Theorem rloc1r 33212

Description: The multiplicative identity of a ring localization. (Contributed by Thierry Arnoux, 4-May-2025.)

Hypotheses

Ref	Expression
rloc0g.1	⊢ 0 = (0g‘𝑅)
rloc0g.2	⊢ 1 = (1r‘𝑅)
rloc0g.3	⊢ 𝐿 = (𝑅 RLocal 𝑆)
rloc0g.4	⊢ ∼ = (𝑅 ~RL 𝑆)
rloc0g.5	⊢ (𝜑 → 𝑅 ∈ CRing)
rloc0g.6	⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
rloc1r.i	⊢ 𝐼 = [⟨ 1 , 1 ⟩] ∼

Assertion

Ref	Expression
rloc1r	⊢ (𝜑 → 𝐼 = (1r‘𝐿))

Proof of Theorem rloc1r

Dummy variables 𝑎 𝑏 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rloc1r.i	. 2 ⊢ 𝐼 = [⟨ 1 , 1 ⟩] ∼
2		eqid 2729	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		eqid 2729	. . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅)
4		eqid 2729	. . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅)
5		rloc0g.3	. . . . 5 ⊢ 𝐿 = (𝑅 RLocal 𝑆)
6		rloc0g.4	. . . . 5 ⊢ ∼ = (𝑅 ~RL 𝑆)
7		rloc0g.5	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing)
8		rloc0g.6	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
9	2, 3, 4, 5, 6, 7, 8	rloccring 33210	. . . 4 ⊢ (𝜑 → 𝐿 ∈ CRing)
10	9	crngringd 20131	. . 3 ⊢ (𝜑 → 𝐿 ∈ Ring)
11		eqid 2729	. . . . . . . . . 10 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
12	11, 2	mgpbas 20030	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
13	12	submss 18683	. . . . . . . 8 ⊢ (𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) → 𝑆 ⊆ (Base‘𝑅))
14	8, 13	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑅))
15		rloc0g.2	. . . . . . . . . 10 ⊢ 1 = (1r‘𝑅)
16	11, 15	ringidval 20068	. . . . . . . . 9 ⊢ 1 = (0g‘(mulGrp‘𝑅))
17	16	subm0cl 18685	. . . . . . . 8 ⊢ (𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) → 1 ∈ 𝑆)
18	8, 17	syl 17	. . . . . . 7 ⊢ (𝜑 → 1 ∈ 𝑆)
19	14, 18	sseldd 3936	. . . . . 6 ⊢ (𝜑 → 1 ∈ (Base‘𝑅))
20	19, 18	opelxpd 5658	. . . . 5 ⊢ (𝜑 → ⟨ 1 , 1 ⟩ ∈ ((Base‘𝑅) × 𝑆))
21	6	ovexi 7383	. . . . . 6 ⊢ ∼ ∈ V
22	21	ecelqsi 8697	. . . . 5 ⊢ (⟨ 1 , 1 ⟩ ∈ ((Base‘𝑅) × 𝑆) → [⟨ 1 , 1 ⟩] ∼ ∈ (((Base‘𝑅) × 𝑆) / ∼ ))
23	20, 22	syl 17	. . . 4 ⊢ (𝜑 → [⟨ 1 , 1 ⟩] ∼ ∈ (((Base‘𝑅) × 𝑆) / ∼ ))
24		rloc0g.1	. . . . 5 ⊢ 0 = (0g‘𝑅)
25		eqid 2729	. . . . 5 ⊢ (-g‘𝑅) = (-g‘𝑅)
26		eqid 2729	. . . . 5 ⊢ ((Base‘𝑅) × 𝑆) = ((Base‘𝑅) × 𝑆)
27	2, 24, 3, 25, 26, 5, 6, 7, 14	rlocbas 33207	. . . 4 ⊢ (𝜑 → (((Base‘𝑅) × 𝑆) / ∼ ) = (Base‘𝐿))
28	23, 27	eleqtrd 2830	. . 3 ⊢ (𝜑 → [⟨ 1 , 1 ⟩] ∼ ∈ (Base‘𝐿))
29	7	ad4antr 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → 𝑅 ∈ CRing)
30	8	ad4antr 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
31	19	ad4antr 732	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → 1 ∈ (Base‘𝑅))
32		simpllr 775	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → 𝑎 ∈ (Base‘𝑅))
33	30, 17	syl 17	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → 1 ∈ 𝑆)
