Theorem aprlring 14709

Description: A ring is a local ring if and only if the relation given by df-apr 14699 is an apartness relation. (Contributed by Jim Kingdon, 28-May-2026.)

Assertion

Ref	Expression
aprlring	⊢ (𝑅 ∈ Ring → (𝑅 ∈ LRing ↔ ( #r ‘𝑅) Ap (Base‘𝑅)))

Proof of Theorem aprlring

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		aprap 14707	. 2 ⊢ (𝑅 ∈ LRing → ( #r ‘𝑅) Ap (Base‘𝑅))
2		aprnzr 14708	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) → 𝑅 ∈ NzRing)
3		simplll 539	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → 𝑅 ∈ Ring)
4		simplrl 541	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
5		simplrr 542	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → 𝑦 ∈ (Base‘𝑅))
6		eqid 2238	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑅) = (Base‘𝑅)
7		eqid 2238	. . . . . . . . . . . . . 14 ⊢ (+g‘𝑅) = (+g‘𝑅)
8	6, 7	ringcom 14443	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)𝑦) = (𝑦(+g‘𝑅)𝑥))
9	3, 4, 5, 8	syl3anc 1278	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → (𝑥(+g‘𝑅)𝑦) = (𝑦(+g‘𝑅)𝑥))
10	9	oveq1d 6103	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → ((𝑥(+g‘𝑅)𝑦)(-g‘𝑅)𝑥) = ((𝑦(+g‘𝑅)𝑥)(-g‘𝑅)𝑥))
11		simpr 110	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅))
12	11	oveq1d 6103	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → ((𝑥(+g‘𝑅)𝑦)(-g‘𝑅)𝑥) = ((1r‘𝑅)(-g‘𝑅)𝑥))
13	3	ringgrpd 14416	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → 𝑅 ∈ Grp)
14		eqid 2238	. . . . . . . . . . . . 13 ⊢ (-g‘𝑅) = (-g‘𝑅)
15	6, 7, 14	grppncan 13955	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → ((𝑦(+g‘𝑅)𝑥)(-g‘𝑅)𝑥) = 𝑦)
16	13, 5, 4, 15	syl3anc 1278	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → ((𝑦(+g‘𝑅)𝑥)(-g‘𝑅)𝑥) = 𝑦)
17	10, 12, 16	3eqtr3d 2279	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → ((1r‘𝑅)(-g‘𝑅)𝑥) = 𝑦)
18	17	adantr 276	. . . . . . . . 9 ⊢ (((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) ∧ (1r‘𝑅)( #r ‘𝑅)𝑥) → ((1r‘𝑅)(-g‘𝑅)𝑥) = 𝑦)
19		eqidd 2239	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → (Base‘𝑅) = (Base‘𝑅))
20		eqidd 2239	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → ( #r ‘𝑅) = ( #r ‘𝑅))
21		eqidd 2239	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → (-g‘𝑅) = (-g‘𝑅))
22		eqidd 2239	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → (Unit‘𝑅) = (Unit‘𝑅))
23		eqid 2238	. . . . . . . . . . . . 13 ⊢ (1r‘𝑅) = (1r‘𝑅)
24	6, 23	ringidcl 14432	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
25	3, 24	syl 14	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → (1r‘𝑅) ∈ (Base‘𝑅))
26	19, 20, 21, 22, 3, 25, 4	aprval 14700	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → ((1r‘𝑅)( #r ‘𝑅)𝑥 ↔ ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ (Unit‘𝑅)))
