Theorem oppr1g 13638

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 2196	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2196	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
3		opprbas.1	. . . . . . . . . . 11 ⊢ 𝑂 = (oppr‘𝑅)
4		eqid 2196	. . . . . . . . . . 11 ⊢ (.r‘𝑂) = (.r‘𝑂)
5	1, 2, 3, 4	opprmulg 13627	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑂)𝑦) = (𝑦(.r‘𝑅)𝑥))
6	5	3expa 1205	. . . . . . . . 9 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑂)𝑦) = (𝑦(.r‘𝑅)𝑥))
7	6	eqeq1d 2205	. . . . . . . 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 13627	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦))
12	8, 9, 10, 11	syl3anc 1249	. . . . . . . . 9 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦))
13	12	eqeq1d 2205	. . . . . . . 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 2493	. . . . 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 5894	. . . 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 5877	. . . 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 5877	. . . 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 2252	. . 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 13629	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → 𝑂 ∈ V)
22		eqid 2196	. . . . . 6 ⊢ (mulGrp‘𝑂) = (mulGrp‘𝑂)
23	22	mgpex 13481	. . . . 5 ⊢ (𝑂 ∈ V → (mulGrp‘𝑂) ∈ V)
24		eqid 2196	. . . . . 6 ⊢ (Base‘(mulGrp‘𝑂)) = (Base‘(mulGrp‘𝑂))
25		eqid 2196	. . . . . 6 ⊢ (+g‘(mulGrp‘𝑂)) = (+g‘(mulGrp‘𝑂))
26		eqid 2196	. . . . . 6 ⊢ (0g‘(mulGrp‘𝑂)) = (0g‘(mulGrp‘𝑂))
27	24, 25, 26	grpidvalg 13016	. . . . 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 13631	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑂))
30		eqid 2196	. . . . . . . . . 10 ⊢ (Base‘𝑂) = (Base‘𝑂)
31	22, 30	mgpbasg 13482	. . . . . . . . 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 2229	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘(mulGrp‘𝑂)))
34	33	eleq2d 2266	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ (Base‘𝑅) ↔ 𝑥 ∈ (Base‘(mulGrp‘𝑂))))
35	22, 4	mgpplusgg 13480	. . . . . . . . . . 11 ⊢ (𝑂 ∈ V → (.r‘𝑂) = (+g‘(mulGrp‘𝑂)))
36	21, 35	syl 14	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → (.r‘𝑂) = (+g‘(mulGrp‘𝑂)))
37	36	oveqd 5939	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑥(.r‘𝑂)𝑦) = (𝑥(+g‘(mulGrp‘𝑂))𝑦))
38	37	eqeq1d 2205	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → ((𝑥(.r‘𝑂)𝑦) = 𝑦 ↔ (𝑥(+g‘(mulGrp‘𝑂))𝑦) = 𝑦))
39	36	oveqd 5939	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑦(.r‘𝑂)𝑥) = (𝑦(+g‘(mulGrp‘𝑂))𝑥))
40	39	eqeq1d 2205	. . . . . . . 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 2709	. . . . . 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 5241	. . . 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 2232	. . 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 2196	. . . . . 6 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
47	46	mgpex 13481	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (mulGrp‘𝑅) ∈ V)
48		eqid 2196	. . . . . 6 ⊢ (Base‘(mulGrp‘𝑅)) = (Base‘(mulGrp‘𝑅))
49		eqid 2196	. . . . . 6 ⊢ (+g‘(mulGrp‘𝑅)) = (+g‘(mulGrp‘𝑅))
50		eqid 2196	. . . . . 6 ⊢ (0g‘(mulGrp‘𝑅)) = (0g‘(mulGrp‘𝑅))
51	48, 49, 50	grpidvalg 13016	. . . . 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 13482	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘(mulGrp‘𝑅)))
54	53	eleq2d 2266	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ (Base‘𝑅) ↔ 𝑥 ∈ (Base‘(mulGrp‘𝑅))))
55	46, 2	mgpplusgg 13480	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → (.r‘𝑅) = (+g‘(mulGrp‘𝑅)))
56	55	oveqd 5939	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑥(.r‘𝑅)𝑦) = (𝑥(+g‘(mulGrp‘𝑅))𝑦))
57	56	eqeq1d 2205	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → ((𝑥(.r‘𝑅)𝑦) = 𝑦 ↔ (𝑥(+g‘(mulGrp‘𝑅))𝑦) = 𝑦))
58	55	oveqd 5939	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑦(.r‘𝑅)𝑥) = (𝑦(+g‘(mulGrp‘𝑅))𝑥))
59	58	eqeq1d 2205	. . . . . . . 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 2709	. . . . . 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 5241	. . . 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 2232	. . 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 2239	. 2 ⊢ (𝑅 ∈ 𝑉 → (0g‘(mulGrp‘𝑂)) = (0g‘(mulGrp‘𝑅)))
66		eqid 2196	. . . 4 ⊢ (1r‘𝑂) = (1r‘𝑂)
67	22, 66	ringidvalg 13517	. . 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 13517	. 2 ⊢ (𝑅 ∈ 𝑉 → 1 = (0g‘(mulGrp‘𝑅)))
71	65, 68, 70	3eqtr4rd 2240	1 ⊢ (𝑅 ∈ 𝑉 → 1 = (1r‘𝑂))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  ∀wral 2475  Vcvv 2763  ℩cio 5217  ‘cfv 5258  ℩crio 5876  (class class class)co 5922  Basecbs 12678  +_gcplusg 12755  ._rcmulr 12756  0_gc0g 12927  mulGrpcmgp 13476  1_rcur 13515  opp_rcoppr 13623

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-tpos 6303  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-plusg 12768  df-mulr 12769  df-0g 12929  df-mgp 13477  df-ur 13516  df-oppr 13624

This theorem is referenced by:  opprunitd  13666  rhmopp  13732  opprnzrbg  13741