xaddeq0 - Mathbox for Thierry Arnoux

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Theorem xaddeq0 33345

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 13245	. . 3 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		simpll 779	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → 𝐴 ∈ ℝ)
3	2	rexrd 11359	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 +_e 𝐵) = 0) → 𝐴 ∈ ℝ^)
4		xnegneg 13344	. . . . . . 7 ⊢ (𝐴 ∈ ℝ^* → -_e -_e𝐴 = 𝐴)
5	3, 4	syl 18	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → -_e -_e𝐴 = 𝐴)
6	3	xnegcld 13430	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 +_e 𝐵) = 0) → -_e𝐴 ∈ ℝ^)
7		xaddlid 13372	. . . . . . . . 9 ⊢ ( -_e𝐴 ∈ ℝ^* → (0 +_e -_e𝐴) = -_e𝐴)
8	6, 7	syl 18	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → (0 +_e -_e𝐴) = -_e𝐴)
9		simplr 781	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 +_e 𝐵) = 0) → 𝐵 ∈ ℝ^)
10		xaddcom 13370	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_e 𝐵) = (𝐵 +_e 𝐴))
11	3, 9, 10	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → (𝐴 +_e 𝐵) = (𝐵 +_e 𝐴))
12	11	oveq1d 7435	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → ((𝐴 +_e 𝐵) +_e -_e𝐴) = ((𝐵 +_e 𝐴) +_e -_e𝐴))
13		simpr 490	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → (𝐴 +_e 𝐵) = 0)
14	13	oveq1d 7435	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → ((𝐴 +_e 𝐵) +_e -_e𝐴) = (0 +_e -_e𝐴))
15		xpncan 13381	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) → ((𝐵 +_e 𝐴) +_e -_e𝐴) = 𝐵)
16	15	ancoms 464	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → ((𝐵 +_e 𝐴) +_e -_e𝐴) = 𝐵)
17	16	adantr 486	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → ((𝐵 +_e 𝐴) +_e -_e𝐴) = 𝐵)
18	12, 14, 17	3eqtr3d 2804	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → (0 +_e -_e𝐴) = 𝐵)
19	8, 18	eqtr3d 2798	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → -_e𝐴 = 𝐵)
20		xnegeq 13337	. . . . . . 7 ⊢ ( -_e𝐴 = 𝐵 → -_e -_e𝐴 = -_e𝐵)
21	19, 20	syl 18	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → -_e -_e𝐴 = -_e𝐵)
22	5, 21	eqtr3d 2798	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → 𝐴 = -_e𝐵)
23	22	ex 418	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_e 𝐵) = 0 → 𝐴 = -_e𝐵))
24		simpll 779	. . . . . 6 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → 𝐴 = +∞)
25		simplr 781	. . . . . . . . . 10 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 +_e 𝐵) = 0) → 𝐵 ∈ ℝ^)
26	24	oveq1d 7435	. . . . . . . . . . . . 13 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → (𝐴 +_e 𝐵) = (+∞ +_e 𝐵))
27		simpr 490	. . . . . . . . . . . . 13 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → (𝐴 +_e 𝐵) = 0)
28	26, 27	eqtr3d 2798	. . . . . . . . . . . 12 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → (+∞ +_e 𝐵) = 0)
29		0re 11310	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
30		renepnf 11357	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℝ → 0 ≠ +∞)
31	29, 30	mp1i 14	. . . . . . . . . . . 12 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → 0 ≠ +∞)
32	28, 31	eqnetrd 3023	. . . . . . . . . . 11 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → (+∞ +_e 𝐵) ≠ +∞)
33	32	neneqd 2961	. . . . . . . . . 10 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → ¬ (+∞ +_e 𝐵) = +∞)
34		xaddpnf2 13357	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (+∞ +_e 𝐵) = +∞)
35	34	stoic1a 1805	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ ¬ (+∞ +_e 𝐵) = +∞) → ¬ 𝐵 ≠ -∞)
36	25, 33, 35	syl2anc 596	. . . . . . . . 9 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → ¬ 𝐵 ≠ -∞)
37		nne 2960	. . . . . . . . 9 ⊢ (¬ 𝐵 ≠ -∞ ↔ 𝐵 = -∞)
38	36, 37	sylib 221	. . . . . . . 8 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → 𝐵 = -∞)
39		xnegeq 13337	. . . . . . . 8 ⊢ (𝐵 = -∞ → -_e𝐵 = -_e-∞)
40	38, 39	syl 18	. . . . . . 7 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → -_e𝐵 = -_e-∞)
41		xnegmnf 13340	. . . . . . 7 ⊢ -_e-∞ = +∞
42	40, 41	eqtr2di 2813	. . . . . 6 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → +∞ = -_e𝐵)
