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Theorem dmncrng 30283
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 30282 . 2  |-  ( R  e.  Dmn  <->  ( R  e.  PrRing  /\  R  e. CRingOps ) )
21simprbi 464 1  |-  ( R  e.  Dmn  ->  R  e. CRingOps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767  CRingOpsccring 30222   PrRingcprrng 30273   Dmncdmn 30274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5551  df-fv 5596  df-crngo 30223  df-prrngo 30275  df-dmn 30276
This theorem is referenced by:  dmnrngo  30284  dmncan2  30304
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