Theorem rloc1r 33333

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 2736	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		eqid 2736	. . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅)
4		eqid 2736	. . . . 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 33331	. . . 4 ⊢ (𝜑 → 𝐿 ∈ CRing)
10	9	crngringd 20227	. . 3 ⊢ (𝜑 → 𝐿 ∈ Ring)
11		eqid 2736	. . . . . . . . . 10 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
12	11, 2	mgpbas 20126	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
13	12	submss 18777	. . . . . . . 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 20164	. . . . . . . . 9 ⊢ 1 = (0g‘(mulGrp‘𝑅))
17	16	subm0cl 18779	. . . . . . . 8 ⊢ (𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) → 1 ∈ 𝑆)
18	8, 17	syl 17	. . . . . . 7 ⊢ (𝜑 → 1 ∈ 𝑆)
19	14, 18	sseldd 3922	. . . . . 6 ⊢ (𝜑 → 1 ∈ (Base‘𝑅))
20	19, 18	opelxpd 5670	. . . . 5 ⊢ (𝜑 → ⟨ 1 , 1 ⟩ ∈ ((Base‘𝑅) × 𝑆))
21	6	ovexi 7401	. . . . . 6 ⊢ ∼ ∈ V
22	21	ecelqsi 8716	. . . . 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 2736	. . . . 5 ⊢ (-g‘𝑅) = (-g‘𝑅)
26		eqid 2736	. . . . 5 ⊢ ((Base‘𝑅) × 𝑆) = ((Base‘𝑅) × 𝑆)
27	2, 24, 3, 25, 26, 5, 6, 7, 14	rlocbas 33328	. . . 4 ⊢ (𝜑 → (((Base‘𝑅) × 𝑆) / ∼ ) = (Base‘𝐿))
28	23, 27	eleqtrd 2838	. . 3 ⊢ (𝜑 → [⟨ 1 , 1 ⟩] ∼ ∈ (Base‘𝐿))
29	7	ad4antr 733	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → 𝑅 ∈ CRing)
30	8	ad4antr 733	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
31	19	ad4antr 733	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → 1 ∈ (Base‘𝑅))
32		simpllr 776	. . . . . . . . 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 769	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → 𝑏 ∈ 𝑆)
35		eqid 2736	. . . . . . . . 9 ⊢ (.r‘𝐿) = (.r‘𝐿)
36	2, 3, 4, 5, 6, 29, 30, 31, 32, 33, 34, 35	rlocmulval 33330	. . . . . . . 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 20227	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → 𝑅 ∈ Ring)
38	2, 3, 15, 37, 32	ringlidmd 20253	. . . . . . . . . 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 3922	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → 𝑏 ∈ (Base‘𝑅))
41	2, 3, 15, 37, 40	ringlidmd 20253	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → ( 1 (.r‘𝑅)𝑏) = 𝑏)
42	38, 41	opeq12d 4824	. . . . . . . . 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 8684	. . . . . . . 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 2771	. . . . . . 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 7383	. . . . . . 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 2781	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → ([⟨ 1 , 1 ⟩] ∼ (.r‘𝐿)𝑥) = 𝑥)
48	27	eqcomd 2742	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐿) = (((Base‘𝑅) × 𝑆) / ∼ ))
49	48	eleq2d 2822	. . . . . . . 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 33327	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) → ∃𝑎 ∈ (Base‘𝑅)∃𝑏 ∈ 𝑆 𝑥 = [⟨𝑎, 𝑏⟩] ∼ )
52	47, 51	r19.29vva 3197	. . . . 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 33330	. . . . . . . 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 20254	. . . . . . . . . 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 20254	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → (𝑏(.r‘𝑅) 1 ) = 𝑏)
56	54, 55	opeq12d 4824	. . . . . . . . 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 8684	. . . . . . . 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 2771	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → ([⟨𝑎, 𝑏⟩] ∼ (.r‘𝐿)[⟨ 1 , 1 ⟩] ∼ ) = [⟨𝑎, 𝑏⟩] ∼ )
59	45	oveq1d 7382	. . . . . . 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 2781	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Base‘𝐿)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑏 ∈ 𝑆) ∧ 𝑥 = [⟨𝑎, 𝑏⟩] ∼ ) → (𝑥(.r‘𝐿)[⟨ 1 , 1 ⟩] ∼ ) = 𝑥)
61	60, 51	r19.29vva 3197	. . . . 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 3129	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐿)(([⟨ 1 , 1 ⟩] ∼ (.r‘𝐿)𝑥) = 𝑥 ∧ (𝑥(.r‘𝐿)[⟨ 1 , 1 ⟩] ∼ ) = 𝑥))
64		eqid 2736	. . . . 5 ⊢ (Base‘𝐿) = (Base‘𝐿)
65		eqid 2736	. . . . 5 ⊢ (1r‘𝐿) = (1r‘𝐿)
66	64, 35, 65	isringid 20252	. . . 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 837	. 2 ⊢ (𝜑 → (1r‘𝐿) = [⟨ 1 , 1 ⟩] ∼ )
69	1, 68	eqtr4id 2790	1 ⊢ (𝜑 → 𝐼 = (1r‘𝐿))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051   ⊆ wss 3889  ⟨cop 4573   × cxp 5629  ‘cfv 6498  (class class class)co 7367  [cec 8641   / cqs 8642  Basecbs 17179  +_gcplusg 17220  ._rcmulr 17221  0_gc0g 17402  SubMndcsubmnd 18750  -_gcsg 18911  mulGrpcmgp 20121  1_rcur 20162  Ringcrg 20214  CRingccrg 20215   ~_RL cerl 33314   RLocal crloc 33315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-ec 8645  df-qs 8649  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-sup 9355  df-inf 9356  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-dec 12645  df-uz 12789  df-fz 13462  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-0g 17404  df-imas 17472  df-qus 17473  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-submnd 18752  df-grp 18912  df-minusg 18913  df-sbg 18914  df-cmn 19757  df-abl 19758  df-mgp 20122  df-rng 20134  df-ur 20163  df-ring 20216  df-cring 20217  df-erl 33316  df-rloc 33317

This theorem is referenced by:  rlocf1  33334  fracfld  33369  zringfrac  33614