Theorem oppr1g 14326

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 2234	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2234	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
3		opprbas.1	. . . . . . . . . . 11 ⊢ 𝑂 = (oppr‘𝑅)
4		eqid 2234	. . . . . . . . . . 11 ⊢ (.r‘𝑂) = (.r‘𝑂)
5	1, 2, 3, 4	opprmulg 14314	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑂)𝑦) = (𝑦(.r‘𝑅)𝑥))
6	5	3expa 1230	. . . . . . . . 9 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑂)𝑦) = (𝑦(.r‘𝑅)𝑥))
7	6	eqeq1d 2243	. . . . . . . 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 529	. . . . . . . . . 10 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
11	1, 2, 3, 4	opprmulg 14314	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦))
12	8, 9, 10, 11	syl3anc 1274	. . . . . . . . 9 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦))
13	12	eqeq1d 2243	. . . . . . . 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 2540	. . . . 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 6029	. . . 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 6011	. . . 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 6011	. . . 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 2290	. . 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 14316	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → 𝑂 ∈ V)
22		eqid 2234	. . . . . 6 ⊢ (mulGrp‘𝑂) = (mulGrp‘𝑂)
23	22	mgpex 14164	. . . . 5 ⊢ (𝑂 ∈ V → (mulGrp‘𝑂) ∈ V)
24		eqid 2234	. . . . . 6 ⊢ (Base‘(mulGrp‘𝑂)) = (Base‘(mulGrp‘𝑂))
25		eqid 2234	. . . . . 6 ⊢ (+g‘(mulGrp‘𝑂)) = (+g‘(mulGrp‘𝑂))
26		eqid 2234	. . . . . 6 ⊢ (0g‘(mulGrp‘𝑂)) = (0g‘(mulGrp‘𝑂))
27	24, 25, 26	grpidvalg 13636	. . . . 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 14318	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑂))
30		eqid 2234	. . . . . . . . . 10 ⊢ (Base‘𝑂) = (Base‘𝑂)
31	22, 30	mgpbasg 14165	. . . . . . . . 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 2267	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘(mulGrp‘𝑂)))
34	33	eleq2d 2304	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ (Base‘𝑅) ↔ 𝑥 ∈ (Base‘(mulGrp‘𝑂))))
35	22, 4	mgpplusgg 14163	. . . . . . . . . . 11 ⊢ (𝑂 ∈ V → (.r‘𝑂) = (+g‘(mulGrp‘𝑂)))
36	21, 35	syl 14	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → (.r‘𝑂) = (+g‘(mulGrp‘𝑂)))
37	36	oveqd 6075	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑥(.r‘𝑂)𝑦) = (𝑥(+g‘(mulGrp‘𝑂))𝑦))
38	37	eqeq1d 2243	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → ((𝑥(.r‘𝑂)𝑦) = 𝑦 ↔ (𝑥(+g‘(mulGrp‘𝑂))𝑦) = 𝑦))
39	36	oveqd 6075	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑦(.r‘𝑂)𝑥) = (𝑦(+g‘(mulGrp‘𝑂))𝑥))
40	39	eqeq1d 2243	. . . . . . . 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 2759	. . . . . 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 5340	. . . 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 2270	. . 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 2234	. . . . . 6 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
47	46	mgpex 14164	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (mulGrp‘𝑅) ∈ V)
48		eqid 2234	. . . . . 6 ⊢ (Base‘(mulGrp‘𝑅)) = (Base‘(mulGrp‘𝑅))
49		eqid 2234	. . . . . 6 ⊢ (+g‘(mulGrp‘𝑅)) = (+g‘(mulGrp‘𝑅))
50		eqid 2234	. . . . . 6 ⊢ (0g‘(mulGrp‘𝑅)) = (0g‘(mulGrp‘𝑅))
51	48, 49, 50	grpidvalg 13636	. . . . 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 14165	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘(mulGrp‘𝑅)))
54	53	eleq2d 2304	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ (Base‘𝑅) ↔ 𝑥 ∈ (Base‘(mulGrp‘𝑅))))
55	46, 2	mgpplusgg 14163	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → (.r‘𝑅) = (+g‘(mulGrp‘𝑅)))
56	55	oveqd 6075	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑥(.r‘𝑅)𝑦) = (𝑥(+g‘(mulGrp‘𝑅))𝑦))
57	56	eqeq1d 2243	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → ((𝑥(.r‘𝑅)𝑦) = 𝑦 ↔ (𝑥(+g‘(mulGrp‘𝑅))𝑦) = 𝑦))
58	55	oveqd 6075	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑦(.r‘𝑅)𝑥) = (𝑦(+g‘(mulGrp‘𝑅))𝑥))
59	58	eqeq1d 2243	. . . . . . . 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 2759	. . . . . 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 5340	. . . 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 2270	. . 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 2277	. 2 ⊢ (𝑅 ∈ 𝑉 → (0g‘(mulGrp‘𝑂)) = (0g‘(mulGrp‘𝑅)))
66		eqid 2234	. . . 4 ⊢ (1r‘𝑂) = (1r‘𝑂)
67	22, 66	ringidvalg 14204	. . 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 14204	. 2 ⊢ (𝑅 ∈ 𝑉 → 1 = (0g‘(mulGrp‘𝑅)))
71	65, 68, 70	3eqtr4rd 2278	1 ⊢ (𝑅 ∈ 𝑉 → 1 = (1r‘𝑂))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205  ∀wral 2522  Vcvv 2815  ℩cio 5315  ‘cfv 5357  ℩crio 6010  (class class class)co 6058  Basecbs 13296  +_gcplusg 13374  ._rcmulr 13375  0_gc0g 13553  mulGrpcmgp 14159  1_rcur 14202  opp_rcoppr 14310

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-tpos 6489  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-plusg 13387  df-mulr 13388  df-0g 13555  df-mgp 14160  df-ur 14203  df-oppr 14311

This theorem is referenced by:  opprunitd  14355  rhmopp  14421  opprnzrbg  14430  opprlring  14442