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Theorem dmncrng 26647
Description: A domain is a commutative ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
Assertion
Ref Expression
dmncrng  |-  ( R  e.  Dmn  ->  R  e. CRingOps )

Proof of Theorem dmncrng
StepHypRef Expression
1 isdmn2 26646 . 2  |-  ( R  e.  Dmn  <->  ( R  e.  PrRing  /\  R  e. CRingOps ) )
21simprbi 451 1  |-  ( R  e.  Dmn  ->  R  e. CRingOps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725  CRingOpsccring 26586   PrRingcprrng 26637   Dmncdmn 26638
This theorem is referenced by:  dmnrngo  26648  dmncan2  26668
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-crngo 26587  df-prrngo 26639  df-dmn 26640
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