Theorem lringuplu 20453

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 20451	. . . . . . . 8 ⊢ (𝑅 ∈ LRing → 𝑅 ∈ Ring)
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring)
4		lring.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5		lring.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝑅))
6	4, 5	eleqtrd 2830	. . . . . . 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 2830	. . . . . . 7 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (Unit‘𝑅))
10		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
11		eqid 2729	. . . . . . . 8 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
12		eqid 2729	. . . . . . . 8 ⊢ (/r‘𝑅) = (/r‘𝑅)
13		eqid 2729	. . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅)
14	10, 11, 12, 13	dvrcan1 20318	. . . . . . 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 1373	. . . . . 6 ⊢ (𝜑 → ((𝑋(/r‘𝑅)(𝑋 + 𝑌))(.r‘𝑅)(𝑋 + 𝑌)) = 𝑋)
16	15	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → ((𝑋(/r‘𝑅)(𝑋 + 𝑌))(.r‘𝑅)(𝑋 + 𝑌)) = 𝑋)
17	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑅 ∈ Ring)
18		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅))
19	9	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → (𝑋 + 𝑌) ∈ (Unit‘𝑅))
20	11, 13	unitmulcl 20289	. . . . . 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 1373	. . . . 5 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → ((𝑋(/r‘𝑅)(𝑋 + 𝑌))(.r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅))
22	16, 21	eqeltrrd 2829	. . . 4 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑋 ∈ (Unit‘𝑅))
23	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑈 = (Unit‘𝑅))
24	22, 23	eleqtrrd 2831	. . 3 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑋 ∈ 𝑈)
25	24	orcd 873	. 2 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈))
26		lring.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
27	26, 5	eleqtrd 2830	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑅))
28	10, 11, 12, 13	dvrcan1 20318	. . . . . . 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 1373	. . . . . 6 ⊢ (𝜑 → ((𝑌(/r‘𝑅)(𝑋 + 𝑌))(.r‘𝑅)(𝑋 + 𝑌)) = 𝑌)
30	29	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → ((𝑌(/r‘𝑅)(𝑋 + 𝑌))(.r‘𝑅)(𝑋 + 𝑌)) = 𝑌)
31	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑅 ∈ Ring)
32		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅))
33	9	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → (𝑋 + 𝑌) ∈ (Unit‘𝑅))
34	11, 13	unitmulcl 20289	. . . . . 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 1373	. . . . 5 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → ((𝑌(/r‘𝑅)(𝑋 + 𝑌))(.r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅))
36	30, 35	eqeltrrd 2829	. . . 4 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑌 ∈ (Unit‘𝑅))
37	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑈 = (Unit‘𝑅))
38	36, 37	eleqtrrd 2831	. . 3 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑌 ∈ 𝑈)
39	38	olcd 874	. 2 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈))
40		eqid 2729	. . . . . 6 ⊢ (+g‘𝑅) = (+g‘𝑅)
41	10, 11, 40, 12	dvrdir 20321	. . . . 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 1374	. . . 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 2735	. . . . . 6 ⊢ (𝜑 → (+g‘𝑅) = + )
45	44	oveqd 7404	. . . . 5 ⊢ (𝜑 → (𝑋(+g‘𝑅)𝑌) = (𝑋 + 𝑌))
46	3	ringgrpd 20151	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Grp)
47	10, 40, 46, 6, 27	grpcld 18879	. . . . . 6 ⊢ (𝜑 → (𝑋(+g‘𝑅)𝑌) ∈ (Base‘𝑅))
48		eqid 2729	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
49	10, 11, 12, 48	dvreq1 20320	. . . . . 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 1373	. . . . 5 ⊢ (𝜑 → (((𝑋(+g‘𝑅)𝑌)(/r‘𝑅)(𝑋 + 𝑌)) = (1r‘𝑅) ↔ (𝑋(+g‘𝑅)𝑌) = (𝑋 + 𝑌)))
51	45, 50	mpbird 257	. . . 4 ⊢ (𝜑 → ((𝑋(+g‘𝑅)𝑌)(/r‘𝑅)(𝑋 + 𝑌)) = (1r‘𝑅))
52	42, 51	eqtr3d 2766	. . 3 ⊢ (𝜑 → ((𝑋(/r‘𝑅)(𝑋 + 𝑌))(+g‘𝑅)(𝑌(/r‘𝑅)(𝑋 + 𝑌))) = (1r‘𝑅))
53		oveq2 7395	. . . . . 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 2731	. . . . 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 2816	. . . . . 6 ⊢ (𝑣 = (𝑌(/r‘𝑅)(𝑋 + 𝑌)) → (𝑣 ∈ (Unit‘𝑅) ↔ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)))
56	55	orbi2d 915	. . . . 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 344	. . . 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 7394	. . . . . . . 8 ⊢ (𝑢 = (𝑋(/r‘𝑅)(𝑋 + 𝑌)) → (𝑢(+g‘𝑅)𝑣) = ((𝑋(/r‘𝑅)(𝑋 + 𝑌))(+g‘𝑅)𝑣))
59	58	eqeq1d 2731	. . . . . . 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 2816	. . . . . . . 8 ⊢ (𝑢 = (𝑋(/r‘𝑅)(𝑋 + 𝑌)) → (𝑢 ∈ (Unit‘𝑅) ↔ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)))
61	60	orbi1d 916	. . . . . . 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 344	. . . . . 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 3156	. . . . 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 20449	. . . . . . 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 218	. . . . . 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 495	. . . . 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 20313	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Base‘𝑅) ∧ (𝑋 + 𝑌) ∈ (Unit‘𝑅)) → (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Base‘𝑅))
68	3, 6, 9, 67	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Base‘𝑅))
69	63, 66, 68	rspcdva 3589	. . . 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 20313	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ (Base‘𝑅) ∧ (𝑋 + 𝑌) ∈ (Unit‘𝑅)) → (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Base‘𝑅))
71	3, 27, 9, 70	syl3anc 1373	. . . 4 ⊢ (𝜑 → (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Base‘𝑅))
72	57, 69, 71	rspcdva 3589	. . 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 15	. 2 ⊢ (𝜑 → ((𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅) ∨ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)))
74	25, 39, 73	mpjaodan 960	1 ⊢ (𝜑 → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  +_gcplusg 17220  ._rcmulr 17221  1_rcur 20090  Ringcrg 20142  Unitcui 20264  /_rcdvr 20309  NzRingcnzr 20421  LRingclring 20447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-tpos 8205  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-0g 17404  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-grp 18868  df-minusg 18869  df-cmn 19712  df-abl 19713  df-mgp 20050  df-rng 20062  df-ur 20091  df-ring 20144  df-oppr 20246  df-dvdsr 20266  df-unit 20267  df-invr 20297  df-dvr 20310  df-nzr 20422  df-lring 20448

This theorem is referenced by: (None)