Theorem rnglidlrng 14818

Description: A (left) ideal of a non-unital ring is a non-unital ring. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption 𝑈 ∈ (SubGrp‘𝑅) is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)

Hypotheses

Ref	Expression
rnglidlabl.l	⊢ 𝐿 = (LIdeal‘𝑅)
rnglidlabl.i	⊢ 𝐼 = (𝑅 ↾s 𝑈)

Assertion

Ref	Expression
rnglidlrng	⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → 𝐼 ∈ Rng)

Proof of Theorem rnglidlrng

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

Step	Hyp	Ref	Expression
1		rngabl 14217	. . . 4 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
2	1	3ad2ant1 1049	. . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → 𝑅 ∈ Abel)
3		simp3 1030	. . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → 𝑈 ∈ (SubGrp‘𝑅))
4		rnglidlabl.i	. . . 4 ⊢ 𝐼 = (𝑅 ↾s 𝑈)
5	4	subgabl 14119	. . 3 ⊢ ((𝑅 ∈ Abel ∧ 𝑈 ∈ (SubGrp‘𝑅)) → 𝐼 ∈ Abel)
6	2, 3, 5	syl2anc 415	. 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → 𝐼 ∈ Abel)
7		eqid 2238	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
8	7	subg0cl 13968	. . 3 ⊢ (𝑈 ∈ (SubGrp‘𝑅) → (0g‘𝑅) ∈ 𝑈)
9		rnglidlabl.l	. . . 4 ⊢ 𝐿 = (LIdeal‘𝑅)
10	9, 4, 7	rnglidlmsgrp 14817	. . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ (0g‘𝑅) ∈ 𝑈) → (mulGrp‘𝐼) ∈ Smgrp)
11	8, 10	syl3an3 1313	. 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → (mulGrp‘𝐼) ∈ Smgrp)
12		simpl1 1031	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → 𝑅 ∈ Rng)
13	9, 4	lidlssbas 14797	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) ⊆ (Base‘𝑅))
14	13	sseld 3247	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → (𝑎 ∈ (Base‘𝐼) → 𝑎 ∈ (Base‘𝑅)))
15	13	sseld 3247	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → (𝑏 ∈ (Base‘𝐼) → 𝑏 ∈ (Base‘𝑅)))
16	13	sseld 3247	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐿 → (𝑐 ∈ (Base‘𝐼) → 𝑐 ∈ (Base‘𝑅)))
17	14, 15, 16	3anim123d 1360	. . . . . . 7 ⊢ (𝑈 ∈ 𝐿 → ((𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼)) → (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅) ∧ 𝑐 ∈ (Base‘𝑅))))
18	17	3ad2ant2 1050	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → ((𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼)) → (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅) ∧ 𝑐 ∈ (Base‘𝑅))))
19	18	imp 124	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅) ∧ 𝑐 ∈ (Base‘𝑅)))
20		eqid 2238	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
21		eqid 2238	. . . . . 6 ⊢ (+g‘𝑅) = (+g‘𝑅)
22		eqid 2238	. . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅)
23	20, 21, 22	rngdi 14222	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅) ∧ 𝑐 ∈ (Base‘𝑅))) → (𝑎(.r‘𝑅)(𝑏(+g‘𝑅)𝑐)) = ((𝑎(.r‘𝑅)𝑏)(+g‘𝑅)(𝑎(.r‘𝑅)𝑐)))
