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Theorem isdmn2 26783
Description: The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)
Assertion
Ref Expression
isdmn2  |-  ( R  e.  Dmn  <->  ( R  e.  PrRing  /\  R  e. CRingOps ) )

Proof of Theorem isdmn2
StepHypRef Expression
1 isdmn 26782 . 2  |-  ( R  e.  Dmn  <->  ( R  e.  PrRing  /\  R  e.  Com2 ) )
2 prrngorngo 26779 . . . 4  |-  ( R  e.  PrRing  ->  R  e.  RingOps )
3 iscrngo 26725 . . . . 5  |-  ( R  e. CRingOps 
<->  ( R  e.  RingOps  /\  R  e.  Com2 ) )
43baibr 872 . . . 4  |-  ( R  e.  RingOps  ->  ( R  e. 
Com2 
<->  R  e. CRingOps ) )
52, 4syl 15 . . 3  |-  ( R  e.  PrRing  ->  ( R  e. 
Com2 
<->  R  e. CRingOps ) )
65pm5.32i 618 . 2  |-  ( ( R  e.  PrRing  /\  R  e.  Com2 )  <->  ( R  e.  PrRing  /\  R  e. CRingOps ) )
71, 6bitri 240 1  |-  ( R  e.  Dmn  <->  ( R  e.  PrRing  /\  R  e. CRingOps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1696   RingOpscrngo 21058   Com2ccm2 21093  CRingOpsccring 26723   PrRingcprrng 26774   Dmncdmn 26775
This theorem is referenced by:  dmncrng  26784  flddmn  26786  isdmn3  26802
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-crngo 26724  df-prrngo 26776  df-dmn 26777
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