issubrg2 - Intuitionistic Logic Explorer

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Theorem issubrg2 14220

Description: Characterize the subrings of a ring by closure properties. (Contributed by Mario Carneiro, 3-Dec-2014.)

**Hypotheses**
Ref	Expression
issubrg2.b	⊢ 𝐵 = (Base‘𝑅)
issubrg2.o	⊢ 1 = (1_r‘𝑅)
issubrg2.t	⊢ · = (._r‘𝑅)

**Assertion**
Ref	Expression
issubrg2	⊢ (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)))

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝑅,𝑦 𝑥, · ,𝑦 Allowed substitution hints: 𝐵(𝑥,𝑦) 1 (𝑥,𝑦)

Proof of Theorem issubrg2 Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		subrgsubg 14206	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
2		issubrg2.o	. . . 4 ⊢ 1 = (1_r‘𝑅)
3	2	subrg1cl 14208	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 1 ∈ 𝐴)
4		issubrg2.t	. . . . . 6 ⊢ · = (._r‘𝑅)
5	4	subrgmcl 14212	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 · 𝑦) ∈ 𝐴)
6	5	3expb 1228	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 · 𝑦) ∈ 𝐴)
7	6	ralrimivva 2612	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)
8	1, 3, 7	3jca 1201	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴))
9		simpl 109	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝑅 ∈ Ring)
10		simpr1 1027	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 ∈ (SubGrp‘𝑅))
11		eqid 2229	. . . . . . 7 ⊢ (𝑅 ↾_s 𝐴) = (𝑅 ↾_s 𝐴)
12	11	subgbas 13730	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → 𝐴 = (Base‘(𝑅 ↾_s 𝐴)))
13	10, 12	syl 14	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 = (Base‘(𝑅 ↾_s 𝐴)))
14		eqidd 2230	. . . . . . 7 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → (𝑅 ↾_s 𝐴) = (𝑅 ↾_s 𝐴))
15		eqidd 2230	. . . . . . 7 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → (+_g‘𝑅) = (+_g‘𝑅))
16		id 19	. . . . . . 7 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
17		subgrcl 13731	. . . . . . 7 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → 𝑅 ∈ Grp)
18	14, 15, 16, 17	ressplusgd 13177	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → (+_g‘𝑅) = (+_g‘(𝑅 ↾_s 𝐴)))
19	10, 18	syl 14	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (+_g‘𝑅) = (+_g‘(𝑅 ↾_s 𝐴)))
20	11, 4	ressmulrg 13193	. . . . . 6 ⊢ ((𝐴 ∈ (SubGrp‘𝑅) ∧ 𝑅 ∈ Grp) → · = (._r‘(𝑅 ↾_s 𝐴)))
21	10, 17, 20	syl2anc2 412	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → · = (._r‘(𝑅 ↾_s 𝐴)))
22	11	subggrp 13729	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → (𝑅 ↾_s 𝐴) ∈ Grp)
23	10, 22	syl 14	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑅 ↾_s 𝐴) ∈ Grp)
24		simpr3 1029	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)
25		oveq1 6014	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → (𝑥 · 𝑦) = (𝑢 · 𝑦))
26	25	eleq1d 2298	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → ((𝑥 · 𝑦) ∈ 𝐴 ↔ (𝑢 · 𝑦) ∈ 𝐴))
27		oveq2 6015	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑢 · 𝑦) = (𝑢 · 𝑣))
28	27	eleq1d 2298	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → ((𝑢 · 𝑦) ∈ 𝐴 ↔ (𝑢 · 𝑣) ∈ 𝐴))
29	26, 28	rspc2v 2920	. . . . . . 7 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴 → (𝑢 · 𝑣) ∈ 𝐴))
30	24, 29	syl5com 29	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (𝑢 · 𝑣) ∈ 𝐴))
31	30	3impib 1225	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (𝑢 · 𝑣) ∈ 𝐴)
32		issubrg2.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑅)
33	32	subgss 13726	. . . . . . . . . 10 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → 𝐴 ⊆ 𝐵)
34	10, 33	syl 14	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 ⊆ 𝐵)
35	34	sseld 3223	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑢 ∈ 𝐴 → 𝑢 ∈ 𝐵))
36	34	sseld 3223	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑣 ∈ 𝐴 → 𝑣 ∈ 𝐵))
37	34	sseld 3223	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑤 ∈ 𝐴 → 𝑤 ∈ 𝐵))