24	12, 19, 23	syl2anc 415	. . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → (𝑎(.r‘𝑅)(𝑏(+g‘𝑅)𝑐)) = ((𝑎(.r‘𝑅)𝑏)(+g‘𝑅)(𝑎(.r‘𝑅)𝑐)))
25	20, 21, 22	rngdir 14223	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅) ∧ 𝑐 ∈ (Base‘𝑅))) → ((𝑎(+g‘𝑅)𝑏)(.r‘𝑅)𝑐) = ((𝑎(.r‘𝑅)𝑐)(+g‘𝑅)(𝑏(.r‘𝑅)𝑐)))
26	12, 19, 25	syl2anc 415	. . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → ((𝑎(+g‘𝑅)𝑏)(.r‘𝑅)𝑐) = ((𝑎(.r‘𝑅)𝑐)(+g‘𝑅)(𝑏(.r‘𝑅)𝑐)))
27		simp2 1029	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → 𝑈 ∈ 𝐿)
28		simp1 1028	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → 𝑅 ∈ Rng)
29	4, 22	ressmulrg 13482	. . . . . . . . . 10 ⊢ ((𝑈 ∈ 𝐿 ∧ 𝑅 ∈ Rng) → (.r‘𝑅) = (.r‘𝐼))
30	27, 28, 29	syl2anc 415	. . . . . . . . 9 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → (.r‘𝑅) = (.r‘𝐼))
31	30	eqcomd 2244	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → (.r‘𝐼) = (.r‘𝑅))
32		eqidd 2239	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → 𝑎 = 𝑎)
33	4	a1i 9	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → 𝐼 = (𝑅 ↾s 𝑈))
34		eqidd 2239	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → (+g‘𝑅) = (+g‘𝑅))
35	33, 34, 27, 28	ressplusgd 13466	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → (+g‘𝑅) = (+g‘𝐼))
36	35	eqcomd 2244	. . . . . . . . 9 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → (+g‘𝐼) = (+g‘𝑅))
37	36	oveqd 6096	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → (𝑏(+g‘𝐼)𝑐) = (𝑏(+g‘𝑅)𝑐))
38	31, 32, 37	oveq123d 6100	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → (𝑎(.r‘𝐼)(𝑏(+g‘𝐼)𝑐)) = (𝑎(.r‘𝑅)(𝑏(+g‘𝑅)𝑐)))
39	31	oveqd 6096	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → (𝑎(.r‘𝐼)𝑏) = (𝑎(.r‘𝑅)𝑏))
40	31	oveqd 6096	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → (𝑎(.r‘𝐼)𝑐) = (𝑎(.r‘𝑅)𝑐))
41	36, 39, 40	oveq123d 6100	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → ((𝑎(.r‘𝐼)𝑏)(+g‘𝐼)(𝑎(.r‘𝐼)𝑐)) = ((𝑎(.r‘𝑅)𝑏)(+g‘𝑅)(𝑎(.r‘𝑅)𝑐)))
42	38, 41	eqeq12d 2253	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → ((𝑎(.r‘𝐼)(𝑏(+g‘𝐼)𝑐)) = ((𝑎(.r‘𝐼)𝑏)(+g‘𝐼)(𝑎(.r‘𝐼)𝑐)) ↔ (𝑎(.r‘𝑅)(𝑏(+g‘𝑅)𝑐)) = ((𝑎(.r‘𝑅)𝑏)(+g‘𝑅)(𝑎(.r‘𝑅)𝑐))))
43	36	oveqd 6096	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → (𝑎(+g‘𝐼)𝑏) = (𝑎(+g‘𝑅)𝑏))
44		eqidd 2239	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → 𝑐 = 𝑐)
45	31, 43, 44	oveq123d 6100	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → ((𝑎(+g‘𝐼)𝑏)(.r‘𝐼)𝑐) = ((𝑎(+g‘𝑅)𝑏)(.r‘𝑅)𝑐))
46	31	oveqd 6096	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → (𝑏(.r‘𝐼)𝑐) = (𝑏(.r‘𝑅)𝑐))
47	36, 40, 46	oveq123d 6100	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → ((𝑎(.r‘𝐼)𝑐)(+g‘𝐼)(𝑏(.r‘𝐼)𝑐)) = ((𝑎(.r‘𝑅)𝑐)(+g‘𝑅)(𝑏(.r‘𝑅)𝑐)))
