Theorem rloc1r 33834

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 2761	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		eqid 2761	. . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅)
4		eqid 2761	. . . . 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 33832	. . . 4 ⊢ (𝜑 → 𝐿 ∈ CRing)
10	9	crngringd 20473	. . 3 ⊢ (𝜑 → 𝐿 ∈ Ring)
11		eqid 2761	. . . . . . . . . 10 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
12	11, 2	mgpbas 20365	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
13	12	submss 19004	. . . . . . . 8 ⊢ (𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) → 𝑆 ⊆ (Base‘𝑅))
14	8, 13	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑅))
15		rloc0g.2	. . . . . . . . . 10 ⊢ 1 = (1r‘𝑅)
16	11, 15	ringidval 20409	. . . . . . . . 9 ⊢ 1 = (0g‘(mulGrp‘𝑅))
17	16	subm0cl 19006	. . . . . . . 8 ⊢ (𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) → 1 ∈ 𝑆)
18	8, 17	syl 18	. . . . . . 7 ⊢ (𝜑 → 1 ∈ 𝑆)
19	14, 18	sseldd 3932	. . . . . 6 ⊢ (𝜑 → 1 ∈ (Base‘𝑅))
20	19, 18	opelxpd 5690	. . . . 5 ⊢ (𝜑 → ⟨ 1 , 1 ⟩ ∈ ((Base‘𝑅) × 𝑆))
21	6	ovexi 7454	. . . . . 6 ⊢ ∼ ∈ V
22	21	ecelqsi 8790	. . . . 5 ⊢ (⟨ 1 , 1 ⟩ ∈ ((Base‘𝑅) × 𝑆) → [⟨ 1 , 1 ⟩] ∼ ∈ (((Base‘𝑅) × 𝑆) / ∼ ))
23	20, 22	syl 18	. . . 4 ⊢ (𝜑 → [⟨ 1 , 1 ⟩] ∼ ∈ (((Base‘𝑅) × 𝑆) / ∼ ))
24		rloc0g.1	. . . . 5 ⊢ 0 = (0g‘𝑅)
25		eqid 2761	. . . . 5 ⊢ (-g‘𝑅) = (-g‘𝑅)
26		eqid 2761	. . . . 5 ⊢ ((Base‘𝑅) × 𝑆) = ((Base‘𝑅) × 𝑆)
27	2, 24, 3, 25, 26, 5, 6, 7, 14	rlocbas 33829	. . . 4 ⊢ (𝜑 → (((Base‘𝑅) × 𝑆) / ∼ ) = (Base‘𝐿))
28	23, 27	eleqtrd 2863	. . 3 ⊢ (𝜑 → [⟨ 1 , 1 ⟩] ∼ ∈ (Base‘𝐿))
29	7	ad4antr 745	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → 𝑅 ∈ CRing)
30	8	ad4antr 745	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
31	19	ad4antr 745	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → 1 ∈ (Base‘𝑅))
32		simpllr 788	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → 𝑎 ∈ (Base‘𝑅))
33	30, 17	syl 18	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → 1 ∈ 𝑆)
34		simplr 781	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → 𝑏 ∈ 𝑆)
35		eqid 2761	. . . . . . . . 9 ⊢ (.r‘𝐿) = (.r‘𝐿)
36	2, 3, 4, 5, 6, 29, 30, 31, 32, 33, 34, 35	rlocmulval 33831	. . . . . . . 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 20473	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → 𝑅 ∈ Ring)
38	2, 3, 15, 37, 32	ringlidmd 20501	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → ( 1 (.r‘𝑅)𝑎) = 𝑎)
39	30, 13	syl 18	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → 𝑆 ⊆ (Base‘𝑅))
40	39, 34	sseldd 3932	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → 𝑏 ∈ (Base‘𝑅))
41	2, 3, 15, 37, 40	ringlidmd 20501	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → ( 1 (.r‘𝑅)𝑏) = 𝑏)
42	38, 41	opeq12d 4841	. . . . . . . . 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 8758	. . . . . . . 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 2796	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → ([⟨ 1 , 1 ⟩] ∼ (.r‘𝐿)[⟨𝑎, 𝑏⟩] ∼ ) = [⟨𝑎, 𝑏⟩] ∼ )
45		simpr 490	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → 𝑥 = [⟨𝑎, 𝑏⟩] ∼ )
46	45	oveq2d 7436	. . . . . . 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 2806	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → ([⟨ 1 , 1 ⟩] ∼ (.r‘𝐿)𝑥) = 𝑥)
