Theorem idlinsubrg 31743

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 4173	. . . 4 ⊢ (𝐼 ∩ 𝐴) ⊆ 𝐴
2		idlinsubrg.s	. . . . 5 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
3	2	subrgbas 20112	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
4	1, 3	sseqtrid 3982	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝐼 ∩ 𝐴) ⊆ (Base‘𝑆))
5	4	adantr 481	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (𝐼 ∩ 𝐴) ⊆ (Base‘𝑆))
6		subrgrcl 20108	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
7		idlinsubrg.u	. . . . . 6 ⊢ 𝑈 = (LIdeal‘𝑅)
8		eqid 2736	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
9	7, 8	lidl0cl 20563	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (0g‘𝑅) ∈ 𝐼)
10	6, 9	sylan 580	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (0g‘𝑅) ∈ 𝐼)
11		subrgsubg 20109	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
12		subgsubm 18850	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → 𝐴 ∈ (SubMnd‘𝑅))
13	8	subm0cl 18524	. . . . . 6 ⊢ (𝐴 ∈ (SubMnd‘𝑅) → (0g‘𝑅) ∈ 𝐴)
14	11, 12, 13	3syl 18	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (0g‘𝑅) ∈ 𝐴)
15	14	adantr 481	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (0g‘𝑅) ∈ 𝐴)
16	10, 15	elind 4138	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (0g‘𝑅) ∈ (𝐼 ∩ 𝐴))
17	16	ne0d 4279	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (𝐼 ∩ 𝐴) ≠ ∅)
18		eqid 2736	. . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑅)
19	2, 18	ressplusg 17074	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (+g‘𝑅) = (+g‘𝑆))
20		eqid 2736	. . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅)
21	2, 20	ressmulr 17091	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆))
22	21	oveqd 7333	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥(.r‘𝑅)𝑎) = (𝑥(.r‘𝑆)𝑎))
23		eqidd 2737	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑏 = 𝑏)
24	19, 22, 23	oveq123d 7337	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((𝑥(.r‘𝑅)𝑎)(+g‘𝑅)𝑏) = ((𝑥(.r‘𝑆)𝑎)(+g‘𝑆)𝑏))
25	24	ad4antr 729	. . . . . 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 729	. . . . . . . 8 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝑅 ∈ Ring)
27		simp-4r 781	. . . . . . . 8 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝐼 ∈ 𝑈)
28		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑅) = (Base‘𝑅)
29	28	subrgss 20104	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
30	3, 29	eqsstrrd 3969	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (Base‘𝑆) ⊆ (Base‘𝑅))
31	30	adantr 481	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (Base‘𝑆) ⊆ (Base‘𝑅))
32	31	sselda 3930	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑅))
33	32	ad2antrr 723	. . . . . . . . 9 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝑥 ∈ (Base‘𝑅))
34		inss1 4172	. . . . . . . . . . . 12 ⊢ (𝐼 ∩ 𝐴) ⊆ 𝐼
35	34	a1i 11	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐼 ∩ 𝐴) ⊆ 𝐼)
36	35	sselda 3930	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) → 𝑎 ∈ 𝐼)
37	36	adantr 481	. . . . . . . . 9 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝑎 ∈ 𝐼)
38	7, 28, 20	lidlmcl 20568	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑎 ∈ 𝐼)) → (𝑥(.r‘𝑅)𝑎) ∈ 𝐼)
39	26, 27, 33, 37, 38	syl22anc 836	. . . . . . . 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 3930	. . . . . . . 8 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝑏 ∈ 𝐼)
42	7, 18	lidlacl 20564	. . . . . . . 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 836	. . . . . . 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 780	. . . . . . . 8 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝐴 ∈ (SubRing‘𝑅))
45		simpr 485	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
46	3	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → 𝐴 = (Base‘𝑆))
47	45, 46	eleqtrrd 2840	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ 𝐴)
48	47	ad2antrr 723	. . . . . . . . 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 3930	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) → 𝑎 ∈ 𝐴)
51	50	adantr 481	. . . . . . . . 9 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝑎 ∈ 𝐴)
52	20	subrgmcl 20115	. . . . . . . . 9 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝐴 ∧ 𝑎 ∈ 𝐴) → (𝑥(.r‘𝑅)𝑎) ∈ 𝐴)
53	44, 48, 51, 52	syl3anc 1370	. . . . . . . 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 3930	. . . . . . . 8 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝑏 ∈ 𝐴)
56	18	subrgacl 20114	. . . . . . . 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 1370	. . . . . . 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 4138	. . . . . 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 2838	. . . . 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 467	. . . 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 3193	. . 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 3139	. 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 2736	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
65		eqid 2736	. . 3 ⊢ (+g‘𝑆) = (+g‘𝑆)
66		eqid 2736	. . 3 ⊢ (.r‘𝑆) = (.r‘𝑆)
67	63, 64, 65, 66	islidl 20562	. 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 1342	1 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (𝐼 ∩ 𝐴) ∈ 𝑉)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1540   ∈ wcel 2105   ≠ wne 2940  ∀wral 3061   ∩ cin 3895   ⊆ wss 3896  ∅c0 4266  ‘cfv 6465  (class class class)co 7316  Basecbs 16986   ↾_s cress 17015  +_gcplusg 17036  ._rcmulr 17037  0_gc0g 17224  SubMndcsubmnd 18503  SubGrpcsubg 18822  Ringcrg 19855  SubRingcsubrg 20099  LIdealclidl 20512