27	26	biimpa 296	. . . . . . . . 9 ⊢ (((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) ∧ (1r‘𝑅)( #r ‘𝑅)𝑥) → ((1r‘𝑅)(-g‘𝑅)𝑥) ∈ (Unit‘𝑅))
28	18, 27	eqeltrrd 2316	. . . . . . . 8 ⊢ (((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) ∧ (1r‘𝑅)( #r ‘𝑅)𝑥) → 𝑦 ∈ (Unit‘𝑅))
29	28	olcd 746	. . . . . . 7 ⊢ (((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) ∧ (1r‘𝑅)( #r ‘𝑅)𝑥) → (𝑥 ∈ (Unit‘𝑅) ∨ 𝑦 ∈ (Unit‘𝑅)))
30		eqid 2238	. . . . . . . . . . . 12 ⊢ (0g‘𝑅) = (0g‘𝑅)
31	6, 30, 14	grpsubid1 13949	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(-g‘𝑅)(0g‘𝑅)) = 𝑥)
32	13, 4, 31	syl2anc 415	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → (𝑥(-g‘𝑅)(0g‘𝑅)) = 𝑥)
33	32	adantr 276	. . . . . . . . 9 ⊢ (((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) ∧ (0g‘𝑅)( #r ‘𝑅)𝑥) → (𝑥(-g‘𝑅)(0g‘𝑅)) = 𝑥)
34		simpllr 540	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → ( #r ‘𝑅) Ap (Base‘𝑅))
35	34	adantr 276	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) ∧ (0g‘𝑅)( #r ‘𝑅)𝑥) → ( #r ‘𝑅) Ap (Base‘𝑅))
36	6, 30	grpidcl 13893	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Grp → (0g‘𝑅) ∈ (Base‘𝑅))
37	13, 36	syl 14	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → (0g‘𝑅) ∈ (Base‘𝑅))
38	37	adantr 276	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) ∧ (0g‘𝑅)( #r ‘𝑅)𝑥) → (0g‘𝑅) ∈ (Base‘𝑅))
39	4	adantr 276	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) ∧ (0g‘𝑅)( #r ‘𝑅)𝑥) → 𝑥 ∈ (Base‘𝑅))
40		simpr 110	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) ∧ (0g‘𝑅)( #r ‘𝑅)𝑥) → (0g‘𝑅)( #r ‘𝑅)𝑥)
41	35, 38, 39, 40	papsym 7618	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) ∧ (0g‘𝑅)( #r ‘𝑅)𝑥) → 𝑥( #r ‘𝑅)(0g‘𝑅))
42	19, 20, 21, 22, 3, 4, 37	aprval 14700	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → (𝑥( #r ‘𝑅)(0g‘𝑅) ↔ (𝑥(-g‘𝑅)(0g‘𝑅)) ∈ (Unit‘𝑅)))
43	42	adantr 276	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) ∧ (0g‘𝑅)( #r ‘𝑅)𝑥) → (𝑥( #r ‘𝑅)(0g‘𝑅) ↔ (𝑥(-g‘𝑅)(0g‘𝑅)) ∈ (Unit‘𝑅)))
44	41, 43	mpbid 147	. . . . . . . . 9 ⊢ (((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) ∧ (0g‘𝑅)( #r ‘𝑅)𝑥) → (𝑥(-g‘𝑅)(0g‘𝑅)) ∈ (Unit‘𝑅))
45	33, 44	eqeltrrd 2316	. . . . . . . 8 ⊢ (((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) ∧ (0g‘𝑅)( #r ‘𝑅)𝑥) → 𝑥 ∈ (Unit‘𝑅))
46	45	orcd 745	. . . . . . 7 ⊢ (((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) ∧ (0g‘𝑅)( #r ‘𝑅)𝑥) → (𝑥 ∈ (Unit‘𝑅) ∨ 𝑦 ∈ (Unit‘𝑅)))
47	6, 30, 14	grpsubid1 13949	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Grp ∧ (1r‘𝑅) ∈ (Base‘𝑅)) → ((1r‘𝑅)(-g‘𝑅)(0g‘𝑅)) = (1r‘𝑅))
48	13, 25, 47	syl2anc 415	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → ((1r‘𝑅)(-g‘𝑅)(0g‘𝑅)) = (1r‘𝑅))
