Theorem oppr1g 13929

Description: Multiplicative identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)

Hypotheses

Ref	Expression
opprbas.1	⊢ 𝑂 = (oppr‘𝑅)
oppr1.2	⊢ 1 = (1r‘𝑅)

Assertion

Ref	Expression
oppr1g	⊢ (𝑅 ∈ 𝑉 → 1 = (1r‘𝑂))

Proof of Theorem oppr1g

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2206	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2206	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
3		opprbas.1	. . . . . . . . . . 11 ⊢ 𝑂 = (oppr‘𝑅)
4		eqid 2206	. . . . . . . . . . 11 ⊢ (.r‘𝑂) = (.r‘𝑂)
5	1, 2, 3, 4	opprmulg 13918	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑂)𝑦) = (𝑦(.r‘𝑅)𝑥))
6	5	3expa 1206	. . . . . . . . 9 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑂)𝑦) = (𝑦(.r‘𝑅)𝑥))
7	6	eqeq1d 2215	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(.r‘𝑂)𝑦) = 𝑦 ↔ (𝑦(.r‘𝑅)𝑥) = 𝑦))
8		simpll 527	. . . . . . . . . 10 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑅 ∈ 𝑉)
9		simpr 110	. . . . . . . . . 10 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 ∈ (Base‘𝑅))
10		simplr 528	. . . . . . . . . 10 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
11	1, 2, 3, 4	opprmulg 13918	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦))
12	8, 9, 10, 11	syl3anc 1250	. . . . . . . . 9 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦))
13	12	eqeq1d 2215	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑦(.r‘𝑂)𝑥) = 𝑦 ↔ (𝑥(.r‘𝑅)𝑦) = 𝑦))
14	7, 13	anbi12d 473	. . . . . . 7 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (((𝑥(.r‘𝑂)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑂)𝑥) = 𝑦) ↔ ((𝑦(.r‘𝑅)𝑥) = 𝑦 ∧ (𝑥(.r‘𝑅)𝑦) = 𝑦)))
15	14	biancomd 271	. . . . . 6 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (((𝑥(.r‘𝑂)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑂)𝑥) = 𝑦) ↔ ((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦)))
16	15	ralbidva 2503	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑅)) → (∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑂)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑂)𝑥) = 𝑦) ↔ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦)))
17	16	riotabidva 5934	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (℩𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑂)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑂)𝑥) = 𝑦)) = (℩𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦)))
18		df-riota 5917	. . . 4 ⊢ (℩𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑂)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑂)𝑥) = 𝑦)) = (℩𝑥(𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑂)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑂)𝑥) = 𝑦)))
19		df-riota 5917	. . . 4 ⊢ (℩𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦)) = (℩𝑥(𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦)))
20	17, 18, 19	3eqtr3g 2262	. . 3 ⊢ (𝑅 ∈ 𝑉 → (℩𝑥(𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑂)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑂)𝑥) = 𝑦))) = (℩𝑥(𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦))))
21	3	opprex 13920	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → 𝑂 ∈ V)
22		eqid 2206	. . . . . 6 ⊢ (mulGrp‘𝑂) = (mulGrp‘𝑂)
23	22	mgpex 13772	. . . . 5 ⊢ (𝑂 ∈ V → (mulGrp‘𝑂) ∈ V)
24		eqid 2206	. . . . . 6 ⊢ (Base‘(mulGrp‘𝑂)) = (Base‘(mulGrp‘𝑂))