48	45, 47	eqeq12d 2253	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → (((𝑎(+g‘𝐼)𝑏)(.r‘𝐼)𝑐) = ((𝑎(.r‘𝐼)𝑐)(+g‘𝐼)(𝑏(.r‘𝐼)𝑐)) ↔ ((𝑎(+g‘𝑅)𝑏)(.r‘𝑅)𝑐) = ((𝑎(.r‘𝑅)𝑐)(+g‘𝑅)(𝑏(.r‘𝑅)𝑐))))
49	42, 48	anbi12d 477	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → (((𝑎(.r‘𝐼)(𝑏(+g‘𝐼)𝑐)) = ((𝑎(.r‘𝐼)𝑏)(+g‘𝐼)(𝑎(.r‘𝐼)𝑐)) ∧ ((𝑎(+g‘𝐼)𝑏)(.r‘𝐼)𝑐) = ((𝑎(.r‘𝐼)𝑐)(+g‘𝐼)(𝑏(.r‘𝐼)𝑐))) ↔ ((𝑎(.r‘𝑅)(𝑏(+g‘𝑅)𝑐)) = ((𝑎(.r‘𝑅)𝑏)(+g‘𝑅)(𝑎(.r‘𝑅)𝑐)) ∧ ((𝑎(+g‘𝑅)𝑏)(.r‘𝑅)𝑐) = ((𝑎(.r‘𝑅)𝑐)(+g‘𝑅)(𝑏(.r‘𝑅)𝑐)))))
50	49	adantr 276	. . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → (((𝑎(.r‘𝐼)(𝑏(+g‘𝐼)𝑐)) = ((𝑎(.r‘𝐼)𝑏)(+g‘𝐼)(𝑎(.r‘𝐼)𝑐)) ∧ ((𝑎(+g‘𝐼)𝑏)(.r‘𝐼)𝑐) = ((𝑎(.r‘𝐼)𝑐)(+g‘𝐼)(𝑏(.r‘𝐼)𝑐))) ↔ ((𝑎(.r‘𝑅)(𝑏(+g‘𝑅)𝑐)) = ((𝑎(.r‘𝑅)𝑏)(+g‘𝑅)(𝑎(.r‘𝑅)𝑐)) ∧ ((𝑎(+g‘𝑅)𝑏)(.r‘𝑅)𝑐) = ((𝑎(.r‘𝑅)𝑐)(+g‘𝑅)(𝑏(.r‘𝑅)𝑐)))))
51	24, 26, 50	mpbir2and 957	. . 3 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → ((𝑎(.r‘𝐼)(𝑏(+g‘𝐼)𝑐)) = ((𝑎(.r‘𝐼)𝑏)(+g‘𝐼)(𝑎(.r‘𝐼)𝑐)) ∧ ((𝑎(+g‘𝐼)𝑏)(.r‘𝐼)𝑐) = ((𝑎(.r‘𝐼)𝑐)(+g‘𝐼)(𝑏(.r‘𝐼)𝑐))))
52	51	ralrimivvva 2633	. 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → ∀𝑎 ∈ (Base‘𝐼)∀𝑏 ∈ (Base‘𝐼)∀𝑐 ∈ (Base‘𝐼)((𝑎(.r‘𝐼)(𝑏(+g‘𝐼)𝑐)) = ((𝑎(.r‘𝐼)𝑏)(+g‘𝐼)(𝑎(.r‘𝐼)𝑐)) ∧ ((𝑎(+g‘𝐼)𝑏)(.r‘𝐼)𝑐) = ((𝑎(.r‘𝐼)𝑐)(+g‘𝐼)(𝑏(.r‘𝐼)𝑐))))
53		eqid 2238	. . 3 ⊢ (Base‘𝐼) = (Base‘𝐼)
54		eqid 2238	. . 3 ⊢ (mulGrp‘𝐼) = (mulGrp‘𝐼)
55		eqid 2238	. . 3 ⊢ (+g‘𝐼) = (+g‘𝐼)
56		eqid 2238	. . 3 ⊢ (.r‘𝐼) = (.r‘𝐼)
57	53, 54, 55, 56	isrng 14216	. 2 ⊢ (𝐼 ∈ Rng ↔ (𝐼 ∈ Abel ∧ (mulGrp‘𝐼) ∈ Smgrp ∧ ∀𝑎 ∈ (Base‘𝐼)∀𝑏 ∈ (Base‘𝐼)∀𝑐 ∈ (Base‘𝐼)((𝑎(.r‘𝐼)(𝑏(+g‘𝐼)𝑐)) = ((𝑎(.r‘𝐼)𝑏)(+g‘𝐼)(𝑎(.r‘𝐼)𝑐)) ∧ ((𝑎(+g‘𝐼)𝑏)(.r‘𝐼)𝑐) = ((𝑎(.r‘𝐼)𝑐)(+g‘𝐼)(𝑏(.r‘𝐼)𝑐)))))
58	6, 11, 52, 57	syl3anbrc 1212	1 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → 𝐼 ∈ Rng)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ‘cfv 5375  (class class class)co 6079  Basecbs 13335   ↾_s cress 13336  +_gcplusg 13414  ._rcmulr 13415  0_gc0g 13593  Smgrpcsgrp 13699  SubGrpcsubg 13953  Abelcabl 14071  mulGrpcmgp 14200  Rngcrng 14214  LIdealclidl 14787

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-pre-ltirr 8285  ax-pre-lttrn 8287  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-pnf 8356  df-mnf 8357  df-ltxr 8359  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-ndx 13338  df-slot 13339  df-base 13341  df-sets 13342  df-iress 13343  df-plusg 13427  df-mulr 13428  df-sca 13430  df-vsca 13431  df-ip 13432  df-0g 13595  df-mgm 13659  df-sgrp 13700  df-mnd 13713  df-grp 13791  df-subg 13956  df-cmn 14072  df-abl 14073  df-mgp 14201  df-rng 14215  df-lssm 14673  df-sra 14755  df-rgmod 14756  df-lidl 14789

This theorem is referenced by:  rng2idlsubgsubrng  14840