34		simplr 768	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → 𝑏 ∈ 𝑆)
35		eqid 2729	. . . . . . . . 9 ⊢ (.r‘𝐿) = (.r‘𝐿)
36	2, 3, 4, 5, 6, 29, 30, 31, 32, 33, 34, 35	rlocmulval 33209	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → ([⟨ 1 , 1 ⟩] ∼ (.r‘𝐿)[⟨𝑎, 𝑏⟩] ∼ ) = [⟨( 1 (.r‘𝑅)𝑎), ( 1 (.r‘𝑅)𝑏)⟩] ∼ )
37	29	crngringd 20131	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → 𝑅 ∈ Ring)
38	2, 3, 15, 37, 32	ringlidmd 20157	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → ( 1 (.r‘𝑅)𝑎) = 𝑎)
39	30, 13	syl 17	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → 𝑆 ⊆ (Base‘𝑅))
40	39, 34	sseldd 3936	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → 𝑏 ∈ (Base‘𝑅))
41	2, 3, 15, 37, 40	ringlidmd 20157	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → ( 1 (.r‘𝑅)𝑏) = 𝑏)
42	38, 41	opeq12d 4832	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → ⟨( 1 (.r‘𝑅)𝑎), ( 1 (.r‘𝑅)𝑏)⟩ = ⟨𝑎, 𝑏⟩)
43	42	eceq1d 8665	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → [⟨( 1 (.r‘𝑅)𝑎), ( 1 (.r‘𝑅)𝑏)⟩] ∼ = [⟨𝑎, 𝑏⟩] ∼ )
44	36, 43	eqtrd 2764	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → ([⟨ 1 , 1 ⟩] ∼ (.r‘𝐿)[⟨𝑎, 𝑏⟩] ∼ ) = [⟨𝑎, 𝑏⟩] ∼ )
45		simpr 484	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → 𝑥 = [⟨𝑎, 𝑏⟩] ∼ )
46	45	oveq2d 7365	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → ([⟨ 1 , 1 ⟩] ∼ (.r‘𝐿)𝑥) = ([⟨ 1 , 1 ⟩] ∼ (.r‘𝐿)[⟨𝑎, 𝑏⟩] ∼ ))
47	44, 46, 45	3eqtr4d 2774	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → ([⟨ 1 , 1 ⟩] ∼ (.r‘𝐿)𝑥) = 𝑥)
48	27	eqcomd 2735	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐿) = (((Base‘𝑅) × 𝑆) / ∼ ))
49	48	eleq2d 2814	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐿) ↔ 𝑥 ∈ (((Base‘𝑅) × 𝑆) / ∼ )))
50	49	biimpa 476	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) → 𝑥 ∈ (((Base‘𝑅) × 𝑆) / ∼ ))
51	50	elrlocbasi 33206	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) → ∃𝑎 ∈ (Base‘𝑅)∃𝑏 ∈ 𝑆 𝑥 = [⟨𝑎, 𝑏⟩] ∼ )
52	47, 51	r19.29vva 3189	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) → ([⟨ 1 , 1 ⟩] ∼ (.r‘𝐿)𝑥) = 𝑥)
53	2, 3, 4, 5, 6, 29, 30, 32, 31, 34, 33, 35	rlocmulval 33209	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → ([⟨𝑎, 𝑏⟩] ∼ (.r‘𝐿)[⟨ 1 , 1 ⟩] ∼ ) = [⟨(𝑎(.r‘𝑅) 1 ), (𝑏(.r‘𝑅) 1 )⟩] ∼ )
54	2, 3, 15, 37, 32	ringridmd 20158	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → (𝑎(.r‘𝑅) 1 ) = 𝑎)
55	2, 3, 15, 37, 40	ringridmd 20158	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → (𝑏(.r‘𝑅) 1 ) = 𝑏)
56	54, 55	opeq12d 4832	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → ⟨(𝑎(.r‘𝑅) 1 ), (𝑏(.r‘𝑅) 1 )⟩ = ⟨𝑎, 𝑏⟩)
57	56	eceq1d 8665	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → [⟨(𝑎(.r‘𝑅) 1 ), (𝑏(.r‘𝑅) 1 )⟩] ∼ = [⟨𝑎, 𝑏⟩] ∼ )
