idlinsubrg - Mathbox for Thierry Arnoux

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Theorem idlinsubrg 33172

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 4232	. . . 4 ⊢ (𝐼 ∩ 𝐴) ⊆ 𝐴
2		idlinsubrg.s	. . . . 5 ⊢ 𝑆 = (𝑅 ↾_s 𝐴)
3	2	subrgbas 20527	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
4	1, 3	sseqtrid 4034	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝐼 ∩ 𝐴) ⊆ (Base‘𝑆))
5	4	adantr 479	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (𝐼 ∩ 𝐴) ⊆ (Base‘𝑆))
6		subrgrcl 20522	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
7		idlinsubrg.u	. . . . . 6 ⊢ 𝑈 = (LIdeal‘𝑅)
8		eqid 2728	. . . . . 6 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
9	7, 8	lidl0cl 21123	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (0_g‘𝑅) ∈ 𝐼)
10	6, 9	sylan 578	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (0_g‘𝑅) ∈ 𝐼)
11		subrgsubg 20523	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
12		subgsubm 19110	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → 𝐴 ∈ (SubMnd‘𝑅))
13	8	subm0cl 18770	. . . . . 6 ⊢ (𝐴 ∈ (SubMnd‘𝑅) → (0_g‘𝑅) ∈ 𝐴)
14	11, 12, 13	3syl 18	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (0_g‘𝑅) ∈ 𝐴)
15	14	adantr 479	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (0_g‘𝑅) ∈ 𝐴)
16	10, 15	elind 4196	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (0_g‘𝑅) ∈ (𝐼 ∩ 𝐴))
17	16	ne0d 4339	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (𝐼 ∩ 𝐴) ≠ ∅)
18		eqid 2728	. . . . . . . . 9 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
19	2, 18	ressplusg 17278	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (+_g‘𝑅) = (+_g‘𝑆))
20		eqid 2728	. . . . . . . . . 10 ⊢ (._r‘𝑅) = (._r‘𝑅)
21	2, 20	ressmulr 17295	. . . . . . . . 9 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (._r‘𝑅) = (._r‘𝑆))
22	21	oveqd 7443	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑥(._r‘𝑅)𝑎) = (𝑥(._r‘𝑆)𝑎))
23		eqidd 2729	. . . . . . . 8 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑏 = 𝑏)
24	19, 22, 23	oveq123d 7447	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) = ((𝑥(._r‘𝑆)𝑎)(+_g‘𝑆)𝑏))
25	24	ad4antr 730	. . . . . 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 730	. . . . . . . 8 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝑅 ∈ Ring)
27		simp-4r 782	. . . . . . . 8 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝐼 ∈ 𝑈)
28		eqid 2728	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑅) = (Base‘𝑅)
29	28	subrgss 20518	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
30	3, 29	eqsstrrd 4021	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (Base‘𝑆) ⊆ (Base‘𝑅))
31	30	adantr 479	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (Base‘𝑆) ⊆ (Base‘𝑅))
32	31	sselda 3982	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑅))
33	32	ad2antrr 724	. . . . . . . . 9 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝑥 ∈ (Base‘𝑅))
34		inss1 4231	. . . . . . . . . . . 12 ⊢ (𝐼 ∩ 𝐴) ⊆ 𝐼
35	34	a1i 11	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐼 ∩ 𝐴) ⊆ 𝐼)
36	35	sselda 3982	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) → 𝑎 ∈ 𝐼)
37	36	adantr 479	. . . . . . . . 9 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝑎 ∈ 𝐼)
38	7, 28, 20	lidlmcl 21128	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑎 ∈ 𝐼)) → (𝑥(._r‘𝑅)𝑎) ∈ 𝐼)
39	26, 27, 33, 37, 38	syl22anc 837	. . . . . . . 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 3982	. . . . . . . 8 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝑏 ∈ 𝐼)
42	7, 18	lidlacl 21124	. . . . . . . 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 837	. . . . . . 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 781	. . . . . . . 8 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝐴 ∈ (SubRing‘𝑅))
45		simpr 483	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
46	3	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → 𝐴 = (Base‘𝑆))
47	45, 46	eleqtrrd 2832	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) → 𝑥 ∈ 𝐴)
48	47	ad2antrr 724	. . . . . . . . 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 3982	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) → 𝑎 ∈ 𝐴)
51	50	adantr 479	. . . . . . . . 9 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝑎 ∈ 𝐴)
52	20	subrgmcl 20530	. . . . . . . . 9 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝐴 ∧ 𝑎 ∈ 𝐴) → (𝑥(._r‘𝑅)𝑎) ∈ 𝐴)
53	44, 48, 51, 52	syl3anc 1368	. . . . . . . 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 3982	. . . . . . . 8 ⊢ (((((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑎 ∈ (𝐼 ∩ 𝐴)) ∧ 𝑏 ∈ (𝐼 ∩ 𝐴)) → 𝑏 ∈ 𝐴)
56	18	subrgacl 20529	. . . . . . . 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 1368	. . . . . . 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 4196	. . . . . 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 2830	. . . . 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 465	. . . 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 3198	. . 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 3143	. 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 2728	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
65		eqid 2728	. . 3 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
66		eqid 2728	. . 3 ⊢ (._r‘𝑆) = (._r‘𝑆)
67	63, 64, 65, 66	islidl 21118	. 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 1340	1 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (𝐼 ∩ 𝐴) ∈ 𝑉)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∀wral 3058 ∩ cin 3948 ⊆ wss 3949 ∅c0 4326 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 ↾_s cress 17216 +_gcplusg 17240 ._rcmulr 17241 0_gc0g 17428 SubMndcsubmnd 18746 SubGrpcsubg 19082 Ringcrg 20180 SubRingcsubrg 20513 LIdealclidl 21109

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-ip 17258 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-grp 18900 df-minusg 18901 df-sbg 18902 df-subg 19085 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-subrng 20490 df-subrg 20515 df-lmod 20752 df-lss 20823 df-sra 21065 df-rgmod 21066 df-lidl 21111

This theorem is referenced by: (None)

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