Theorem idlinsubrg 31510

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 4160	. . . 4 ⊢ (𝐼 ∩ 𝐴) ⊆ 𝐴
2		idlinsubrg.s	. . . . 5 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
3	2	subrgbas 19948	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
4	1, 3	sseqtrid 3969	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝐼 ∩ 𝐴) ⊆ (Base‘𝑆))
5	4	adantr 480	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (𝐼 ∩ 𝐴) ⊆ (Base‘𝑆))
6		subrgrcl 19944	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
7		idlinsubrg.u	. . . . . 6 ⊢ 𝑈 = (LIdeal‘𝑅)
8		eqid 2738	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
9	7, 8	lidl0cl 20396	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (0g‘𝑅) ∈ 𝐼)
10	6, 9	sylan 579	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (0g‘𝑅) ∈ 𝐼)
11		subrgsubg 19945	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
12		subgsubm 18692	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → 𝐴 ∈ (SubMnd‘𝑅))
13	8	subm0cl 18365	. . . . . 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 4124	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (0g‘𝑅) ∈ (𝐼 ∩ 𝐴))
17	16	ne0d 4266	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (𝐼 ∩ 𝐴) ≠ ∅)
18		eqid 2738	. . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑅)
19	2, 18	ressplusg 16926	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (+g‘𝑅) = (+g‘𝑆))
20		eqid 2738	. . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅)
21	2, 20	ressmulr 16943	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆))
22	21	oveqd 7272	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥(.r‘𝑅)𝑎) = (𝑥(.r‘𝑆)𝑎))
23		eqidd 2739	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑏 = 𝑏)
24	19, 22, 23	oveq123d 7276	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((𝑥(.r‘𝑅)𝑎)(+g‘𝑅)𝑏) = ((𝑥(.r‘𝑆)𝑎)(+g‘𝑆)𝑏))
25	24	ad4antr 728	. . . . . 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 728	. . . . . . . 8 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝑅 ∈ Ring)
27		simp-4r 780	. . . . . . . 8 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝐼 ∈ 𝑈)
28		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑅) = (Base‘𝑅)
29	28	subrgss 19940	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
30	3, 29	eqsstrrd 3956	. . . . . . . . . . . 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 3917	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑅))
33	32	ad2antrr 722	. . . . . . . . 9 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝑥 ∈ (Base‘𝑅))
34		inss1 4159	. . . . . . . . . . . 12 ⊢ (𝐼 ∩ 𝐴) ⊆ 𝐼
35	34	a1i 11	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐼 ∩ 𝐴) ⊆ 𝐼)
36	35	sselda 3917	. . . . . . . . . 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 20401	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑎 ∈ 𝐼)) → (𝑥(.r‘𝑅)𝑎) ∈ 𝐼)
39	26, 27, 33, 37, 38	syl22anc 835	. . . . . . . 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 3917	. . . . . . . 8 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝑏 ∈ 𝐼)
42	7, 18	lidlacl 20397	. . . . . . . 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 835	. . . . . . 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 779	. . . . . . . 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 722	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → 𝐴 = (Base‘𝑆))
47	45, 46	eleqtrrd 2842	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ 𝐴)
48	47	ad2antrr 722	. . . . . . . . 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 3917	. . . . . . . . . 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 19951	. . . . . . . . 9 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝐴 ∧ 𝑎 ∈ 𝐴) → (𝑥(.r‘𝑅)𝑎) ∈ 𝐴)
53	44, 48, 51, 52	syl3anc 1369	. . . . . . . 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 3917	. . . . . . . 8 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝑏 ∈ 𝐴)
56	18	subrgacl 19950	. . . . . . . 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 1369	. . . . . . 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 4124	. . . . . 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 2840	. . . . 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 3114	. . 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 3107	. 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 2738	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
65		eqid 2738	. . 3 ⊢ (+g‘𝑆) = (+g‘𝑆)
66		eqid 2738	. . 3 ⊢ (.r‘𝑆) = (.r‘𝑆)
67	63, 64, 65, 66	islidl 20395	. 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 1341	1 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (𝐼 ∩ 𝐴) ∈ 𝑉)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  ‘cfv 6418  (class class class)co 7255  Basecbs 16840   ↾_s cress 16867  +_gcplusg 16888  ._rcmulr 16889  0_gc0g 17067  SubMndcsubmnd 18344  SubGrpcsubg 18664  Ringcrg 19698  SubRingcsubrg 19935  LIdealclidl 20347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-sca 16904  df-vsca 16905  df-ip 16906  df-0g 17069  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-submnd 18346  df-grp 18495  df-minusg 18496  df-sbg 18497  df-subg 18667  df-mgp 19636  df-ur 19653  df-ring 19700  df-subrg 19937  df-lmod 20040  df-lss 20109  df-sra 20349  df-rgmod 20350  df-lidl 20351

This theorem is referenced by: (None)