49		eqid 2238	. . . . . . . . . . . 12 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
50	49, 23	1unit 14523	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Unit‘𝑅))
51	3, 50	syl 14	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → (1r‘𝑅) ∈ (Unit‘𝑅))
52	48, 51	eqeltrd 2315	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → ((1r‘𝑅)(-g‘𝑅)(0g‘𝑅)) ∈ (Unit‘𝑅))
53	19, 20, 21, 22, 3, 25, 37	aprval 14700	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → ((1r‘𝑅)( #r ‘𝑅)(0g‘𝑅) ↔ ((1r‘𝑅)(-g‘𝑅)(0g‘𝑅)) ∈ (Unit‘𝑅)))
54	52, 53	mpbird 167	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → (1r‘𝑅)( #r ‘𝑅)(0g‘𝑅))
55	34, 25, 37, 54, 4	papcotr 7619	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → ((1r‘𝑅)( #r ‘𝑅)𝑥 ∨ (0g‘𝑅)( #r ‘𝑅)𝑥))
56	29, 46, 55	mpjaodan 810	. . . . . 6 ⊢ ((((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → (𝑥 ∈ (Unit‘𝑅) ∨ 𝑦 ∈ (Unit‘𝑅)))
57	56	ex 115	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((𝑥(+g‘𝑅)𝑦) = (1r‘𝑅) → (𝑥 ∈ (Unit‘𝑅) ∨ 𝑦 ∈ (Unit‘𝑅))))
58	57	ralrimivva 2632	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(+g‘𝑅)𝑦) = (1r‘𝑅) → (𝑥 ∈ (Unit‘𝑅) ∨ 𝑦 ∈ (Unit‘𝑅))))
59	6, 7, 23, 49	islring 14608	. . . 4 ⊢ (𝑅 ∈ LRing ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(+g‘𝑅)𝑦) = (1r‘𝑅) → (𝑥 ∈ (Unit‘𝑅) ∨ 𝑦 ∈ (Unit‘𝑅)))))
60	2, 58, 59	sylanbrc 421	. . 3 ⊢ ((𝑅 ∈ Ring ∧ ( #r ‘𝑅) Ap (Base‘𝑅)) → 𝑅 ∈ LRing)
61	60	ex 115	. 2 ⊢ (𝑅 ∈ Ring → (( #r ‘𝑅) Ap (Base‘𝑅) → 𝑅 ∈ LRing))
62	1, 61	impbid2 143	1 ⊢ (𝑅 ∈ Ring → (𝑅 ∈ LRing ↔ ( #r ‘𝑅) Ap (Base‘𝑅)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720   = wceq 1402   ∈ wcel 2209  ∀wral 2528   class class class wbr 4131  ‘cfv 5380  (class class class)co 6088   Ap wap 7613  Basecbs 13410  +_gcplusg 13490  0_gc0g 13669  Grpcgrp 13864  -_gcsg 13866  1_rcur 14369  Ringcrg 14407  Unitcui 14502  NzRingcnzr 14595  LRingclring 14606   #_r capr 14698

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-pre-ltirr 8297  ax-pre-lttrn 8299  ax-pre-ltadd 8301

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-tpos 6520  df-pap 7614  df-pnf 8368  df-mnf 8369  df-ltxr 8371  df-inn 9314  df-2 9372  df-3 9373  df-ndx 13413  df-slot 13414  df-base 13416  df-sets 13417  df-iress 13418  df-plusg 13503  df-mulr 13504  df-0g 13671  df-mgm 13735  df-sgrp 13776  df-mnd 13789  df-grp 13867  df-minusg 13868  df-sbg 13869  df-cmn 14179  df-abl 14180  df-mgp 14325  df-ur 14370  df-srg 14375  df-ring 14409  df-oppr 14482  df-dvdsr 14504  df-unit 14505  df-invr 14537  df-dvr 14548  df-nzr 14596  df-lring 14607  df-apr 14699

This theorem is used by:  drnglring  14716  opprdrng  14729