Theorem lringuplu 14505

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 14503	. . . . . . . 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 2317	. . . . . . 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 2317	. . . . . . 7 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (Unit‘𝑅))
10		eqid 2238	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
11		eqid 2238	. . . . . . . 8 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
12		eqid 2238	. . . . . . . 8 ⊢ (/r‘𝑅) = (/r‘𝑅)
13		eqid 2238	. . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅)
14	10, 11, 12, 13	dvrcan1 14449	. . . . . . 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 1278	. . . . . 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 14422	. . . . . 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 1278	. . . . 5 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → ((𝑋(/r‘𝑅)(𝑋 + 𝑌))(.r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅))
22	16, 21	eqeltrrd 2316	. . . 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 2318	. . 3 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑋 ∈ 𝑈)
25	24	orcd 745	. 2 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈))
26		lring.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
27	26, 5	eleqtrd 2317	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑅))
28	10, 11, 12, 13	dvrcan1 14449	. . . . . . 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 1278	. . . . . 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 14422	. . . . . 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 1278	. . . . 5 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → ((𝑌(/r‘𝑅)(𝑋 + 𝑌))(.r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅))
36	30, 35	eqeltrrd 2316	. . . 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 2318	. . 3 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑌 ∈ 𝑈)
39	38	olcd 746	. 2 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈))
40		eqid 2238	. . . . . 6 ⊢ (+g‘𝑅) = (+g‘𝑅)
41	10, 11, 40, 12	dvrdir 14452	. . . . 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 1280	. . . 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 2244	. . . . . 6 ⊢ (𝜑 → (+g‘𝑅) = + )
45	44	oveqd 6102	. . . . 5 ⊢ (𝜑 → (𝑋(+g‘𝑅)𝑌) = (𝑋 + 𝑌))
46	3	ringgrpd 14311	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Grp)
47	10, 40, 46, 6, 27	grpcld 13821	. . . . . 6 ⊢ (𝜑 → (𝑋(+g‘𝑅)𝑌) ∈ (Base‘𝑅))
48		eqid 2238	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
49	10, 11, 12, 48	dvreq1 14451	. . . . . 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 1278	. . . . 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 2273	. . 3 ⊢ (𝜑 → ((𝑋(/r‘𝑅)(𝑋 + 𝑌))(+g‘𝑅)(𝑌(/r‘𝑅)(𝑋 + 𝑌))) = (1r‘𝑅))
53		oveq2 6093	. . . . . 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 2247	. . . . 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 2301	. . . . . 6 ⊢ (𝑣 = (𝑌(/r‘𝑅)(𝑋 + 𝑌)) → (𝑣 ∈ (Unit‘𝑅) ↔ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)))
56	55	orbi2d 802	. . . . 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 6092	. . . . . . . 8 ⊢ (𝑢 = (𝑋(/r‘𝑅)(𝑋 + 𝑌)) → (𝑢(+g‘𝑅)𝑣) = ((𝑋(/r‘𝑅)(𝑋 + 𝑌))(+g‘𝑅)𝑣))
59	58	eqeq1d 2247	. . . . . . 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 2301	. . . . . . . 8 ⊢ (𝑢 = (𝑋(/r‘𝑅)(𝑋 + 𝑌)) → (𝑢 ∈ (Unit‘𝑅) ↔ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)))
61	60	orbi1d 803	. . . . . . 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 2550	. . . . 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 14501	. . . . . . 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 14444	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Base‘𝑅) ∧ (𝑋 + 𝑌) ∈ (Unit‘𝑅)) → (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Base‘𝑅))
68	3, 6, 9, 67	syl3anc 1278	. . . . 5 ⊢ (𝜑 → (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Base‘𝑅))
69	63, 66, 68	rspcdva 2934	. . . 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 14444	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ (Base‘𝑅) ∧ (𝑋 + 𝑌) ∈ (Unit‘𝑅)) → (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Base‘𝑅))
71	3, 27, 9, 70	syl3anc 1278	. . . 4 ⊢ (𝜑 → (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Base‘𝑅))
72	57, 69, 71	rspcdva 2934	. . 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 810	1 ⊢ (𝜑 → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ‘cfv 5377  (class class class)co 6085  Basecbs 13354  +_gcplusg 13433  ._rcmulr 13434  1_rcur 14264  Ringcrg 14302  Unitcui 14395  /_rcdvr 14440  NzRingcnzr 14488  LRingclring 14499

This proof depends on 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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-tpos 6516  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9306  df-2 9364  df-3 9365  df-ndx 13357  df-slot 13358  df-base 13360  df-sets 13361  df-iress 13362  df-plusg 13446  df-mulr 13447  df-0g 13614  df-mgm 13678  df-sgrp 13719  df-mnd 13732  df-grp 13810  df-minusg 13811  df-cmn 14091  df-abl 14092  df-mgp 14220  df-ur 14265  df-srg 14270  df-ring 14304  df-oppr 14375  df-dvdsr 14397  df-unit 14398  df-invr 14430  df-dvr 14441  df-nzr 14489  df-lring 14500

This theorem is used by:  aprcotr  14599