Theorem lringuplu 13342

Description: If the sum of two elements of a local ring is invertible, then at least one of the summands must be invertible. (Contributed by Jim Kingdon, 18-Feb-2025.) (Revised by SN, 23-Feb-2025.)

Hypotheses

Ref	Expression
lring.b	⊢ (𝜑 → 𝐵 = (Base‘𝑅))
lring.u	⊢ (𝜑 → 𝑈 = (Unit‘𝑅))
lring.p	⊢ (𝜑 → + = (+g‘𝑅))
lring.l	⊢ (𝜑 → 𝑅 ∈ LRing)
lring.s	⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑈)
lring.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
lring.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
lringuplu	⊢ (𝜑 → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈))

Proof of Theorem lringuplu

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lring.l	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ LRing)
2		lringring 13340	. . . . . . . 8 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ Ring)
3	1, 2	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring)
4		lring.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5		lring.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝑅))
6	4, 5	eleqtrd 2256	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑅))
7		lring.s	. . . . . . . 8 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑈)
8		lring.u	. . . . . . . 8 ⊢ (𝜑 → 𝑈 = (Unit‘𝑅))
9	7, 8	eleqtrd 2256	. . . . . . 7 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (Unit‘𝑅))
10		eqid 2177	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
11		eqid 2177	. . . . . . . 8 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
12		eqid 2177	. . . . . . . 8 ⊢ (/r‘𝑅) = (/r‘𝑅)
13		eqid 2177	. . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅)
14	10, 11, 12, 13	dvrcan1 13314	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Base‘𝑅) ∧ (𝑋 + 𝑌) ∈ (Unit‘𝑅)) → ((𝑋(/r‘𝑅)(𝑋 + 𝑌))(.r‘𝑅)(𝑋 + 𝑌)) = 𝑋)
15	3, 6, 9, 14	syl3anc 1238	. . . . . 6 ⊢ (𝜑 → ((𝑋(/r‘𝑅)(𝑋 + 𝑌))(.r‘𝑅)(𝑋 + 𝑌)) = 𝑋)
16	15	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → ((𝑋(/r‘𝑅)(𝑋 + 𝑌))(.r‘𝑅)(𝑋 + 𝑌)) = 𝑋)
17	3	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑅 ∈ Ring)
18		simpr 110	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅))
19	9	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → (𝑋 + 𝑌) ∈ (Unit‘𝑅))
20	11, 13	unitmulcl 13287	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅) ∧ (𝑋 + 𝑌) ∈ (Unit‘𝑅)) → ((𝑋(/r‘𝑅)(𝑋 + 𝑌))(.r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅))
21	17, 18, 19, 20	syl3anc 1238	. . . . 5 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → ((𝑋(/r‘𝑅)(𝑋 + 𝑌))(.r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅))
