0)
50		renemnf 11181	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℝ → 0 ≠ -∞)
51	29, 50	mp1i 13	. . . . . . . . . . . 12 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 0 ≠ -∞)
52	49, 51	eqnetrd 2999	. . . . . . . . . . 11 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (-∞ +_𝑒 𝐵) ≠ -∞)
53	52	neneqd 2937	. . . . . . . . . 10 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ¬ (-∞ +_𝑒 𝐵) = -∞)
54		xaddmnf2 13144	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (-∞ +_𝑒 𝐵) = -∞)
55	54	stoic1a 1773	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ ¬ (-∞ +_𝑒 𝐵) = -∞) → ¬ 𝐵 ≠ +∞)
56	46, 53, 55	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ¬ 𝐵 ≠ +∞)
57		nne 2936	. . . . . . . . 9 ⊢ (¬ 𝐵 ≠ +∞ ↔ 𝐵 = +∞)
58	56, 57	sylib 218	. . . . . . . 8 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐵 = +∞)
59		xnegeq 13122	. . . . . . . 8 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -_𝑒+∞)
60	58, 59	syl 17	. . . . . . 7 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → -_𝑒𝐵 = -_𝑒+∞)
61		xnegpnf 13124	. . . . . . 7 ⊢ -_𝑒+∞ = -∞
62	60, 61	eqtr2di 2788	. . . . . 6 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → -∞ = -_𝑒𝐵)
63	45, 62	eqtrd 2771	. . . . 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 7373	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 = -_𝑒𝐵) → (𝐴 +_𝑒 𝐵) = (-_𝑒𝐵 +_𝑒 𝐵))
69		xnegcl 13128	. . . . . 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 13155	. . . . 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 13153	. . . . 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 2775	. . 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 1541 ∈ wcel 2113 ≠ wne 2932 (class class class)co 7358 ℝcr 11025 0cc0 11026 +∞cpnf 11163 -∞cmnf 11164 ℝ^*cxr 11165 -_𝑒cxne 13023 +_𝑒 cxad 13024

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-po 5532 df-so 5533 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7933 df-2nd 7934 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-sub 11366 df-neg 11367 df-xneg 13026 df-xadd 13027

This theorem is referenced by: xrsinvgval 33090

W3C validator