58	53, 57	eqtrd 2764	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → ([⟨𝑎, 𝑏⟩] ∼ (.r‘𝐿)[⟨ 1 , 1 ⟩] ∼ ) = [⟨𝑎, 𝑏⟩] ∼ )
59	45	oveq1d 7364	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → (𝑥(.r‘𝐿)[⟨ 1 , 1 ⟩] ∼ ) = ([⟨𝑎, 𝑏⟩] ∼ (.r‘𝐿)[⟨ 1 , 1 ⟩] ∼ ))
60	58, 59, 45	3eqtr4d 2774	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → (𝑥(.r‘𝐿)[⟨ 1 , 1 ⟩] ∼ ) = 𝑥)
61	60, 51	r19.29vva 3189	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) → (𝑥(.r‘𝐿)[⟨ 1 , 1 ⟩] ∼ ) = 𝑥)
62	52, 61	jca 511	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) → (([⟨ 1 , 1 ⟩] ∼ (.r‘𝐿)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐿)[⟨ 1 , 1 ⟩] ∼ ) = 𝑥))
63	62	ralrimiva 3121	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐿)(([⟨ 1 , 1 ⟩] ∼ (.r‘𝐿)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐿)[⟨ 1 , 1 ⟩] ∼ ) = 𝑥))
64		eqid 2729	. . . . 5 ⊢ (Base‘𝐿) = (Base‘𝐿)
65		eqid 2729	. . . . 5 ⊢ (1r‘𝐿) = (1r‘𝐿)
66	64, 35, 65	isringid 20156	. . . 4 ⊢ (𝐿 ∈ Ring → (([⟨ 1 , 1 ⟩] ∼ ∈ (Base‘𝐿) ∧ ∀𝑥 ∈ (Base‘𝐿)(([⟨ 1 , 1 ⟩] ∼ (.r‘𝐿)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐿)[⟨ 1 , 1 ⟩] ∼ ) = 𝑥)) ↔ (1r‘𝐿) = [⟨ 1 , 1 ⟩] ∼ ))
67	66	biimpa 476	. . 3 ⊢ ((𝐿 ∈ Ring ∧ ([⟨ 1 , 1 ⟩] ∼ ∈ (Base‘𝐿) ∧ ∀𝑥 ∈ (Base‘𝐿)(([⟨ 1 , 1 ⟩] ∼ (.r‘𝐿)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐿)[⟨ 1 , 1 ⟩] ∼ ) = 𝑥))) → (1r‘𝐿) = [⟨ 1 , 1 ⟩] ∼ )
68	10, 28, 63, 67	syl12anc 836	. 2 ⊢ (𝜑 → (1r‘𝐿) = [⟨ 1 , 1 ⟩] ∼ )
69	1, 68	eqtr4id 2783	1 ⊢ (𝜑 → 𝐼 = (1r‘𝐿))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3903  ⟨cop 4583   × cxp 5617  ‘cfv 6482  (class class class)co 7349  [cec 8623   / cqs 8624  Basecbs 17120  +_gcplusg 17161  ._rcmulr 17162  0_gc0g 17343  SubMndcsubmnd 18656  -_gcsg 18814  mulGrpcmgp 20025  1_rcur 20066  Ringcrg 20118  CRingccrg 20119   ~_RL cerl 33193   RLocal crloc 33194

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 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

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 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-er 8625  df-ec 8627  df-qs 8631  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-sup 9332  df-inf 9333  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-z 12472  df-dec 12592  df-uz 12736  df-fz 13411  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-sca 17177  df-vsca 17178  df-ip 17179  df-tset 17180  df-ple 17181  df-ds 17183  df-0g 17345  df-imas 17412  df-qus 17413  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-submnd 18658  df-grp 18815  df-minusg 18816  df-sbg 18817  df-cmn 19661  df-abl 19662  df-mgp 20026  df-rng 20038  df-ur 20067  df-ring 20120  df-cring 20121  df-erl 33195  df-rloc 33196

This theorem is referenced by:  rlocf1  33213  fracfld  33247  zringfrac  33491