25		eqid 2206	. . . . . 6 ⊢ (+g‘(mulGrp‘𝑂)) = (+g‘(mulGrp‘𝑂))
26		eqid 2206	. . . . . 6 ⊢ (0g‘(mulGrp‘𝑂)) = (0g‘(mulGrp‘𝑂))
27	24, 25, 26	grpidvalg 13290	. . . . 5 ⊢ ((mulGrp‘𝑂) ∈ V → (0g‘(mulGrp‘𝑂)) = (℩𝑥(𝑥 ∈ (Base‘(mulGrp‘𝑂)) ∧ ∀𝑦 ∈ (Base‘(mulGrp‘𝑂))((𝑥(+g‘(mulGrp‘𝑂))𝑦) = 𝑦 ∧ (𝑦(+g‘(mulGrp‘𝑂))𝑥) = 𝑦))))
28	21, 23, 27	3syl 17	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (0g‘(mulGrp‘𝑂)) = (℩𝑥(𝑥 ∈ (Base‘(mulGrp‘𝑂)) ∧ ∀𝑦 ∈ (Base‘(mulGrp‘𝑂))((𝑥(+g‘(mulGrp‘𝑂))𝑦) = 𝑦 ∧ (𝑦(+g‘(mulGrp‘𝑂))𝑥) = 𝑦))))
29	3, 1	opprbasg 13922	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑂))
30		eqid 2206	. . . . . . . . . 10 ⊢ (Base‘𝑂) = (Base‘𝑂)
31	22, 30	mgpbasg 13773	. . . . . . . . 9 ⊢ (𝑂 ∈ V → (Base‘𝑂) = (Base‘(mulGrp‘𝑂)))
32	21, 31	syl 14	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑂) = (Base‘(mulGrp‘𝑂)))
33	29, 32	eqtrd 2239	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘(mulGrp‘𝑂)))
34	33	eleq2d 2276	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ (Base‘𝑅) ↔ 𝑥 ∈ (Base‘(mulGrp‘𝑂))))
35	22, 4	mgpplusgg 13771	. . . . . . . . . . 11 ⊢ (𝑂 ∈ V → (.r‘𝑂) = (+g‘(mulGrp‘𝑂)))
36	21, 35	syl 14	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → (.r‘𝑂) = (+g‘(mulGrp‘𝑂)))
37	36	oveqd 5979	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑥(.r‘𝑂)𝑦) = (𝑥(+g‘(mulGrp‘𝑂))𝑦))
38	37	eqeq1d 2215	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → ((𝑥(.r‘𝑂)𝑦) = 𝑦 ↔ (𝑥(+g‘(mulGrp‘𝑂))𝑦) = 𝑦))
39	36	oveqd 5979	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑦(.r‘𝑂)𝑥) = (𝑦(+g‘(mulGrp‘𝑂))𝑥))
40	39	eqeq1d 2215	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → ((𝑦(.r‘𝑂)𝑥) = 𝑦 ↔ (𝑦(+g‘(mulGrp‘𝑂))𝑥) = 𝑦))
41	38, 40	anbi12d 473	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (((𝑥(.r‘𝑂)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑂)𝑥) = 𝑦) ↔ ((𝑥(+g‘(mulGrp‘𝑂))𝑦) = 𝑦 ∧ (𝑦(+g‘(mulGrp‘𝑂))𝑥) = 𝑦)))
42	33, 41	raleqbidv 2719	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑂)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑂)𝑥) = 𝑦) ↔ ∀𝑦 ∈ (Base‘(mulGrp‘𝑂))((𝑥(+g‘(mulGrp‘𝑂))𝑦) = 𝑦 ∧ (𝑦(+g‘(mulGrp‘𝑂))𝑥) = 𝑦)))
43	34, 42	anbi12d 473	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → ((𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑂)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑂)𝑥) = 𝑦)) ↔ (𝑥 ∈ (Base‘(mulGrp‘𝑂)) ∧ ∀𝑦 ∈ (Base‘(mulGrp‘𝑂))((𝑥(+g‘(mulGrp‘𝑂))𝑦) = 𝑦 ∧ (𝑦(+g‘(mulGrp‘𝑂))𝑥) = 𝑦))))
44	43	iotabidv 5268	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (℩𝑥(𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑂)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑂)𝑥) = 𝑦))) = (℩𝑥(𝑥 ∈ (Base‘(mulGrp‘𝑂)) ∧ ∀𝑦 ∈ (Base‘(mulGrp‘𝑂))((𝑥(+g‘(mulGrp‘𝑂))𝑦) = 𝑦 ∧ (𝑦(+g‘(mulGrp‘𝑂))𝑥) = 𝑦))))
45	28, 44	eqtr4d 2242	. . 3 ⊢ (𝑅 ∈ 𝑉 → (0g‘(mulGrp‘𝑂)) = (℩𝑥(𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑂)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑂)𝑥) = 𝑦))))
46		eqid 2206	. . . . . 6 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
47	46	mgpex 13772	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (mulGrp‘𝑅) ∈ V)
48		eqid 2206	. . . . . 6 ⊢ (Base‘(mulGrp‘𝑅)) = (Base‘(mulGrp‘𝑅))
49		eqid 2206	. . . . . 6 ⊢ (+g‘(mulGrp‘𝑅)) = (+g‘(mulGrp‘𝑅))
50		eqid 2206	. . . . . 6 ⊢ (0g‘(mulGrp‘𝑅)) = (0g‘(mulGrp‘𝑅))