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5223  ax-sep 5237  ax-nul 5244  ax-pow 5302  ax-pr 5366  ax-un 7629  ax-cnex 11006  ax-resscn 11007  ax-1cn 11008  ax-icn 11009  ax-addcl 11010  ax-addrcl 11011  ax-mulcl 11012  ax-mulrcl 11013  ax-mulcom 11014  ax-addass 11015  ax-mulass 11016  ax-distr 11017  ax-i2m1 11018  ax-1ne0 11019  ax-1rid 11020  ax-rnegex 11021  ax-rrecex 11022  ax-cnre 11023  ax-pre-lttri 11024  ax-pre-lttrn 11025  ax-pre-ltadd 11026  ax-pre-mulgt0 11027

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3442  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-pss 3915  df-nul 4267  df-if 4471  df-pw 4546  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4850  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5170  df-tr 5204  df-id 5506  df-eprel 5512  df-po 5520  df-so 5521  df-fr 5562  df-we 5564  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-pred 6224  df-ord 6291  df-on 6292  df-lim 6293  df-suc 6294  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473  df-riota 7273  df-ov 7319  df-oprab 7320  df-mpo 7321  df-om 7759  df-1st 7877  df-2nd 7878  df-frecs 8145  df-wrecs 8176  df-recs 8250  df-rdg 8289  df-er 8547  df-en 8783  df-dom 8784  df-sdom 8785  df-pnf 11090  df-mnf 11091  df-xr 11092  df-ltxr 11093  df-le 11094  df-sub 11286  df-neg 11287  df-nn 12053  df-2 12115  df-3 12116  df-4 12117  df-5 12118  df-6 12119  df-7 12120  df-8 12121  df-sets 16939  df-slot 16957  df-ndx 16969  df-base 16987  df-ress 17016  df-plusg 17049  df-mulr 17050  df-sca 17052  df-vsca 17053  df-ip 17054  df-0g 17226  df-mgm 18400  df-sgrp 18449  df-mnd 18460  df-submnd 18505  df-grp 18653  df-minusg 18654  df-sbg 18655  df-subg 18825  df-mgp 19793  df-ur 19810  df-ring 19857  df-subrg 20101  df-lmod 20205  df-lss 20274  df-sra 20514  df-rgmod 20515  df-lidl 20516

This theorem is referenced by: (None)