22	16, 21	eqeltrrd 2255	. . . 4 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑋 ∈ (Unit‘𝑅))
23	8	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑈 = (Unit‘𝑅))
24	22, 23	eleqtrrd 2257	. . 3 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑋 ∈ 𝑈)
25	24	orcd 733	. 2 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈))
26		lring.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
27	26, 5	eleqtrd 2256	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑅))
28	10, 11, 12, 13	dvrcan1 13314	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ (Base‘𝑅) ∧ (𝑋 + 𝑌) ∈ (Unit‘𝑅)) → ((𝑌(/r‘𝑅)(𝑋 + 𝑌))(.r‘𝑅)(𝑋 + 𝑌)) = 𝑌)
29	3, 27, 9, 28	syl3anc 1238	. . . . . 6 ⊢ (𝜑 → ((𝑌(/r‘𝑅)(𝑋 + 𝑌))(.r‘𝑅)(𝑋 + 𝑌)) = 𝑌)
30	29	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → ((𝑌(/r‘𝑅)(𝑋 + 𝑌))(.r‘𝑅)(𝑋 + 𝑌)) = 𝑌)
31	3	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑅 ∈ Ring)
32		simpr 110	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅))
33	9	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → (𝑋 + 𝑌) ∈ (Unit‘𝑅))
34	11, 13	unitmulcl 13287	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅) ∧ (𝑋 + 𝑌) ∈ (Unit‘𝑅)) → ((𝑌(/r‘𝑅)(𝑋 + 𝑌))(.r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅))
35	31, 32, 33, 34	syl3anc 1238	. . . . 5 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → ((𝑌(/r‘𝑅)(𝑋 + 𝑌))(.r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅))
36	30, 35	eqeltrrd 2255	. . . 4 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑌 ∈ (Unit‘𝑅))
37	8	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑈 = (Unit‘𝑅))
38	36, 37	eleqtrrd 2257	. . 3 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑌 ∈ 𝑈)
39	38	olcd 734	. 2 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈))
40		eqid 2177	. . . . . 6 ⊢ (+g‘𝑅) = (+g‘𝑅)
41	10, 11, 40, 12	dvrdir 13317	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ (Base‘𝑅) ∧ 𝑌 ∈ (Base‘𝑅) ∧ (𝑋 + 𝑌) ∈ (Unit‘𝑅))) → ((𝑋(+g‘𝑅)𝑌)(/r‘𝑅)(𝑋 + 𝑌)) = ((𝑋(/r‘𝑅)(𝑋 + 𝑌))(+g‘𝑅)(𝑌(/r‘𝑅)(𝑋 + 𝑌))))
42	3, 6, 27, 9, 41	syl13anc 1240	. . . 4 ⊢ (𝜑 → ((𝑋(+g‘𝑅)𝑌)(/r‘𝑅)(𝑋 + 𝑌)) = ((𝑋(/r‘𝑅)(𝑋 + 𝑌))(+g‘𝑅)(𝑌(/r‘𝑅)(𝑋 + 𝑌))))
43		lring.p	. . . . . . 7 ⊢ (𝜑 → + = (+g‘𝑅))
44	43	eqcomd 2183	. . . . . 6 ⊢ (𝜑 → (+g‘𝑅) = + )
45	44	oveqd 5894	. . . . 5 ⊢ (𝜑 → (𝑋(+g‘𝑅)𝑌) = (𝑋 + 𝑌))
46	3	ringgrpd 13193	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Grp)
47	10, 40, 46, 6, 27	grpcld 12895	. . . . . 6 ⊢ (𝜑 → (𝑋(+g‘𝑅)𝑌) ∈ (Base‘𝑅))
