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Theorem idomsubr 33311
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 (𝜑𝑅 ∈ IDomn)
Assertion
Ref Expression
idomsubr (𝜑 → ∃𝑓 ∈ Field ∃𝑠 ∈ (SubRing‘𝑓)𝑅𝑟 (𝑓s 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.
StepHypRef Expression
1 fveq2 6906 . . 3 (𝑓 = ( Frac ‘𝑅) → (SubRing‘𝑓) = (SubRing‘( Frac ‘𝑅)))
2 oveq1 7438 . . . 4 (𝑓 = ( Frac ‘𝑅) → (𝑓s 𝑠) = (( Frac ‘𝑅) ↾s 𝑠))
32breq2d 5155 . . 3 (𝑓 = ( Frac ‘𝑅) → (𝑅𝑟 (𝑓s 𝑠) ↔ 𝑅𝑟 (( Frac ‘𝑅) ↾s 𝑠)))
41, 3rexeqbidv 3347 . 2 (𝑓 = ( Frac ‘𝑅) → (∃𝑠 ∈ (SubRing‘𝑓)𝑅𝑟 (𝑓s 𝑠) ↔ ∃𝑠 ∈ (SubRing‘( Frac ‘𝑅))𝑅𝑟 (( Frac ‘𝑅) ↾s 𝑠)))
5 idomsubr.1 . . 3 (𝜑𝑅 ∈ IDomn)
65fracfld 33310 . 2 (𝜑 → ( Frac ‘𝑅) ∈ Field)
7 oveq2 7439 . . . 4 (𝑠 = ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) → (( Frac ‘𝑅) ↾s 𝑠) = (( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))))
87breq2d 5155 . . 3 (𝑠 = ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) → (𝑅𝑟 (( Frac ‘𝑅) ↾s 𝑠) ↔ 𝑅𝑟 (( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))))
9 eqid 2737 . . . . 5 (Base‘𝑅) = (Base‘𝑅)
10 eqid 2737 . . . . 5 (RLReg‘𝑅) = (RLReg‘𝑅)
11 eqid 2737 . . . . 5 (1r𝑅) = (1r𝑅)
125idomcringd 20727 . . . . 5 (𝜑𝑅 ∈ CRing)
13 eqid 2737 . . . . 5 (𝑅 ~RL (RLReg‘𝑅)) = (𝑅 ~RL (RLReg‘𝑅))
14 opeq1 4873 . . . . . . 7 (𝑥 = 𝑦 → ⟨𝑥, (1r𝑅)⟩ = ⟨𝑦, (1r𝑅)⟩)
1514eceq1d 8785 . . . . . 6 (𝑥 = 𝑦 → [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)) = [⟨𝑦, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))
1615cbvmptv 5255 . . . . 5 (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) = (𝑦 ∈ (Base‘𝑅) ↦ [⟨𝑦, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))
179, 10, 11, 12, 13, 16fracf1 33309 . . . 4 (𝜑 → ((𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))):(Base‘𝑅)–1-1→(((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))) ∧ (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (𝑅 RingHom ( Frac ‘𝑅))))
18 rnrhmsubrg 20605 . . . 4 ((𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (𝑅 RingHom ( Frac ‘𝑅)) → ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (SubRing‘( Frac ‘𝑅)))
1917, 18simpl2im 503 . . 3 (𝜑 → ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (SubRing‘( Frac ‘𝑅)))
20 ssidd 4007 . . . . . 6 (𝜑 → ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ⊆ ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))
2117simprd 495 . . . . . 6 (𝜑 → (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (𝑅 RingHom ( Frac ‘𝑅)))
22 eqid 2737 . . . . . . . 8 (( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))) = (( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))
2322resrhm2b 20602 . . . . . . 7 ((ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (SubRing‘( Frac ‘𝑅)) ∧ ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ⊆ ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))) → ((𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (𝑅 RingHom ( Frac ‘𝑅)) ↔ (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (𝑅 RingHom (( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))))))
2423biimpa 476 . . . . . 6 (((ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (SubRing‘( Frac ‘𝑅)) ∧ ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ⊆ ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))) ∧ (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (𝑅 RingHom ( Frac ‘𝑅))) → (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (𝑅 RingHom (( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))))
2519, 20, 21, 24syl21anc 838 . . . . 5 (𝜑 → (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (𝑅 RingHom (( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))))
2617simpld 494 . . . . . . 7 (𝜑 → (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))):(Base‘𝑅)–1-1→(((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))))
27 f1f1orn 6859 . . . . . . 7 ((𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))):(Base‘𝑅)–1-1→(((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))) → (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))):(Base‘𝑅)–1-1-onto→ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))
2826, 27syl 17 . . . . . 6 (𝜑 → (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))):(Base‘𝑅)–1-1-onto→ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))
