Theorem idomsubr 33095

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 6896	. . 3 ⊢ (𝑓 = ( Frac ‘𝑅) → (SubRing‘𝑓) = (SubRing‘( Frac ‘𝑅)))
2		oveq1 7426	. . . 4 ⊢ (𝑓 = ( Frac ‘𝑅) → (𝑓 ↾s 𝑠) = (( Frac ‘𝑅) ↾s 𝑠))
3	2	breq2d 5161	. . 3 ⊢ (𝑓 = ( Frac ‘𝑅) → (𝑅 ≃𝑟 (𝑓 ↾s 𝑠) ↔ 𝑅 ≃𝑟 (( Frac ‘𝑅) ↾s 𝑠)))
4	1, 3	rexeqbidv 3330	. 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 33094	. 2 ⊢ (𝜑 → ( Frac ‘𝑅) ∈ Field)
7		oveq2 7427	. . . 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 5161	. . 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 2725	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
10		eqid 2725	. . . . 5 ⊢ (RLReg‘𝑅) = (RLReg‘𝑅)
11		eqid 2725	. . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅)
12	5	idomcringd 21273	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing)
13		eqid 2725	. . . . 5 ⊢ (𝑅 ~RL (RLReg‘𝑅)) = (𝑅 ~RL (RLReg‘𝑅))
14		opeq1 4875	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ⟨𝑥, (1r‘𝑅)⟩ = ⟨𝑦, (1r‘𝑅)⟩)
15	14	eceq1d 8764	. . . . . 6 ⊢ (𝑥 = 𝑦 → [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨𝑦, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))
16	15	cbvmptv 5262	. . . . 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 33093	. . . 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 20556	. . . 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 502	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (SubRing‘( Frac ‘𝑅)))
20		ssidd 4000	. . . . . 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 494	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (𝑅 RingHom ( Frac ‘𝑅)))
22		eqid 2725	. . . . . . . 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 20553	. . . . . . 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 475	. . . . . 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 836	. . . . 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 493	. . . . . . 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 6849	. . . . . . 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 6793	. . . . . . . . . . 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 6731	. . . . . . . . 9 ⊢ (𝜑 → ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ⊆ (((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))))
32		eqid 2725	. . . . . . . . . 10 ⊢ ( Frac ‘𝑅) = ( Frac ‘𝑅)
33	9, 10, 32, 13	fracbas 33091	. . . . . . . . 9 ⊢ (((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))) = (Base‘( Frac ‘𝑅))
34	31, 33	sseqtrdi 4027	. . . . . . . 8 ⊢ (𝜑 → ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ⊆ (Base‘( Frac ‘𝑅)))
35		eqid 2725	. . . . . . . . 9 ⊢ (Base‘( Frac ‘𝑅)) = (Base‘( Frac ‘𝑅))
36	22, 35	ressbas2 17221	. . . . . . . 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 6835	. . . . . 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 231	. . . . 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 2725	. . . . . 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 20443	. . . . 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 581	. . . 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 20456	. . . 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 3609	. 2 ⊢ (𝜑 → ∃𝑠 ∈ (SubRing‘( Frac ‘𝑅))𝑅 ≃𝑟 (( Frac ‘𝑅) ↾s 𝑠))
46	4, 6, 45	rspcedvdw 3609	1 ⊢ (𝜑 → ∃𝑓 ∈ Field ∃𝑠 ∈ (SubRing‘𝑓)𝑅 ≃𝑟 (𝑓 ↾s 𝑠))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∃wrex 3059   ⊆ wss 3944  ⟨cop 4636   class class class wbr 5149   ↦ cmpt 5232   × cxp 5676  ran crn 5679  ⟶wf 6545  –1-1→wf1 6546  –1-1-onto→wf1o 6548  ‘cfv 6549  (class class class)co 7419  [cec 8723   / cqs 8724  Basecbs 17183   ↾_s cress 17212  1_rcur 20133   RingHom crh 20420   RingIso crs 20421   ≃_𝑟 cric 20422  SubRingcsubrg 20518  Fieldcfield 20637  RLRegcrlreg 21243  IDomncidom 21245   ~_RL cerl 33043   Frac cfrac 33088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-tpos 8232  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-ec 8727  df-qs 8731  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9467  df-inf 9468  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12506  df-z 12592  df-dec 12711  df-uz 12856  df-fz 13520  df-struct 17119  df-sets 17136  df-slot 17154  df-ndx 17166  df-base 17184  df-ress 17213  df-plusg 17249  df-mulr 17250  df-sca 17252  df-vsca 17253  df-ip 17254  df-tset 17255  df-ple 17256  df-ds 17258  df-0g 17426  df-imas 17493  df-qus 17494  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-mhm 18743  df-submnd 18744  df-grp 18901  df-minusg 18902  df-sbg 18903  df-subg 19086  df-ghm 19176  df-cmn 19749  df-abl 19750  df-mgp 20087  df-rng 20105  df-ur 20134  df-ring 20187  df-cring 20188  df-oppr 20285  df-dvdsr 20308  df-unit 20309  df-invr 20339  df-rhm 20423  df-rim 20424  df-ric 20426  df-nzr 20464  df-subrng 20495  df-subrg 20520  df-drng 20638  df-field 20639  df-rlreg 21247  df-domn 21248  df-idom 21249  df-erl 33045  df-rloc 33046  df-frac 33089

This theorem is referenced by: (None)