Theorem lringuplu 20523

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 20521	. . . . . . . 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 2842	. . . . . . 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 2842	. . . . . . 7 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (Unit‘𝑅))
10		eqid 2740	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
11		eqid 2740	. . . . . . . 8 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
12		eqid 2740	. . . . . . . 8 ⊢ (/r‘𝑅) = (/r‘𝑅)
13		eqid 2740	. . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅)
14	10, 11, 12, 13	dvrcan1 20387	. . . . . . 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 1379	. . . . . 6 ⊢ (𝜑 → ((𝑋(/r‘𝑅)(𝑋 + 𝑌))(.r‘𝑅)(𝑋 + 𝑌)) = 𝑋)
16	15	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → ((𝑋(/r‘𝑅)(𝑋 + 𝑌))(.r‘𝑅)(𝑋 + 𝑌)) = 𝑋)
17	3	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑅 ∈ Ring)
18		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅))
19	9	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → (𝑋 + 𝑌) ∈ (Unit‘𝑅))
20	11, 13	unitmulcl 20358	. . . . . 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 1379	. . . . 5 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → ((𝑋(/r‘𝑅)(𝑋 + 𝑌))(.r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅))
22	16, 21	eqeltrrd 2841	. . . 4 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑋 ∈ (Unit‘𝑅))
23	8	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑈 = (Unit‘𝑅))
24	22, 23	eleqtrrd 2843	. . 3 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑋 ∈ 𝑈)
25	24	orcd 879	. 2 ⊢ ((𝜑 ∧ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈))
26		lring.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
27	26, 5	eleqtrd 2842	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑅))
28	10, 11, 12, 13	dvrcan1 20387	. . . . . . 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 1379	. . . . . 6 ⊢ (𝜑 → ((𝑌(/r‘𝑅)(𝑋 + 𝑌))(.r‘𝑅)(𝑋 + 𝑌)) = 𝑌)
30	29	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → ((𝑌(/r‘𝑅)(𝑋 + 𝑌))(.r‘𝑅)(𝑋 + 𝑌)) = 𝑌)
31	3	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑅 ∈ Ring)
32		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅))
33	9	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → (𝑋 + 𝑌) ∈ (Unit‘𝑅))
34	11, 13	unitmulcl 20358	. . . . . 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 1379	. . . . 5 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → ((𝑌(/r‘𝑅)(𝑋 + 𝑌))(.r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅))
36	30, 35	eqeltrrd 2841	. . . 4 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑌 ∈ (Unit‘𝑅))
37	8	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑈 = (Unit‘𝑅))
38	36, 37	eleqtrrd 2843	. . 3 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → 𝑌 ∈ 𝑈)
39	38	olcd 880	. 2 ⊢ ((𝜑 ∧ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)) → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈))
40		eqid 2740	. . . . . 6 ⊢ (+g‘𝑅) = (+g‘𝑅)
41	10, 11, 40, 12	dvrdir 20390	. . . . 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 1380	. . . 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 2746	. . . . . 6 ⊢ (𝜑 → (+g‘𝑅) = + )
45	44	oveqd 7380	. . . . 5 ⊢ (𝜑 → (𝑋(+g‘𝑅)𝑌) = (𝑋 + 𝑌))
46	3	ringgrpd 20221	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Grp)
47	10, 40, 46, 6, 27	grpcld 18921	. . . . . 6 ⊢ (𝜑 → (𝑋(+g‘𝑅)𝑌) ∈ (Base‘𝑅))
48		eqid 2740	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
49	10, 11, 12, 48	dvreq1 20389	. . . . . 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 1379	. . . . 5 ⊢ (𝜑 → (((𝑋(+g‘𝑅)𝑌)(/r‘𝑅)(𝑋 + 𝑌)) = (1r‘𝑅) ↔ (𝑋(+g‘𝑅)𝑌) = (𝑋 + 𝑌)))
51	45, 50	mpbird 258	. . . 4 ⊢ (𝜑 → ((𝑋(+g‘𝑅)𝑌)(/r‘𝑅)(𝑋 + 𝑌)) = (1r‘𝑅))
52	42, 51	eqtr3d 2777	. . 3 ⊢ (𝜑 → ((𝑋(/r‘𝑅)(𝑋 + 𝑌))(+g‘𝑅)(𝑌(/r‘𝑅)(𝑋 + 𝑌))) = (1r‘𝑅))
53		oveq2 7371	. . . . . 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 2742	. . . . 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 2828	. . . . . 6 ⊢ (𝑣 = (𝑌(/r‘𝑅)(𝑋 + 𝑌)) → (𝑣 ∈ (Unit‘𝑅) ↔ (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)))
56	55	orbi2d 921	. . . . 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 345	. . . 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 7370	. . . . . . . 8 ⊢ (𝑢 = (𝑋(/r‘𝑅)(𝑋 + 𝑌)) → (𝑢(+g‘𝑅)𝑣) = ((𝑋(/r‘𝑅)(𝑋 + 𝑌))(+g‘𝑅)𝑣))
59	58	eqeq1d 2742	. . . . . . 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 2828	. . . . . . . 8 ⊢ (𝑢 = (𝑋(/r‘𝑅)(𝑋 + 𝑌)) → (𝑢 ∈ (Unit‘𝑅) ↔ (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Unit‘𝑅)))
61	60	orbi1d 922	. . . . . . 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 345	. . . . . 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 3163	. . . . 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 20519	. . . . . . 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 219	. . . . . 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 496	. . . . 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 20382	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Base‘𝑅) ∧ (𝑋 + 𝑌) ∈ (Unit‘𝑅)) → (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Base‘𝑅))
68	3, 6, 9, 67	syl3anc 1379	. . . . 5 ⊢ (𝜑 → (𝑋(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Base‘𝑅))
69	63, 66, 68	rspcdva 3568	. . . 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 20382	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ (Base‘𝑅) ∧ (𝑋 + 𝑌) ∈ (Unit‘𝑅)) → (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Base‘𝑅))
71	3, 27, 9, 70	syl3anc 1379	. . . 4 ⊢ (𝜑 → (𝑌(/r‘𝑅)(𝑋 + 𝑌)) ∈ (Base‘𝑅))
72	57, 69, 71	rspcdva 3568	. . 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 966	1 ⊢ (𝜑 → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  +_gcplusg 17218  ._rcmulr 17219  1_rcur 20160  Ringcrg 20212  Unitcui 20333  /_rcdvr 20378  NzRingcnzr 20491  LRingclring 20517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-tpos 8173  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-2 12242  df-3 12243  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17178  df-ress 17199  df-plusg 17231  df-mulr 17232  df-0g 17402  df-mgm 18606  df-sgrp 18685  df-mnd 18701  df-grp 18910  df-minusg 18911  df-cmn 19755  df-abl 19756  df-mgp 20120  df-rng 20132  df-ur 20161  df-ring 20214  df-oppr 20315  df-dvdsr 20335  df-unit 20336  df-invr 20366  df-dvr 20379  df-nzr 20492  df-lring 20518

This theorem is referenced by: (None)