Theorem idlinsubrg 33461

Description: The intersection between an ideal and a subring is an ideal of the subring. (Contributed by Thierry Arnoux, 6-Jul-2024.)

Hypotheses

Ref	Expression
idlinsubrg.s	⊢ 𝑆 = (𝑅 ↾s 𝐴)
idlinsubrg.u	⊢ 𝑈 = (LIdeal‘𝑅)
idlinsubrg.v	⊢ 𝑉 = (LIdeal‘𝑆)

Assertion

Ref	Expression
idlinsubrg	⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (𝐼 ∩ 𝐴) ∈ 𝑉)

Proof of Theorem idlinsubrg

Dummy variables 𝑎 𝑏 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss2 4188	. . . 4 ⊢ (𝐼 ∩ 𝐴) ⊆ 𝐴
2		idlinsubrg.s	. . . . 5 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
3	2	subrgbas 20512	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
4	1, 3	sseqtrid 3974	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝐼 ∩ 𝐴) ⊆ (Base‘𝑆))
5	4	adantr 480	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (𝐼 ∩ 𝐴) ⊆ (Base‘𝑆))
6		subrgrcl 20507	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
7		idlinsubrg.u	. . . . . 6 ⊢ 𝑈 = (LIdeal‘𝑅)
8		eqid 2734	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
9	7, 8	lidl0cl 21173	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (0g‘𝑅) ∈ 𝐼)
10	6, 9	sylan 580	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (0g‘𝑅) ∈ 𝐼)
11		subrgsubg 20508	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
12		subgsubm 19076	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → 𝐴 ∈ (SubMnd‘𝑅))
13	8	subm0cl 18734	. . . . . 6 ⊢ (𝐴 ∈ (SubMnd‘𝑅) → (0g‘𝑅) ∈ 𝐴)
14	11, 12, 13	3syl 18	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (0g‘𝑅) ∈ 𝐴)
15	14	adantr 480	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (0g‘𝑅) ∈ 𝐴)
16	10, 15	elind 4150	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (0g‘𝑅) ∈ (𝐼 ∩ 𝐴))
17	16	ne0d 4292	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (𝐼 ∩ 𝐴) ≠ ∅)
18		eqid 2734	. . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑅)
19	2, 18	ressplusg 17209	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (+g‘𝑅) = (+g‘𝑆))
20		eqid 2734	. . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅)
21	2, 20	ressmulr 17225	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆))
22	21	oveqd 7373	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥(.r‘𝑅)𝑎) = (𝑥(.r‘𝑆)𝑎))
23		eqidd 2735	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑏 = 𝑏)
24	19, 22, 23	oveq123d 7377	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((𝑥(.r‘𝑅)𝑎)(+g‘𝑅)𝑏) = ((𝑥(.r‘𝑆)𝑎)(+g‘𝑆)𝑏))
25	24	ad4antr 732	. . . . . 6 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → ((𝑥(.r‘𝑅)𝑎)(+g‘𝑅)𝑏) = ((𝑥(.r‘𝑆)𝑎)(+g‘𝑆)𝑏))
26	6	ad4antr 732	. . . . . . . 8 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝑅 ∈ Ring)
27		simp-4r 783	. . . . . . . 8 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝐼 ∈ 𝑈)
28		eqid 2734	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑅) = (Base‘𝑅)
29	28	subrgss 20503	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
30	3, 29	eqsstrrd 3967	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (Base‘𝑆) ⊆ (Base‘𝑅))
31	30	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (Base‘𝑆) ⊆ (Base‘𝑅))
32	31	sselda 3931	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑅))
33	32	ad2antrr 726	. . . . . . . . 9 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝑥 ∈ (Base‘𝑅))
34		inss1 4187	. . . . . . . . . . . 12 ⊢ (𝐼 ∩ 𝐴) ⊆ 𝐼
35	34	a1i 11	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐼 ∩ 𝐴) ⊆ 𝐼)
36	35	sselda 3931	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) → 𝑎 ∈ 𝐼)
37	36	adantr 480	. . . . . . . . 9 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝑎 ∈ 𝐼)
38	7, 28, 20	lidlmcl 21178	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑎 ∈ 𝐼)) → (𝑥(.r‘𝑅)𝑎) ∈ 𝐼)
39	26, 27, 33, 37, 38	syl22anc 838	. . . . . . . 8 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → (𝑥(.r‘𝑅)𝑎) ∈ 𝐼)
40	34	a1i 11	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) → (𝐼 ∩ 𝐴) ⊆ 𝐼)
41	40	sselda 3931	. . . . . . . 8 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝑏 ∈ 𝐼)
