Theorem aprlring 14426

Description: A ring is a local ring if and only if the relation given by df-apr 14419 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 14424	. 2 ⊢ (𝑅 ∈ LRing → (#r‘𝑅) Ap (Base‘𝑅))
2		aprnzr 14425	. . . 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 2232	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑅) = (Base‘𝑅)
7		eqid 2232	. . . . . . . . . . . . . 14 ⊢ (+g‘𝑅) = (+g‘𝑅)
8	6, 7	ringcom 14167	. . . . . . . . . . . . 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 6064	. . . . . . . . . . 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 6064	. . . . . . . . . . 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 14141	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ (#r‘𝑅) Ap (Base‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) ∧ (𝑥(+g‘𝑅)𝑦) = (1r‘𝑅)) → 𝑅 ∈ Grp)
14		eqid 2232	. . . . . . . . . . . . 13 ⊢ (-g‘𝑅) = (-g‘𝑅)
15	6, 7, 14	grppncan 13796	. . . . . . . . . . . 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 2273	. . . . . . . . . 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 2233	. . . . . . . . . . 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 2233	. . . . . . . . . . 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 2233	. . . . . . . . . . 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 2233	. . . . . . . . . . 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 2232	. . . . . . . . . . . . 13 ⊢ (1r‘𝑅) = (1r‘𝑅)
24	6, 23	ringidcl 14156	. . . . . . . . . . . 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 14420	. . . . . . . . . 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 2310	. . . . . . . 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 2232	. . . . . . . . . . . 12 ⊢ (0g‘𝑅) = (0g‘𝑅)
31	6, 30, 14	grpsubid1 13790	. . . . . . . . . . 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 13734	. . . . . . . . . . . . 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 7560	. . . . . . . . . 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 14420	. . . . . . . . . . 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 2310	. . . . . . . 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 13790	. . . . . . . . . . 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 2232	. . . . . . . . . . . 12 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
50	49, 23	1unit 14244	. . . . . . . . . . 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 2309	. . . . . . . . 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 14420	. . . . . . . . 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 7561	. . . . . . 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 2624	. . . 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 14329	. . . 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 2203  ∀wral 2520   class class class wbr 4108  ‘cfv 5351  (class class class)co 6049   Ap wap 7557  Basecbs 13204  +_gcplusg 13282  0_gc0g 13461  Grpcgrp 13705  -_gcsg 13707  1_rcur 14095  Ringcrg 14132  Unitcui 14223  NzRingcnzr 14316  LRingclring 14327  #_rcapr 14418

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-addass 8228  ax-i2m1 8231  ax-0lt1 8232  ax-0id 8234  ax-rnegex 8235  ax-pre-ltirr 8238  ax-pre-lttrn 8240  ax-pre-ltadd 8242

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-tpos 6475  df-pap 7558  df-pnf 8309  df-mnf 8310  df-ltxr 8312  df-inn 9237  df-2 9295  df-3 9296  df-ndx 13207  df-slot 13208  df-base 13210  df-sets 13211  df-iress 13212  df-plusg 13295  df-mulr 13296  df-0g 13463  df-mgm 13561  df-sgrp 13607  df-mnd 13622  df-grp 13708  df-minusg 13709  df-sbg 13710  df-cmn 13995  df-abl 13996  df-mgp 14057  df-ur 14096  df-srg 14100  df-ring 14134  df-oppr 14204  df-dvdsr 14225  df-unit 14226  df-invr 14258  df-dvr 14269  df-nzr 14317  df-lring 14328  df-apr 14419

This theorem is referenced by: (None)