Theorem idomsubr 33291

Description: Every integral domain is isomorphic with a subring of some field. (Proposed by Gerard Lang, 10-May-2025.) (Contributed by Thierry Arnoux, 10-May-2025.)

Hypothesis

Ref	Expression
idomsubr.1	⊢ (𝜑 → 𝑅 ∈ IDomn)

Assertion

Ref	Expression
idomsubr	⊢ (𝜑 → ∃𝑓 ∈ Field ∃𝑠 ∈ (SubRing‘𝑓)𝑅 ≃𝑟 (𝑓 ↾s 𝑠))

Distinct variable groups:   𝑅,𝑓,𝑠   𝜑,𝑓,𝑠

Proof of Theorem idomsubr

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6907	. . 3 ⊢ (𝑓 = ( Frac ‘𝑅) → (SubRing‘𝑓) = (SubRing‘( Frac ‘𝑅)))
2		oveq1 7438	. . . 4 ⊢ (𝑓 = ( Frac ‘𝑅) → (𝑓 ↾s 𝑠) = (( Frac ‘𝑅) ↾s 𝑠))
3	2	breq2d 5160	. . 3 ⊢ (𝑓 = ( Frac ‘𝑅) → (𝑅 ≃𝑟 (𝑓 ↾s 𝑠) ↔ 𝑅 ≃𝑟 (( Frac ‘𝑅) ↾s 𝑠)))
4	1, 3	rexeqbidv 3345	. 2 ⊢ (𝑓 = ( Frac ‘𝑅) → (∃𝑠 ∈ (SubRing‘𝑓)𝑅 ≃𝑟 (𝑓 ↾s 𝑠) ↔ ∃𝑠 ∈ (SubRing‘( Frac ‘𝑅))𝑅 ≃𝑟 (( Frac ‘𝑅) ↾s 𝑠)))
5		idomsubr.1	. . 3 ⊢ (𝜑 → 𝑅 ∈ IDomn)
6	5	fracfld 33290	. 2 ⊢ (𝜑 → ( Frac ‘𝑅) ∈ Field)
7		oveq2 7439	. . . 4 ⊢ (𝑠 = ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) → (( Frac ‘𝑅) ↾s 𝑠) = (( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))))
8	7	breq2d 5160	. . 3 ⊢ (𝑠 = ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) → (𝑅 ≃𝑟 (( Frac ‘𝑅) ↾s 𝑠) ↔ 𝑅 ≃𝑟 (( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))))
9		eqid 2735	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
10		eqid 2735	. . . . 5 ⊢ (RLReg‘𝑅) = (RLReg‘𝑅)
11		eqid 2735	. . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅)
12	5	idomcringd 20744	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing)
13		eqid 2735	. . . . 5 ⊢ (𝑅 ~RL (RLReg‘𝑅)) = (𝑅 ~RL (RLReg‘𝑅))
14		opeq1 4878	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ⟨𝑥, (1r‘𝑅)⟩ = ⟨𝑦, (1r‘𝑅)⟩)
15	14	eceq1d 8784	. . . . . 6 ⊢ (𝑥 = 𝑦 → [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨𝑦, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))
16	15	cbvmptv 5261	. . . . 5 ⊢ (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) = (𝑦 ∈ (Base‘𝑅) ↦ [⟨𝑦, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))
17	9, 10, 11, 12, 13, 16	fracf1 33289	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))):(Base‘𝑅)–1-1→(((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))) ∧ (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (𝑅 RingHom ( Frac ‘𝑅))))
18		rnrhmsubrg 20622	. . . 4 ⊢ ((𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (𝑅 RingHom ( Frac ‘𝑅)) → ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (SubRing‘( Frac ‘𝑅)))
19	17, 18	simpl2im 503	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (SubRing‘( Frac ‘𝑅)))
20		ssidd 4019	. . . . . 6 ⊢ (𝜑 → ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ⊆ ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))
21	17	simprd 495	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (𝑅 RingHom ( Frac ‘𝑅)))
22		eqid 2735	. . . . . . . 8 ⊢ (( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))) = (( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))
23	22	resrhm2b 20619	. . . . . . 7 ⊢ ((ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (SubRing‘( Frac ‘𝑅)) ∧ ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ⊆ ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))) → ((𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (𝑅 RingHom ( Frac ‘𝑅)) ↔ (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (𝑅 RingHom (( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))))))
24	23	biimpa 476	. . . . . 6 ⊢ (((ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (SubRing‘( Frac ‘𝑅)) ∧ ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ⊆ ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))) ∧ (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (𝑅 RingHom ( Frac ‘𝑅))) → (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (𝑅 RingHom (( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))))
25	19, 20, 21, 24	syl21anc 838	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (𝑅 RingHom (( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))))
26	17	simpld 494	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))):(Base‘𝑅)–1-1→(((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))))
27		f1f1orn 6860	. . . . . . 7 ⊢ ((𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))):(Base‘𝑅)–1-1→(((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))) → (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))):(Base‘𝑅)–1-1-onto→ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))