29 f1f 6804 . . . . . . . . . . 11 ((𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))):(Base‘𝑅)–1-1→(((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))) → (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))):(Base‘𝑅)⟶(((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))))
3026, 29syl 17 . . . . . . . . . 10 (𝜑 → (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))):(Base‘𝑅)⟶(((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))))
3130frnd 6744 . . . . . . . . 9 (𝜑 → ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ⊆ (((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))))
32 eqid 2737 . . . . . . . . . 10 ( Frac ‘𝑅) = ( Frac ‘𝑅)
339, 10, 32, 13fracbas 33307 . . . . . . . . 9 (((Base‘𝑅) × (RLReg‘𝑅)) / (𝑅 ~RL (RLReg‘𝑅))) = (Base‘( Frac ‘𝑅))
3431, 33sseqtrdi 4024 . . . . . . . 8 (𝜑 → ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ⊆ (Base‘( Frac ‘𝑅)))
35 eqid 2737 . . . . . . . . 9 (Base‘( Frac ‘𝑅)) = (Base‘( Frac ‘𝑅))
3622, 35ressbas2 17283 . . . . . . . 8 (ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ⊆ (Base‘( Frac ‘𝑅)) → ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) = (Base‘(( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))))
3734, 36syl 17 . . . . . . 7 (𝜑 → ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) = (Base‘(( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))))
3837f1oeq3d 6845 . . . . . 6 (𝜑 → ((𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))):(Base‘𝑅)–1-1-onto→ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ↔ (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))):(Base‘𝑅)–1-1-onto→(Base‘(( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))))))
3928, 38mpbid 232 . . . . 5 (𝜑 → (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))):(Base‘𝑅)–1-1-onto→(Base‘(( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))))
40 eqid 2737 . . . . . 6 (Base‘(( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))) = (Base‘(( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))))
419, 40isrim 20492 . . . . 5 ((𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (𝑅 RingIso (( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))) ↔ ((𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (𝑅 RingHom (( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))) ∧ (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))):(Base‘𝑅)–1-1-onto→(Base‘(( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))))))
4225, 39, 41sylanbrc 583 . . . 4 (𝜑 → (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (𝑅 RingIso (( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))))
43 brrici 20505 . . . 4 ((𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))) ∈ (𝑅 RingIso (( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅))))) → 𝑅𝑟 (( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))))
4442, 43syl 17 . . 3 (𝜑𝑅𝑟 (( Frac ‘𝑅) ↾s ran (𝑥 ∈ (Base‘𝑅) ↦ [⟨𝑥, (1r𝑅)⟩](𝑅 ~RL (RLReg‘𝑅)))))
458, 19, 44rspcedvdw 3625 . 2 (𝜑 → ∃𝑠 ∈ (SubRing‘( Frac ‘𝑅))𝑅𝑟 (( Frac ‘𝑅) ↾s 𝑠))
464, 6, 45rspcedvdw 3625 1 (𝜑 → ∃𝑓 ∈ Field ∃𝑠 ∈ (SubRing‘𝑓)𝑅𝑟 (𝑓s 𝑠))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wrex 3070  wss 3951  cop 4632   class class class wbr 5143  cmpt 5225   × cxp 5683  ran crn 5686  wf 6557  1-1wf1 6558  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  [cec 8743   / cqs 8744  Basecbs 17247  s cress 17274  1rcur 20178   RingHom crh 20469   RingIso crs 20470  𝑟 cric 20471  SubRingcsubrg 20569  RLRegcrlreg 20691  IDomncidom 20693  Fieldcfield 20730   ~RL cerl 33257   Frac cfrac 33304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-ec 8747  df-qs 8751  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-0g 17486  df-imas 17553  df-qus 17554  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-subg 19141  df-ghm 19231  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-cring 20233  df-oppr 20334  df-dvdsr 20357  df-unit 20358  df-invr 20388  df-rhm 20472  df-rim 20473  df-ric 20475  df-nzr 20513  df-subrng 20546  df-subrg 20570  df-rlreg 20694  df-domn 20695  df-idom 20696  df-drng 20731  df-field 20732  df-erl 33259  df-rloc 33260  df-frac 33305
This theorem is referenced by: (None)
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