Theorem aprlring 14523

Description: A ring is a local ring if and only if the relation given by df-apr 14513 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 14521	. 2 ⊢ (𝑅 ∈ LRing → (#r‘𝑅) Ap (Base‘𝑅))
2		aprnzr 14522	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (#r‘𝑅) Ap (Base‘𝑅)) → 𝑅 ∈ NzRing)
3		simplll 535	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ (#r‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → 𝑅 ∈ Ring)
4		simplrl 537	. . . . . . . . . . . . 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 538	. . . . . . . . . . . . 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 2234	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑅) = (Base‘𝑅)
7		eqid 2234	. . . . . . . . . . . . . 14 ⊢ (+g‘𝑅) = (+g‘𝑅)
8	6, 7	ringcom 14259	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)𝑦) = (𝑦(+g‘𝑅)𝑥))
9	3, 4, 5, 8	syl3anc 1274	. . . . . . . . . . . 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 6073	. . . . . . . . . . 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 6073	. . . . . . . . . . 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 14233	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ (#r‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → 𝑅 ∈ Grp)
14		eqid 2234	. . . . . . . . . . . . 13 ⊢ (-g‘𝑅) = (-g‘𝑅)
15	6, 7, 14	grppncan 13888	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → ((𝑦(+g‘𝑅)𝑥)(-g‘𝑅)𝑥) = 𝑦)
16	13, 5, 4, 15	syl3anc 1274	. . . . . . . . . . 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 2275	. . . . . . . . . 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 2235	. . . . . . . . . . 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 2235	. . . . . . . . . . 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 2235	. . . . . . . . . . 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 2235	. . . . . . . . . . 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 2234	. . . . . . . . . . . . 13 ⊢ (1r‘𝑅) = (1r‘𝑅)
24	6, 23	ringidcl 14248	. . . . . . . . . . . 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 14514	. . . . . . . . . 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 2312	. . . . . . . 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 742	. . . . . . 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 2234	. . . . . . . . . . . 12 ⊢ (0g‘𝑅) = (0g‘𝑅)
31	6, 30, 14	grpsubid1 13882	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(-g‘𝑅)(0g‘𝑅)) = 𝑥)
32	13, 4, 31	syl2anc 411	. . . . . . . . . 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 536	. . . . . . . . . . . 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 13826	. . . . . . . . . . . . 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 7576	. . . . . . . . . 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 14514	. . . . . . . . . . 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 2312	. . . . . . . 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 741	. . . . . . 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 13882	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Grp ∧ (1r‘𝑅) ∈ (Base‘𝑅)) → ((1r‘𝑅)(-g‘𝑅)(0g‘𝑅)) = (1r‘𝑅))
48	13, 25, 47	syl2anc 411	. . . . . . . . . 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 2234	. . . . . . . . . . . 12 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
50	49, 23	1unit 14337	. . . . . . . . . . 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 2311	. . . . . . . . 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 14514	. . . . . . . . 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 7577	. . . . . . 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 806	. . . . . 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 2626	. . . 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 14422	. . . 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 417	. . 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)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   = wceq 1398   ∈ wcel 2205  ∀wral 2522   class class class wbr 4114  ‘cfv 5357  (class class class)co 6058   Ap wap 7571  Basecbs 13296  +_gcplusg 13374  0_gc0g 13553  Grpcgrp 13797  -_gcsg 13799  1_rcur 14187  Ringcrg 14224  Unitcui 14316  NzRingcnzr 14409  LRingclring 14420  #_rcapr 14512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-tpos 6489  df-pap 7572  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-iress 13304  df-plusg 13387  df-mulr 13388  df-0g 13555  df-mgm 13653  df-sgrp 13699  df-mnd 13714  df-grp 13800  df-minusg 13801  df-sbg 13802  df-cmn 14087  df-abl 14088  df-mgp 14149  df-ur 14188  df-srg 14192  df-ring 14226  df-oppr 14296  df-dvdsr 14318  df-unit 14319  df-invr 14351  df-dvr 14362  df-nzr 14410  df-lring 14421  df-apr 14513

This theorem is referenced by:  drnglring  14530  opprdrng  14543