42	7, 18	lidlacl 21174	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ ((𝑥(.r‘𝑅)𝑎) ∈ 𝐼 ∧ 𝑏 ∈ 𝐼)) → ((𝑥(.r‘𝑅)𝑎)(+g‘𝑅)𝑏) ∈ 𝐼)
43	26, 27, 39, 41, 42	syl22anc 838	. . . . . . 7 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → ((𝑥(.r‘𝑅)𝑎)(+g‘𝑅)𝑏) ∈ 𝐼)
44		simp-4l 782	. . . . . . . 8 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝐴 ∈ (SubRing‘𝑅))
45		simpr 484	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
46	3	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → 𝐴 = (Base‘𝑆))
47	45, 46	eleqtrrd 2837	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ 𝐴)
48	47	ad2antrr 726	. . . . . . . . 9 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝑥 ∈ 𝐴)
49	1	a1i 11	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐼 ∩ 𝐴) ⊆ 𝐴)
50	49	sselda 3931	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) → 𝑎 ∈ 𝐴)
51	50	adantr 480	. . . . . . . . 9 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝑎 ∈ 𝐴)
52	20	subrgmcl 20515	. . . . . . . . 9 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝐴 ∧ 𝑎 ∈ 𝐴) → (𝑥(.r‘𝑅)𝑎) ∈ 𝐴)
53	44, 48, 51, 52	syl3anc 1373	. . . . . . . 8 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → (𝑥(.r‘𝑅)𝑎) ∈ 𝐴)
54	1	a1i 11	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) → (𝐼 ∩ 𝐴) ⊆ 𝐴)
55	54	sselda 3931	. . . . . . . 8 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝑏 ∈ 𝐴)
56	18	subrgacl 20514	. . . . . . . 8 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑥(.r‘𝑅)𝑎) ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((𝑥(.r‘𝑅)𝑎)(+g‘𝑅)𝑏) ∈ 𝐴)
57	44, 53, 55, 56	syl3anc 1373	. . . . . . 7 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → ((𝑥(.r‘𝑅)𝑎)(+g‘𝑅)𝑏) ∈ 𝐴)
58	43, 57	elind 4150	. . . . . 6 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → ((𝑥(.r‘𝑅)𝑎)(+g‘𝑅)𝑏) ∈ (𝐼 ∩ 𝐴))
59	25, 58	eqeltrrd 2835	. . . . 5 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → ((𝑥(.r‘𝑆)𝑎)(+g‘𝑆)𝑏) ∈ (𝐼 ∩ 𝐴))
60	59	anasss 466	. . . 4 ⊢ ((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ (𝑎 ∈ (𝐼 ∩ 𝐴) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴))) → ((𝑥(.r‘𝑆)𝑎)(+g‘𝑆)𝑏) ∈ (𝐼 ∩ 𝐴))
61	60	ralrimivva 3177	. . 3 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → ∀𝑎 ∈ (𝐼 ∩ 𝐴)∀𝑏 ∈ (𝐼 ∩ 𝐴)((𝑥(.r‘𝑆)𝑎)(+g‘𝑆)𝑏) ∈ (𝐼 ∩ 𝐴))
62	61	ralrimiva 3126	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → ∀𝑥 ∈ (Base‘𝑆)∀𝑎 ∈ (𝐼 ∩ 𝐴)∀𝑏 ∈ (𝐼 ∩ 𝐴)((𝑥(.r‘𝑆)𝑎)(+g‘𝑆)𝑏) ∈ (𝐼 ∩ 𝐴))
63		idlinsubrg.v	. . 3 ⊢ 𝑉 = (LIdeal‘𝑆)
64		eqid 2734	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
65		eqid 2734	. . 3 ⊢ (+g‘𝑆) = (+g‘𝑆)
66		eqid 2734	. . 3 ⊢ (.r‘𝑆) = (.r‘𝑆)
67	63, 64, 65, 66	islidl 21168	. 2 ⊢ ((𝐼 ∩ 𝐴) ∈ 𝑉 ↔ ((𝐼 ∩ 𝐴) ⊆ (Base‘𝑆) ∧ (𝐼 ∩ 𝐴) ≠ ∅ ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑎 ∈ (𝐼 ∩ 𝐴)∀𝑏 ∈ (𝐼 ∩ 𝐴)((𝑥(.r‘𝑆)𝑎)(+g‘𝑆)𝑏) ∈ (𝐼 ∩ 𝐴)))
68	5, 17, 62, 67	syl3anbrc 1344	1 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (𝐼 ∩ 𝐴) ∈ 𝑉)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  ∀wral 3049   ∩ cin 3898   ⊆ wss 3899  ∅c0 4283  ‘cfv 6490  (class class class)co 7356  Basecbs 17134   ↾_s cress 17155  +_gcplusg 17175  ._rcmulr 17176  0_gc0g 17357  SubMndcsubmnd 18705  SubGrpcsubg 19048  Ringcrg 20166  SubRingcsubrg 20500  LIdealclidl 21159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8633  df-en 8882  df-dom 8883  df-sdom 8884  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-nn 12144  df-2 12206  df-3 12207  df-4 12208  df-5 12209  df-6 12210  df-7 12211  df-8 12212  df-sets 17089  df-slot 17107  df-ndx 17119  df-base 17135  df-ress 17156  df-plusg 17188  df-mulr 17189  df-sca 17191  df-vsca 17192  df-ip 17193  df-0g 17359  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-submnd 18707  df-grp 18864  df-minusg 18865  df-sbg 18866  df-subg 19051  df-cmn 19709  df-abl 19710  df-mgp 20074  df-rng 20086  df-ur 20115  df-ring 20168  df-subrng 20477  df-subrg 20501  df-lmod 20811  df-lss 20881  df-sra 21123  df-rgmod 21124  df-lidl 21161

This theorem is referenced by: (None)