48	27	eqcomd 2767	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐿) = (((Base‘𝑅) × 𝑆) / ∼ ))
49	48	eleq2d 2847	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐿) ↔ 𝑥 ∈ (((Base‘𝑅) × 𝑆) / ∼ )))
50	49	biimpa 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) → 𝑥 ∈ (((Base‘𝑅) × 𝑆) / ∼ ))
51	50	elrlocbasi 33828	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) → ∃𝑎 ∈ (Base‘𝑅)∃𝑏 ∈ 𝑆 𝑥 = [⟨𝑎, 𝑏⟩] ∼ )
52	47, 51	r19.29vva 3223	. . . . 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 33831	. . . . . . . 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 20502	. . . . . . . . . 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 20502	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → (𝑏(.r‘𝑅) 1 ) = 𝑏)
56	54, 55	opeq12d 4841	. . . . . . . . 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 8758	. . . . . . . 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 2796	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → ([⟨𝑎, 𝑏⟩] ∼ (.r‘𝐿)[⟨ 1 , 1 ⟩] ∼ ) = [⟨𝑎, 𝑏⟩] ∼ )
59	45	oveq1d 7435	. . . . . . 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 2806	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → (𝑥(.r‘𝐿)[⟨ 1 , 1 ⟩] ∼ ) = 𝑥)
61	60, 51	r19.29vva 3223	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) → (𝑥(.r‘𝐿)[⟨ 1 , 1 ⟩] ∼ ) = 𝑥)
62	52, 61	jca 521	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) → (([⟨ 1 , 1 ⟩] ∼ (.r‘𝐿)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐿)[⟨ 1 , 1 ⟩] ∼ ) = 𝑥))
63	62	ralrimiva 3155	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐿)(([⟨ 1 , 1 ⟩] ∼ (.r‘𝐿)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐿)[⟨ 1 , 1 ⟩] ∼ ) = 𝑥))
64		eqid 2761	. . . . 5 ⊢ (Base‘𝐿) = (Base‘𝐿)
65		eqid 2761	. . . . 5 ⊢ (1r‘𝐿) = (1r‘𝐿)
66	64, 35, 65	isringid 20500	. . . 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 482	. . 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 850	. 2 ⊢ (𝜑 → (1r‘𝐿) = [⟨ 1 , 1 ⟩] ∼ )
69	1, 68	eqtr4id 2815	1 ⊢ (𝜑 → 𝐼 = (1r‘𝐿))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077   ⊆ wss 3899  ⟨cop 4590   × cxp 5649  ‘cfv 6538  (class class class)co 7420  [cec 8715   / cqs 8716  Basecbs 17387  +_gcplusg 17428  ._rcmulr 17429  0_gc0g 17610  SubMndcsubmnd 18977  -_gcsg 19146  mulGrpcmgp 20360  1_rcur 20407  Ringcrg 20459  CRingccrg 20460   ~_RL cerl 33814   RLocal crloc 33815

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-ec 8719  df-qs 8723  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-sup 9434  df-inf 9435  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-struct 17325  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-mulr 17442  df-sca 17444  df-vsca 17445  df-ip 17446  df-tset 17447  df-ple 17448  df-ds 17450  df-0g 17612  df-imas 17680  df-qus 17681  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-submnd 18979  df-grp 19147  df-minusg 19148  df-sbg 19149  df-cmn 19996  df-abl 19997  df-mgp 20361  df-rng 20375  df-ur 20408  df-ring 20461  df-cring 20462  df-erl 33816  df-rloc 33817

This theorem is used by:  rlocf1  33835  rlocinvunit  33836  rlocisunit  33837  fracfld  33870  zringfrac  34086