Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isdmn2 Structured version   Unicode version

Theorem isdmn2 26656
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 26655 . 2  |-  ( R  e.  Dmn  <->  ( R  e.  PrRing  /\  R  e.  Com2 ) )
2 prrngorngo 26652 . . . 4  |-  ( R  e.  PrRing  ->  R  e.  RingOps )
3 iscrngo 26598 . . . . 5  |-  ( R  e. CRingOps 
<->  ( R  e.  RingOps  /\  R  e.  Com2 ) )
43baibr 873 . . . 4  |-  ( R  e.  RingOps  ->  ( R  e. 
Com2 
<->  R  e. CRingOps ) )
52, 4syl 16 . . 3  |-  ( R  e.  PrRing  ->  ( R  e. 
Com2 
<->  R  e. CRingOps ) )
65pm5.32i 619 . 2  |-  ( ( R  e.  PrRing  /\  R  e.  Com2 )  <->  ( R  e.  PrRing  /\  R  e. CRingOps ) )
71, 6bitri 241 1  |-  ( R  e.  Dmn  <->  ( R  e.  PrRing  /\  R  e. CRingOps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    e. wcel 1725   RingOpscrngo 21955   Com2ccm2 21990  CRingOpsccring 26596   PrRingcprrng 26647   Dmncdmn 26648
This theorem is referenced by:  dmncrng  26657  flddmn  26659  isdmn3  26675
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 26597  df-prrngo 26649  df-dmn 26650
  Copyright terms: Public domain W3C validator