48		eqid 2177	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
49	10, 11, 12, 48	dvreq1 13316	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝑋(+g‘𝑅)𝑌) ∈ (Base‘𝑅) ∧ (𝑋 + 𝑌) ∈ (Unit‘𝑅)) → (((𝑋(+g‘𝑅)𝑌)(/r‘𝑅)(𝑋 + 𝑌)) = (1r‘𝑅) ↔ (𝑋(+g‘𝑅)𝑌) = (𝑋 + 𝑌)))
50	3, 47, 9, 49	syl3anc 1238	. . . . 5 ⊢ (𝜑 → (((𝑋(+g‘𝑅)𝑌)(/r‘𝑅)(𝑋 + 𝑌)) = (1r‘𝑅) ↔ (𝑋(+g‘𝑅)𝑌) = (𝑋 + 𝑌)))
51	45, 50	mpbird 167	. . . 4 ⊢ (𝜑 → ((𝑋(+g‘𝑅)𝑌)(/r‘𝑅)(𝑋 + 𝑌)) = (1r‘𝑅))
52	42, 51	eqtr3d 2212	. . 3 ⊢ (𝜑 → ((𝑋(/r‘𝑅)(𝑋 + 𝑌))(+g‘𝑅)(𝑌(/r‘𝑅)(𝑋 + 𝑌))) = (1r‘𝑅))
53		oveq2 5885	. . . . . 6 ⊢ (𝑣 = (𝑌(/r‘𝑅)(𝑋 + 𝑌)) → ((𝑋(/r‘𝑅)(𝑋 + 𝑌))(+g‘𝑅)𝑣) = ((𝑋(/r‘𝑅)(𝑋 + 𝑌))(+g‘𝑅)(𝑌(/r‘𝑅)(𝑋 + 𝑌))))
54	53	eqeq1d 2186	. . . . 5 ⊢ (𝑣 = (𝑌(/r‘𝑅)(𝑋 + 𝑌)) → (((𝑋(/r‘𝑅)(𝑋 + 𝑌))(+g‘𝑅)𝑣) = (1r‘𝑅) ↔ ((𝑋(/r‘𝑅)(𝑋 + 𝑌))(+g‘𝑅)(𝑌(/r‘𝑅)(𝑋 + 𝑌))) = (1r‘𝑅)))
55		eleq1 2240	. . . . . 6 ⊢ (𝑣 = (𝑌(/r‘𝑅)(𝑋 + 𝑌)) → (𝑣 ∈ (Unit‘𝑅) ↔ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)))
56	55	orbi2d 790	. . . . 5 ⊢ (𝑣 = (𝑌(/r‘𝑅)(𝑋 + 𝑌)) → (((𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅) ∨ 𝑣 ∈ (Unit‘𝑅)) ↔ ((𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅) ∨ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅))))
57	54, 56	imbi12d 234	. . . 4 ⊢ (𝑣 = (𝑌(/r‘𝑅)(𝑋 + 𝑌)) → ((((𝑋(/r‘𝑅)(𝑋 + 𝑌))(+g‘𝑅)𝑣) = (1r‘𝑅) → ((𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅) ∨ 𝑣 ∈ (Unit‘𝑅))) ↔ (((𝑋(/r‘𝑅)(𝑋 + 𝑌))(+g‘𝑅)(𝑌(/r‘𝑅)(𝑋 + 𝑌))) = (1r‘𝑅) → ((𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅) ∨ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)))))
58		oveq1 5884	. . . . . . . 8 ⊢ (𝑢 = (𝑋(/r‘𝑅)(𝑋 + 𝑌)) → (𝑢(+g‘𝑅)𝑣) = ((𝑋(/r‘𝑅)(𝑋 + 𝑌))(+g‘𝑅)𝑣))
59	58	eqeq1d 2186	. . . . . . 7 ⊢ (𝑢 = (𝑋(/r‘𝑅)(𝑋 + 𝑌)) → ((𝑢(+g‘𝑅)𝑣) = (1r‘𝑅) ↔ ((𝑋(/r‘𝑅)(𝑋 + 𝑌))(+g‘𝑅)𝑣) = (1r‘𝑅)))
60		eleq1 2240	. . . . . . . 8 ⊢ (𝑢 = (𝑋(/r‘𝑅)(𝑋 + 𝑌)) → (𝑢 ∈ (Unit‘𝑅) ↔ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)))
61	60	orbi1d 791	. . . . . . 7 ⊢ (𝑢 = (𝑋(/r‘𝑅)(𝑋 + 𝑌)) → ((𝑢 ∈ (Unit‘𝑅) ∨ 𝑣 ∈ (Unit‘𝑅)) ↔ ((𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅) ∨ 𝑣 ∈ (Unit‘𝑅))))
62	59, 61	imbi12d 234	. . . . . 6 ⊢ (𝑢 = (𝑋(/r‘𝑅)(𝑋 + 𝑌)) → (((𝑢(+g‘𝑅)𝑣) = (1r‘𝑅) → (𝑢 ∈ (Unit‘𝑅) ∨ 𝑣 ∈ (Unit‘𝑅))) ↔ (((𝑋(/r‘𝑅)(𝑋 + 𝑌))(+g‘𝑅)𝑣) = (1r‘𝑅) → ((𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅) ∨ 𝑣 ∈ (Unit‘𝑅)))))
63	62	ralbidv 2477	. . . . 5 ⊢ (𝑢 = (𝑋(/r‘𝑅)(𝑋 + 𝑌)) → (∀𝑣 ∈ (Base‘𝑅)((𝑢(+g‘𝑅)𝑣) = (1r‘𝑅) → (𝑢 ∈ (Unit‘𝑅) ∨ 𝑣 ∈ (Unit‘𝑅))) ↔ ∀𝑣 ∈ (Base‘𝑅)(((𝑋(/r‘𝑅)(𝑋 + 𝑌))(+g‘𝑅)𝑣) = (1r‘𝑅) → ((𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅) ∨ 𝑣 ∈ (Unit‘𝑅)))))
