Theorem idomsubr 33409

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 6844	. . 3 ⊢ (𝑓 = ( Frac ‘𝑅) → (SubRing‘𝑓) = (SubRing‘( Frac ‘𝑅)))
2		oveq1 7377	. . . 4 ⊢ (𝑓 = ( Frac ‘𝑅) → (𝑓 ↾s 𝑠) = (( Frac ‘𝑅) ↾s 𝑠))
3	2	breq2d 5112	. . 3 ⊢ (𝑓 = ( Frac ‘𝑅) → (𝑅 ≃𝑟 (𝑓 ↾s 𝑠) ↔ 𝑅 ≃𝑟 (( Frac ‘𝑅) ↾s 𝑠)))
4	1, 3	rexeqbidv 3319	. 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 33408	. 2 ⊢ (𝜑 → ( Frac ‘𝑅) ∈ Field)
7		oveq2 7378	. . . 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 5112	. . 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 20677	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing)
13		eqid 2737	. . . . 5 ⊢ (𝑅 ~RL (RLReg‘𝑅)) = (𝑅 ~RL (RLReg‘𝑅))
14		opeq1 4831	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ⟨𝑥, (1r‘𝑅)⟩ = ⟨𝑦, (1r‘𝑅)⟩)
15	14	eceq1d 8688	. . . . . 6 ⊢ (𝑥 = 𝑦 → [⟨𝑥, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨𝑦, (1r‘𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))
16	15	cbvmptv 5204	. . . . 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 33407	. . . 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 20555	. . . 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 3959	. . . . . 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 20552	. . . . . . 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 6795	. . . . . . 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 6740	. . . . . . . . . . 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 6680	. . . . . . . . 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 33405	. . . . . . . . 9 ⊢ (((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))) = (Base‘( Frac ‘𝑅))
34	31, 33	sseqtrdi 3976	. . . . . . . 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 17179	. . . . . . . 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 6781	. . . . . 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 20444	. . . . 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 20455	. . . 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 3581	. 2 ⊢ (𝜑 → ∃𝑠 ∈ (SubRing‘( Frac ‘𝑅))𝑅 ≃𝑟 (( Frac ‘𝑅) ↾s 𝑠))
46	4, 6, 45	rspcedvdw 3581	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 3903  ⟨cop 4588   class class class wbr 5100   ↦ cmpt 5181   × cxp 5632  ran crn 5635  ⟶wf 6498  –1-1→wf1 6499  –1-1-onto→wf1o 6501  ‘cfv 6502  (class class class)co 7370  [cec 8645   / cqs 8646  Basecbs 17150   ↾_s cress 17171  1_rcur 20133   RingHom crh 20422   RingIso crs 20423   ≃_𝑟 cric 20424  SubRingcsubrg 20519  RLRegcrlreg 20641  IDomncidom 20643  Fieldcfield 20680   ~_RL cerl 33353   Frac cfrac 33402

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 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117

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 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-1st 7945  df-2nd 7946  df-tpos 8180  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-er 8647  df-ec 8649  df-qs 8653  df-map 8779  df-en 8898  df-dom 8899  df-sdom 8900  df-fin 8901  df-sup 9359  df-inf 9360  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-nn 12160  df-2 12222  df-3 12223  df-4 12224  df-5 12225  df-6 12226  df-7 12227  df-8 12228  df-9 12229  df-n0 12416  df-z 12503  df-dec 12622  df-uz 12766  df-fz 13438  df-struct 17088  df-sets 17105  df-slot 17123  df-ndx 17135  df-base 17151  df-ress 17172  df-plusg 17204  df-mulr 17205  df-sca 17207  df-vsca 17208  df-ip 17209  df-tset 17210  df-ple 17211  df-ds 17213  df-0g 17375  df-imas 17443  df-qus 17444  df-mgm 18579  df-sgrp 18658  df-mnd 18674  df-mhm 18722  df-submnd 18723  df-grp 18883  df-minusg 18884  df-sbg 18885  df-subg 19070  df-ghm 19159  df-cmn 19728  df-abl 19729  df-mgp 20093  df-rng 20105  df-ur 20134  df-ring 20187  df-cring 20188  df-oppr 20290  df-dvdsr 20310  df-unit 20311  df-invr 20341  df-rhm 20425  df-rim 20426  df-ric 20428  df-nzr 20463  df-subrng 20496  df-subrg 20520  df-rlreg 20644  df-domn 20645  df-idom 20646  df-drng 20681  df-field 20682  df-erl 33355  df-rloc 33356  df-frac 33403

This theorem is referenced by: (None)