xaddeq0 - Mathbox for Thierry Arnoux

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Theorem xaddeq0 31076

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 12852	. . 3 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		simpll 764	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐴 ∈ ℝ)
3	2	rexrd 11025	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐴 ∈ ℝ^)
4		xnegneg 12948	. . . . . . 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 13034	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 +_𝑒 𝐵) = 0) → -_𝑒𝐴 ∈ ℝ^)
7		xaddid2 12976	. . . . . . . . 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 766	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐵 ∈ ℝ^)
10		xaddcom 12974	. . . . . . . . . . 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 7290	. . . . . . . . 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 7290	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ((𝐴 +_𝑒 𝐵) +_𝑒 -_𝑒𝐴) = (0 +_𝑒 -_𝑒𝐴))
15		xpncan 12985	. . . . . . . . . . 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 2786	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (0 +_𝑒 -_𝑒𝐴) = 𝐵)
19	8, 18	eqtr3d 2780	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → -_𝑒𝐴 = 𝐵)
20		xnegeq 12941	. . . . . . 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 2780	. . . . 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 764	. . . . . 6 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐴 = +∞)
25		simplr 766	. . . . . . . . . 10 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐵 ∈ ℝ^)
26	24	oveq1d 7290	. . . . . . . . . . . . 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 2780	. . . . . . . . . . . 12 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (+∞ +_𝑒 𝐵) = 0)
29		0re 10977	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
30		renepnf 11023	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℝ → 0 ≠ +∞)
31	29, 30	mp1i 13	. . . . . . . . . . . 12 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 0 ≠ +∞)
32	28, 31	eqnetrd 3011	. . . . . . . . . . 11 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (+∞ +_𝑒 𝐵) ≠ +∞)
33	32	neneqd 2948	. . . . . . . . . 10 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ¬ (+∞ +_𝑒 𝐵) = +∞)
34		xaddpnf2 12961	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (+∞ +_𝑒 𝐵) = +∞)
35	34	stoic1a 1775	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ ¬ (+∞ +_𝑒 𝐵) = +∞) → ¬ 𝐵 ≠ -∞)
36	25, 33, 35	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ¬ 𝐵 ≠ -∞)
37		nne 2947	. . . . . . . . 9 ⊢ (¬ 𝐵 ≠ -∞ ↔ 𝐵 = -∞)
38	36, 37	sylib 217	. . . . . . . 8 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐵 = -∞)
39		xnegeq 12941	. . . . . . . 8 ⊢ (𝐵 = -∞ → -_𝑒𝐵 = -_𝑒-∞)
40	38, 39	syl 17	. . . . . . 7 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → -_𝑒𝐵 = -_𝑒-∞)
41		xnegmnf 12944	. . . . . . 7 ⊢ -_𝑒-∞ = +∞
42	40, 41	eqtr2di 2795	. . . . . 6 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → +∞ = -_𝑒𝐵)
43	24, 42	eqtrd 2778	. . . . 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 764	. . . . . 6 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐴 = -∞)
46		simplr 766	. . . . . . . . . 10 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐵 ∈ ℝ^)
47	45	oveq1d 7290	. . . . . . . . . . . . 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 2780	. . . . . . . . . . . 12 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (-∞ +_𝑒 𝐵) = 0)
50		renemnf 11024	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℝ → 0 ≠ -∞)
51	29, 50	mp1i 13	. . . . . . . . . . . 12 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 0 ≠ -∞)
52	49, 51	eqnetrd 3011	. . . . . . . . . . 11 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → (-∞ +_𝑒 𝐵) ≠ -∞)
53	52	neneqd 2948	. . . . . . . . . 10 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ¬ (-∞ +_𝑒 𝐵) = -∞)
54		xaddmnf2 12963	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (-∞ +_𝑒 𝐵) = -∞)
55	54	stoic1a 1775	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ ¬ (-∞ +_𝑒 𝐵) = -∞) → ¬ 𝐵 ≠ +∞)
56	46, 53, 55	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → ¬ 𝐵 ≠ +∞)
57		nne 2947	. . . . . . . . 9 ⊢ (¬ 𝐵 ≠ +∞ ↔ 𝐵 = +∞)
58	56, 57	sylib 217	. . . . . . . 8 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → 𝐵 = +∞)
59		xnegeq 12941	. . . . . . . 8 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -_𝑒+∞)
60	58, 59	syl 17	. . . . . . 7 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → -_𝑒𝐵 = -_𝑒+∞)
61		xnegpnf 12943	. . . . . . 7 ⊢ -_𝑒+∞ = -∞
62	60, 61	eqtr2di 2795	. . . . . 6 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_𝑒 𝐵) = 0) → -∞ = -_𝑒𝐵)
63	45, 62	eqtrd 2778	. . . . 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 1428	. . 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 485	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 = -_𝑒𝐵) → 𝐴 = -_𝑒𝐵)
68	67	oveq1d 7290	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 = -_𝑒𝐵) → (𝐴 +_𝑒 𝐵) = (-_𝑒𝐵 +_𝑒 𝐵))
69		xnegcl 12947	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → -_𝑒𝐵 ∈ ℝ^*)
70	69	ad2antlr 724	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 = -_𝑒𝐵) → -_𝑒𝐵 ∈ ℝ^)
71		simplr 766	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 = -_𝑒𝐵) → 𝐵 ∈ ℝ^)
72		xaddcom 12974	. . . . 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 12972	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → (𝐵 +_𝑒 -_𝑒𝐵) = 0)
75	74	ad2antlr 724	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 = -_𝑒𝐵) → (𝐵 +_𝑒 -_𝑒𝐵) = 0)
76	68, 73, 75	3eqtrd 2782	. . 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 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 396 ∨ w3o 1085 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 (class class class)co 7275 ℝcr 10870 0cc0 10871 +∞cpnf 11006 -∞cmnf 11007 ℝ^*cxr 11008 -_𝑒cxne 12845 +_𝑒 cxad 12846

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-sub 11207 df-neg 11208 df-xneg 12848 df-xadd 12849

This theorem is referenced by: xrsinvgval 31286

W3C validator