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Theorem dmncrng 35215
Description: A domain is a commutative ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
Assertion
Ref Expression
dmncrng (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)

Proof of Theorem dmncrng
StepHypRef Expression
1 isdmn2 35214 . 2 (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))
21simprbi 497 1 (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  CRingOpsccring 35152  PrRingcprrng 35205  Dmncdmn 35206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-crngo 35153  df-prrngo 35207  df-dmn 35208
This theorem is referenced by:  dmnrngo  35216  dmncan2  35236
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