Theorem idomsubr 33394

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 6828	. . 3 ⊢ (𝑓 = ( Frac ‘𝑅) → (SubRing‘𝑓) = (SubRing‘( Frac ‘𝑅)))
2		oveq1 7364	. . . 4 ⊢ (𝑓 = ( Frac ‘𝑅) → (𝑓 ↾s 𝑠) = (( Frac ‘𝑅) ↾s 𝑠))
3	2	breq2d 5085	. . 3 ⊢ (𝑓 = ( Frac ‘𝑅) → (𝑅 ≃𝑟 (𝑓 ↾s 𝑠) ↔ 𝑅 ≃𝑟 (( Frac ‘𝑅) ↾s 𝑠)))
4	1, 3	rexeqbidv 3314	. 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 33393	. 2 ⊢ (𝜑 → ( Frac ‘𝑅) ∈ Field)
7		oveq2 7365	. . . 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 5085	. . 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 2739	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
10		eqid 2739	. . . . 5 ⊢ (RLReg‘𝑅) = (RLReg‘𝑅)
11		eqid 2739	. . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅)
12	5	idomcringd 20700	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing)
13		eqid 2739	. . . . 5 ⊢ (𝑅 ~RL (RLReg‘𝑅)) = (𝑅 ~RL (RLReg‘𝑅))
14		opeq1 4805	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ⟨𝑥, (1r‘𝑅)⟩ = ⟨𝑦, (1r‘𝑅)⟩)
15	14	eceq1d 8675	. . . . . 6 ⊢ (𝑥 = 𝑦 → [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨𝑦, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))
16	15	cbvmptv 5177	. . . . 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 33392	. . . 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 20578	. . . 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 508	. . 3 ⊢ (𝜑 → ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (SubRing‘( Frac ‘𝑅)))
20		ssidd 3938	. . . . . 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 496	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (𝑅 RingHom ( Frac ‘𝑅)))
22		eqid 2739	. . . . . . . 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 20575	. . . . . . 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 477	. . . . . 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 843	. . . . 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 495	. . . . . . 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 6779	. . . . . . 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 6724	. . . . . . . . . . 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 6664	. . . . . . . . 9 ⊢ (𝜑 → ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ⊆ (((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))))
32		eqid 2739	. . . . . . . . . 10 ⊢ ( Frac ‘𝑅) = ( Frac ‘𝑅)
33	9, 10, 32, 13	fracbas 33390	. . . . . . . . 9 ⊢ (((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))) = (Base‘( Frac ‘𝑅))
34	31, 33	sseqtrdi 3955	. . . . . . . 8 ⊢ (𝜑 → ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ⊆ (Base‘( Frac ‘𝑅)))
35		eqid 2739	. . . . . . . . 9 ⊢ (Base‘( Frac ‘𝑅)) = (Base‘( Frac ‘𝑅))
36	22, 35	ressbas2 17200	. . . . . . . 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 6765	. . . . . 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 233	. . . . 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 2739	. . . . . 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 20464	. . . . 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 589	. . . 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 3563	. 2 ⊢ (𝜑 → ∃𝑠 ∈ (SubRing‘( Frac ‘𝑅))𝑅 ≃𝑟 (( Frac ‘𝑅) ↾s 𝑠))
46	4, 6, 45	rspcedvdw 3563	1 ⊢ (𝜑 → ∃𝑓 ∈ Field ∃𝑠 ∈ (SubRing‘𝑓)𝑅 ≃𝑟 (𝑓 ↾s 𝑠))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∃wrex 3063   ⊆ wss 3883  ⟨cop 4562   class class class wbr 5073   ↦ cmpt 5154   × cxp 5617  ran crn 5620  ⟶wf 6482  –1-1→wf1 6483  –1-1-onto→wf1o 6485  ‘cfv 6486  (class class class)co 7357  [cec 8632   / cqs 8633  Basecbs 17171   ↾_s cress 17192  1_rcur 20154   RingHom crh 20441   RingIso crs 20442   ≃_𝑟 cric 20443  SubRingcsubrg 20542  RLRegcrlreg 20664  IDomncidom 20666  Fieldcfield 20703   ~_RL cerl 33335   Frac cfrac 33387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-tp 4561  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-tr 5181  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7314  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7808  df-1st 7932  df-2nd 7933  df-tpos 8167  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-er 8634  df-ec 8636  df-qs 8640  df-map 8766  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-sup 9346  df-inf 9347  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372  df-nn 12167  df-2 12236  df-3 12237  df-4 12238  df-5 12239  df-6 12240  df-7 12241  df-8 12242  df-9 12243  df-n0 12430  df-z 12517  df-dec 12637  df-uz 12781  df-fz 13454  df-struct 17109  df-sets 17126  df-slot 17144  df-ndx 17156  df-base 17172  df-ress 17193  df-plusg 17225  df-mulr 17226  df-sca 17228  df-vsca 17229  df-ip 17230  df-tset 17231  df-ple 17232  df-ds 17234  df-0g 17396  df-imas 17464  df-qus 17465  df-mgm 18600  df-sgrp 18679  df-mnd 18695  df-mhm 18743  df-submnd 18744  df-grp 18904  df-minusg 18905  df-sbg 18906  df-subg 19091  df-ghm 19180  df-cmn 19749  df-abl 19750  df-mgp 20114  df-rng 20126  df-ur 20155  df-ring 20208  df-cring 20209  df-oppr 20309  df-dvdsr 20329  df-unit 20330  df-invr 20360  df-rhm 20444  df-rim 20445  df-ric 20447  df-nzr 20486  df-subrng 20519  df-subrg 20543  df-rlreg 20667  df-domn 20668  df-idom 20669  df-drng 20704  df-field 20705  df-erl 33337  df-rloc 33338  df-frac 33388

This theorem is referenced by: (None)