Theorem idomsubr 33389

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 6836	. . 3 ⊢ (𝑓 = ( Frac ‘𝑅) → (SubRing‘𝑓) = (SubRing‘( Frac ‘𝑅)))
2		oveq1 7369	. . . 4 ⊢ (𝑓 = ( Frac ‘𝑅) → (𝑓 ↾s 𝑠) = (( Frac ‘𝑅) ↾s 𝑠))
3	2	breq2d 5098	. . 3 ⊢ (𝑓 = ( Frac ‘𝑅) → (𝑅 ≃𝑟 (𝑓 ↾s 𝑠) ↔ 𝑅 ≃𝑟 (( Frac ‘𝑅) ↾s 𝑠)))
4	1, 3	rexeqbidv 3313	. 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 33388	. 2 ⊢ (𝜑 → ( Frac ‘𝑅) ∈ Field)
7		oveq2 7370	. . . 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 5098	. . 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 2737	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
10		eqid 2737	. . . . 5 ⊢ (RLReg‘𝑅) = (RLReg‘𝑅)
11		eqid 2737	. . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅)
12	5	idomcringd 20699	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing)
13		eqid 2737	. . . . 5 ⊢ (𝑅 ~RL (RLReg‘𝑅)) = (𝑅 ~RL (RLReg‘𝑅))
14		opeq1 4817	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ⟨𝑥, (1r‘𝑅)⟩ = ⟨𝑦, (1r‘𝑅)⟩)
15	14	eceq1d 8679	. . . . . 6 ⊢ (𝑥 = 𝑦 → [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨𝑦, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))
16	15	cbvmptv 5190	. . . . 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 33387	. . . 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 20577	. . . 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 3946	. . . . . 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 2737	. . . . . . . 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 20574	. . . . . . 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 6787	. . . . . . 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 6732	. . . . . . . . . . 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 6672	. . . . . . . . 9 ⊢ (𝜑 → ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ⊆ (((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))))
32		eqid 2737	. . . . . . . . . 10 ⊢ ( Frac ‘𝑅) = ( Frac ‘𝑅)
33	9, 10, 32, 13	fracbas 33385	. . . . . . . . 9 ⊢ (((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))) = (Base‘( Frac ‘𝑅))
34	31, 33	sseqtrdi 3963	. . . . . . . 8 ⊢ (𝜑 → ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ⊆ (Base‘( Frac ‘𝑅)))
35		eqid 2737	. . . . . . . . 9 ⊢ (Base‘( Frac ‘𝑅)) = (Base‘( Frac ‘𝑅))
36	22, 35	ressbas2 17203	. . . . . . . 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 6773	. . . . . 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 2737	. . . . . 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 20466	. . . . 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 584	. . . 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 20477	. . . 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 3568	. 2 ⊢ (𝜑 → ∃𝑠 ∈ (SubRing‘( Frac ‘𝑅))𝑅 ≃𝑟 (( Frac ‘𝑅) ↾s 𝑠))
46	4, 6, 45	rspcedvdw 3568	1 ⊢ (𝜑 → ∃𝑓 ∈ Field ∃𝑠 ∈ (SubRing‘𝑓)𝑅 ≃𝑟 (𝑓 ↾s 𝑠))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3062   ⊆ wss 3890  ⟨cop 4574   class class class wbr 5086   ↦ cmpt 5167   × cxp 5624  ran crn 5627  ⟶wf 6490  –1-1→wf1 6491  –1-1-onto→wf1o 6493  ‘cfv 6494  (class class class)co 7362  [cec 8636   / cqs 8637  Basecbs 17174   ↾_s cress 17195  1_rcur 20157   RingHom crh 20444   RingIso crs 20445   ≃_𝑟 cric 20446  SubRingcsubrg 20541  RLRegcrlreg 20663  IDomncidom 20665  Fieldcfield 20702   ~_RL cerl 33333   Frac cfrac 33382

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 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-tpos 8171  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-er 8638  df-ec 8640  df-qs 8644  df-map 8770  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-sup 9350  df-inf 9351  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-2 12239  df-3 12240  df-4 12241  df-5 12242  df-6 12243  df-7 12244  df-8 12245  df-9 12246  df-n0 12433  df-z 12520  df-dec 12640  df-uz 12784  df-fz 13457  df-struct 17112  df-sets 17129  df-slot 17147  df-ndx 17159  df-base 17175  df-ress 17196  df-plusg 17228  df-mulr 17229  df-sca 17231  df-vsca 17232  df-ip 17233  df-tset 17234  df-ple 17235  df-ds 17237  df-0g 17399  df-imas 17467  df-qus 17468  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-mhm 18746  df-submnd 18747  df-grp 18907  df-minusg 18908  df-sbg 18909  df-subg 19094  df-ghm 19183  df-cmn 19752  df-abl 19753  df-mgp 20117  df-rng 20129  df-ur 20158  df-ring 20211  df-cring 20212  df-oppr 20312  df-dvdsr 20332  df-unit 20333  df-invr 20363  df-rhm 20447  df-rim 20448  df-ric 20450  df-nzr 20485  df-subrng 20518  df-subrg 20542  df-rlreg 20666  df-domn 20667  df-idom 20668  df-drng 20703  df-field 20704  df-erl 33335  df-rloc 33336  df-frac 33383

This theorem is referenced by: (None)