43	24, 42	eqtrd 2796	. . . . 5 ⊢ (((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → 𝐴 = -_e𝐵)
44	43	ex 418	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_e 𝐵) = 0 → 𝐴 = -_e𝐵))
45		simpll 779	. . . . . 6 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → 𝐴 = -∞)
46		simplr 781	. . . . . . . . . 10 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 +_e 𝐵) = 0) → 𝐵 ∈ ℝ^)
47	45	oveq1d 7435	. . . . . . . . . . . . 13 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → (𝐴 +_e 𝐵) = (-∞ +_e 𝐵))
48		simpr 490	. . . . . . . . . . . . 13 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → (𝐴 +_e 𝐵) = 0)
49	47, 48	eqtr3d 2798	. . . . . . . . . . . 12 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → (-∞ +_e 𝐵) = 0)
50		renemnf 11358	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℝ → 0 ≠ -∞)
51	29, 50	mp1i 14	. . . . . . . . . . . 12 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → 0 ≠ -∞)
52	49, 51	eqnetrd 3023	. . . . . . . . . . 11 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → (-∞ +_e 𝐵) ≠ -∞)
53	52	neneqd 2961	. . . . . . . . . 10 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → ¬ (-∞ +_e 𝐵) = -∞)
54		xaddmnf2 13359	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (-∞ +_e 𝐵) = -∞)
55	54	stoic1a 1805	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ ¬ (-∞ +_e 𝐵) = -∞) → ¬ 𝐵 ≠ +∞)
56	46, 53, 55	syl2anc 596	. . . . . . . . 9 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → ¬ 𝐵 ≠ +∞)
57		nne 2960	. . . . . . . . 9 ⊢ (¬ 𝐵 ≠ +∞ ↔ 𝐵 = +∞)
58	56, 57	sylib 221	. . . . . . . 8 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → 𝐵 = +∞)
59		xnegeq 13337	. . . . . . . 8 ⊢ (𝐵 = +∞ → -_e𝐵 = -_e+∞)
60	58, 59	syl 18	. . . . . . 7 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → -_e𝐵 = -_e+∞)
61		xnegpnf 13339	. . . . . . 7 ⊢ -_e+∞ = -∞
62	60, 61	eqtr2di 2813	. . . . . 6 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → -∞ = -_e𝐵)
63	45, 62	eqtrd 2796	. . . . 5 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 +_e 𝐵) = 0) → 𝐴 = -_e𝐵)
64	63	ex 418	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_e 𝐵) = 0 → 𝐴 = -_e𝐵))
65	23, 44, 64	3jaoian 1457	. . 3 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_e 𝐵) = 0 → 𝐴 = -_e𝐵))
66	1, 65	sylanb 593	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_e 𝐵) = 0 → 𝐴 = -_e𝐵))
67		simpr 490	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 = -_e𝐵) → 𝐴 = -_e𝐵)
68	67	oveq1d 7435	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 = -_e𝐵) → (𝐴 +_e 𝐵) = ( -_e𝐵 +_e 𝐵))
69		xnegcl 13343	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → -_e𝐵 ∈ ℝ^*)
70	69	ad2antlr 740	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 = -_e𝐵) → -_e𝐵 ∈ ℝ^)
71		simplr 781	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 = -_e𝐵) → 𝐵 ∈ ℝ^)
72		xaddcom 13370	. . . . 5 ⊢ (( -_e𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ( -_e𝐵 +_e 𝐵) = (𝐵 +_e -_e𝐵))
73	70, 71, 72	syl2anc 596	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 = -_e𝐵) → ( -_e𝐵 +_e 𝐵) = (𝐵 +_e -_e𝐵))
74		xnegid 13368	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → (𝐵 +_e -_e𝐵) = 0)
75	74	ad2antlr 740	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 = -_e𝐵) → (𝐵 +_e -_e𝐵) = 0)
76	68, 73, 75	3eqtrd 2800	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 = -_e𝐵) → (𝐴 +_e 𝐵) = 0)
77	76	ex 418	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 = -_e𝐵 → (𝐴 +_e 𝐵) = 0))
78	66, 77	impbid 215	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_e 𝐵) = 0 ↔ 𝐴 = -_e𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ w3o 1102 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 (class class class)co 7420 ℝcr 11199 0cc0 11200 +∞cpnf 11340 -∞cmnf 11341 ℝ^*cxr 11342 -_ecxne 13238 +_e cxad 13239

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-po 5559 df-so 5560 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 8001 df-2nd 8002 df-er 8717 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-sub 11543 df-neg 11544 df-xneg 13241 df-xadd 13242

This theorem is used by: xrsinvgval 33569

W3C validator