38	35, 36, 37	3anim123d 1353	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)))
39	38	imp 124	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵))
40	32, 4	ringass 13994	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
41	40	adantlr 477	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
42	39, 41	syldan 282	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
43		eqid 2229	. . . . . . . 8 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
44	32, 43, 4	ringdi 13996	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑢 · (𝑣(+_g‘𝑅)𝑤)) = ((𝑢 · 𝑣)(+_g‘𝑅)(𝑢 · 𝑤)))
45	44	adantlr 477	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑢 · (𝑣(+_g‘𝑅)𝑤)) = ((𝑢 · 𝑣)(+_g‘𝑅)(𝑢 · 𝑤)))
46	39, 45	syldan 282	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑢 · (𝑣(+_g‘𝑅)𝑤)) = ((𝑢 · 𝑣)(+_g‘𝑅)(𝑢 · 𝑤)))
47	32, 43, 4	ringdir 13997	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢(+_g‘𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+_g‘𝑅)(𝑣 · 𝑤)))
48	47	adantlr 477	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢(+_g‘𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+_g‘𝑅)(𝑣 · 𝑤)))
49	39, 48	syldan 282	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑢(+_g‘𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+_g‘𝑅)(𝑣 · 𝑤)))
50		simpr2 1028	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 1 ∈ 𝐴)
51	35	imp 124	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢 ∈ 𝐴) → 𝑢 ∈ 𝐵)
52	32, 4, 2	ringlidm 14001	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑢 ∈ 𝐵) → ( 1 · 𝑢) = 𝑢)
53	52	adantlr 477	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢 ∈ 𝐵) → ( 1 · 𝑢) = 𝑢)
54	51, 53	syldan 282	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢 ∈ 𝐴) → ( 1 · 𝑢) = 𝑢)
55	32, 4, 2	ringridm 14002	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑢 ∈ 𝐵) → (𝑢 · 1 ) = 𝑢)
56	55	adantlr 477	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢 ∈ 𝐵) → (𝑢 · 1 ) = 𝑢)
57	51, 56	syldan 282	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢 ∈ 𝐴) → (𝑢 · 1 ) = 𝑢)
58	13, 19, 21, 23, 31, 42, 46, 49, 50, 54, 57	isringd 14019	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑅 ↾_s 𝐴) ∈ Ring)
59	34, 50	jca 306	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴))
60	32, 2	issubrg 14200	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾_s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴)))
61	9, 58, 59, 60	syl21anbrc 1206	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 ∈ (SubRing‘𝑅))
62	61	ex 115	. 2 ⊢ (𝑅 ∈ Ring → ((𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴) → 𝐴 ∈ (SubRing‘𝑅)))
63	8, 62	impbid2 143	1 ⊢ (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1002 = wceq 1395 ∈ wcel 2200 ∀wral 2508 ⊆ wss 3197 ‘cfv 5318 (class class class)co 6007 Basecbs 13047 ↾_s cress 13048 +_gcplusg 13125 ._rcmulr 13126 Grpcgrp 13548 SubGrpcsubg 13719 1_rcur 13937 Ringcrg 13974 SubRingcsubrg 14196

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-cnex 8101 ax-resscn 8102 ax-1cn 8103 ax-1re 8104 ax-icn 8105 ax-addcl 8106 ax-addrcl 8107 ax-mulcl 8108 ax-addcom 8110 ax-addass 8112 ax-i2m1 8115 ax-0lt1 8116 ax-0id 8118 ax-rnegex 8119 ax-pre-ltirr 8122 ax-pre-lttrn 8124 ax-pre-ltadd 8126

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-br 4084 df-opab 4146 df-mpt 4147 df-id 4384 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-fv 5326 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-pnf 8194 df-mnf 8195 df-ltxr 8197 df-inn 9122 df-2 9180 df-3 9181 df-ndx 13050 df-slot 13051 df-base 13053 df-sets 13054 df-iress 13055 df-plusg 13138 df-mulr 13139 df-0g 13306 df-mgm 13404 df-sgrp 13450 df-mnd 13465 df-subg 13722 df-mgp 13899 df-ur 13938 df-ring 13976 df-subrg 14198

This theorem is referenced by: subrgintm 14222 issubrg3 14226 issubrgd 14431 cnsubrglem 14559

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