28	26, 27	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))):(Base‘𝑅)–1-1-onto→ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))
29		f1f 6805	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))):(Base‘𝑅)–1-1→(((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))) → (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))):(Base‘𝑅)⟶(((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))))
30	26, 29	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))):(Base‘𝑅)⟶(((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))))
31	30	frnd 6745	. . . . . . . . 9 ⊢ (𝜑 → ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ⊆ (((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))))
32		eqid 2735	. . . . . . . . . 10 ⊢ ( Frac ‘𝑅) = ( Frac ‘𝑅)
33	9, 10, 32, 13	fracbas 33287	. . . . . . . . 9 ⊢ (((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))) = (Base‘( Frac ‘𝑅))
34	31, 33	sseqtrdi 4046	. . . . . . . 8 ⊢ (𝜑 → ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ⊆ (Base‘( Frac ‘𝑅)))
35		eqid 2735	. . . . . . . . 9 ⊢ (Base‘( Frac ‘𝑅)) = (Base‘( Frac ‘𝑅))
36	22, 35	ressbas2 17283	. . . . . . . 8 ⊢ (ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ⊆ (Base‘( Frac ‘𝑅)) → ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) = (Base‘(( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))))
37	34, 36	syl 17	. . . . . . 7 ⊢ (𝜑 → ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) = (Base‘(( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))))
38	37	f1oeq3d 6846	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))):(Base‘𝑅)–1-1-onto→ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ↔ (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))):(Base‘𝑅)–1-1-onto→(Base‘(( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))))))
39	28, 38	mpbid 232	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))):(Base‘𝑅)–1-1-onto→(Base‘(( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))))
40		eqid 2735	. . . . . 6 ⊢ (Base‘(( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))) = (Base‘(( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))))
41	9, 40	isrim 20509	. . . . 5 ⊢ ((𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (𝑅 RingIso (( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))) ↔ ((𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (𝑅 RingHom (( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))) ∧ (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))):(Base‘𝑅)–1-1-onto→(Base‘(( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))))))
42	25, 39, 41	sylanbrc 583	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (𝑅 RingIso (( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))))
43		brrici 20522	. . . 4 ⊢ ((𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (𝑅 RingIso (( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))) → 𝑅 ≃𝑟 (( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))))
44	42, 43	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ≃𝑟 (( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))))
45	8, 19, 44	rspcedvdw 3625	. 2 ⊢ (𝜑 → ∃𝑠 ∈ (SubRing‘( Frac ‘𝑅))𝑅 ≃𝑟 (( Frac ‘𝑅) ↾s 𝑠))
46	4, 6, 45	rspcedvdw 3625	1 ⊢ (𝜑 → ∃𝑓 ∈ Field ∃𝑠 ∈ (SubRing‘𝑓)𝑅 ≃𝑟 (𝑓 ↾s 𝑠))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∃wrex 3068   ⊆ wss 3963  ⟨cop 4637   class class class wbr 5148   ↦ cmpt 5231   × cxp 5687  ran crn 5690  ⟶wf 6559  –1-1→wf1 6560  –1-1-onto→wf1o 6562  ‘cfv 6563  (class class class)co 7431  [cec 8742   / cqs 8743  Basecbs 17245   ↾_s cress 17274  1_rcur 20199   RingHom crh 20486   RingIso crs 20487   ≃_𝑟 cric 20488  SubRingcsubrg 20586  RLRegcrlreg 20708  IDomncidom 20710  Fieldcfield 20747   ~_RL cerl 33240   Frac cfrac 33284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-ec 8746  df-qs 8750  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-0g 17488  df-imas 17555  df-qus 17556  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-ghm 19244  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-cring 20254  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-invr 20405  df-rhm 20489  df-rim 20490  df-ric 20492  df-nzr 20530  df-subrng 20563  df-subrg 20587  df-rlreg 20711  df-domn 20712  df-idom 20713  df-drng 20748  df-field 20749  df-erl 33242  df-rloc 33243  df-frac 33285

This theorem is referenced by: (None)