64	10, 40, 48, 11	islring 13338	. . . . . . 7 ⊢ (𝑅 ∈ LRing ↔ (𝑅 ∈ NzRing ∧ ∀𝑢 ∈ (Base‘𝑅)∀𝑣 ∈ (Base‘𝑅)((𝑢(+g‘𝑅)𝑣) = (1r‘𝑅) → (𝑢 ∈ (Unit‘𝑅) ∨ 𝑣 ∈ (Unit‘𝑅)))))
65	1, 64	sylib 122	. . . . . 6 ⊢ (𝜑 → (𝑅 ∈ NzRing ∧ ∀𝑢 ∈ (Base‘𝑅)∀𝑣 ∈ (Base‘𝑅)((𝑢(+g‘𝑅)𝑣) = (1r‘𝑅) → (𝑢 ∈ (Unit‘𝑅) ∨ 𝑣 ∈ (Unit‘𝑅)))))
66	65	simprd 114	. . . . 5 ⊢ (𝜑 → ∀𝑢 ∈ (Base‘𝑅)∀𝑣 ∈ (Base‘𝑅)((𝑢(+g‘𝑅)𝑣) = (1r‘𝑅) → (𝑢 ∈ (Unit‘𝑅) ∨ 𝑣 ∈ (Unit‘𝑅))))
67	10, 11, 12	dvrcl 13309	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Base‘𝑅) ∧ (𝑋 + 𝑌) ∈ (Unit‘𝑅)) → (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Base‘𝑅))
68	3, 6, 9, 67	syl3anc 1238	. . . . 5 ⊢ (𝜑 → (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Base‘𝑅))
69	63, 66, 68	rspcdva 2848	. . . 4 ⊢ (𝜑 → ∀𝑣 ∈ (Base‘𝑅)(((𝑋(/r‘𝑅)(𝑋 + 𝑌))(+g‘𝑅)𝑣) = (1r‘𝑅) → ((𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅) ∨ 𝑣 ∈ (Unit‘𝑅))))
70	10, 11, 12	dvrcl 13309	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ (Base‘𝑅) ∧ (𝑋 + 𝑌) ∈ (Unit‘𝑅)) → (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Base‘𝑅))
71	3, 27, 9, 70	syl3anc 1238	. . . 4 ⊢ (𝜑 → (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Base‘𝑅))
72	57, 69, 71	rspcdva 2848	. . 3 ⊢ (𝜑 → (((𝑋(/r‘𝑅)(𝑋 + 𝑌))(+g‘𝑅)(𝑌(/r‘𝑅)(𝑋 + 𝑌))) = (1r‘𝑅) → ((𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅) ∨ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅))))
73	52, 72	mpd 13	. 2 ⊢ (𝜑 → ((𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅) ∨ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)))
74	25, 39, 73	mpjaodan 798	1 ⊢ (𝜑 → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708   = wceq 1353   ∈ wcel 2148  ∀wral 2455  ‘cfv 5218  (class class class)co 5877  Basecbs 12464  +_gcplusg 12538  ._rcmulr 12539  1_rcur 13147  Ringcrg 13184  Unitcui 13261  /_rcdvr 13305  NzRingcnzr 13328  LRingclring 13336

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-pre-ltirr 7925  ax-pre-lttrn 7927  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-tpos 6248  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-inn 8922  df-2 8980  df-3 8981  df-ndx 12467  df-slot 12468  df-base 12470  df-sets 12471  df-iress 12472  df-plusg 12551  df-mulr 12552  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-grp 12885  df-minusg 12886  df-cmn 13095  df-abl 13096  df-mgp 13136  df-ur 13148  df-srg 13152  df-ring 13186  df-oppr 13245  df-dvdsr 13263  df-unit 13264  df-invr 13295  df-dvr 13306  df-nzr 13329  df-lring 13337

This theorem is referenced by:  aprcotr  13380