51	48, 49, 50	grpidvalg 13290	. . . . 5 ⊢ ((mulGrp‘𝑅) ∈ V → (0g‘(mulGrp‘𝑅)) = (℩𝑥(𝑥 ∈ (Base‘(mulGrp‘𝑅)) ∧ ∀𝑦 ∈ (Base‘(mulGrp‘𝑅))((𝑥(+g‘(mulGrp‘𝑅))𝑦) = 𝑦 ∧ (𝑦(+g‘(mulGrp‘𝑅))𝑥) = 𝑦))))
52	47, 51	syl 14	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (0g‘(mulGrp‘𝑅)) = (℩𝑥(𝑥 ∈ (Base‘(mulGrp‘𝑅)) ∧ ∀𝑦 ∈ (Base‘(mulGrp‘𝑅))((𝑥(+g‘(mulGrp‘𝑅))𝑦) = 𝑦 ∧ (𝑦(+g‘(mulGrp‘𝑅))𝑥) = 𝑦))))
53	46, 1	mgpbasg 13773	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘(mulGrp‘𝑅)))
54	53	eleq2d 2276	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ (Base‘𝑅) ↔ 𝑥 ∈ (Base‘(mulGrp‘𝑅))))
55	46, 2	mgpplusgg 13771	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → (.r‘𝑅) = (+g‘(mulGrp‘𝑅)))
56	55	oveqd 5979	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑥(.r‘𝑅)𝑦) = (𝑥(+g‘(mulGrp‘𝑅))𝑦))
57	56	eqeq1d 2215	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → ((𝑥(.r‘𝑅)𝑦) = 𝑦 ↔ (𝑥(+g‘(mulGrp‘𝑅))𝑦) = 𝑦))
58	55	oveqd 5979	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑦(.r‘𝑅)𝑥) = (𝑦(+g‘(mulGrp‘𝑅))𝑥))
59	58	eqeq1d 2215	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → ((𝑦(.r‘𝑅)𝑥) = 𝑦 ↔ (𝑦(+g‘(mulGrp‘𝑅))𝑥) = 𝑦))
60	57, 59	anbi12d 473	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦) ↔ ((𝑥(+g‘(mulGrp‘𝑅))𝑦) = 𝑦 ∧ (𝑦(+g‘(mulGrp‘𝑅))𝑥) = 𝑦)))
61	53, 60	raleqbidv 2719	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦) ↔ ∀𝑦 ∈ (Base‘(mulGrp‘𝑅))((𝑥(+g‘(mulGrp‘𝑅))𝑦) = 𝑦 ∧ (𝑦(+g‘(mulGrp‘𝑅))𝑥) = 𝑦)))
62	54, 61	anbi12d 473	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → ((𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦)) ↔ (𝑥 ∈ (Base‘(mulGrp‘𝑅)) ∧ ∀𝑦 ∈ (Base‘(mulGrp‘𝑅))((𝑥(+g‘(mulGrp‘𝑅))𝑦) = 𝑦 ∧ (𝑦(+g‘(mulGrp‘𝑅))𝑥) = 𝑦))))
63	62	iotabidv 5268	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (℩𝑥(𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦))) = (℩𝑥(𝑥 ∈ (Base‘(mulGrp‘𝑅)) ∧ ∀𝑦 ∈ (Base‘(mulGrp‘𝑅))((𝑥(+g‘(mulGrp‘𝑅))𝑦) = 𝑦 ∧ (𝑦(+g‘(mulGrp‘𝑅))𝑥) = 𝑦))))
64	52, 63	eqtr4d 2242	. . 3 ⊢ (𝑅 ∈ 𝑉 → (0g‘(mulGrp‘𝑅)) = (℩𝑥(𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦))))
65	20, 45, 64	3eqtr4d 2249	. 2 ⊢ (𝑅 ∈ 𝑉 → (0g‘(mulGrp‘𝑂)) = (0g‘(mulGrp‘𝑅)))
66		eqid 2206	. . . 4 ⊢ (1r‘𝑂) = (1r‘𝑂)
67	22, 66	ringidvalg 13808	. . 3 ⊢ (𝑂 ∈ V → (1r‘𝑂) = (0g‘(mulGrp‘𝑂)))
68	21, 67	syl 14	. 2 ⊢ (𝑅 ∈ 𝑉 → (1r‘𝑂) = (0g‘(mulGrp‘𝑂)))
69		oppr1.2	. . 3 ⊢ 1 = (1r‘𝑅)
70	46, 69	ringidvalg 13808	. 2 ⊢ (𝑅 ∈ 𝑉 → 1 = (0g‘(mulGrp‘𝑅)))
71	65, 68, 70	3eqtr4rd 2250	1 ⊢ (𝑅 ∈ 𝑉 → 1 = (1r‘𝑂))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2177  ∀wral 2485  Vcvv 2773  ℩cio 5244  ‘cfv 5285  ℩crio 5916  (class class class)co 5962  Basecbs 12917  +_gcplusg 12994  ._rcmulr 12995  0_gc0g 13173  mulGrpcmgp 13767  1_rcur 13806  opp_rcoppr 13914

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-addcom 8055  ax-addass 8057  ax-i2m1 8060  ax-0lt1 8061  ax-0id 8063  ax-rnegex 8064  ax-pre-ltirr 8067  ax-pre-lttrn 8069  ax-pre-ltadd 8071

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-br 4055  df-opab 4117  df-mpt 4118  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-fv 5293  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-tpos 6349  df-pnf 8139  df-mnf 8140  df-ltxr 8142  df-inn 9067  df-2 9125  df-3 9126  df-ndx 12920  df-slot 12921  df-base 12923  df-sets 12924  df-plusg 13007  df-mulr 13008  df-0g 13175  df-mgp 13768  df-ur 13807  df-oppr 13915

This theorem is referenced by:  opprunitd  13957  rhmopp  14023  opprnzrbg  14032