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

Theorem isdmn2 26349
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 26348 . 2  |-  ( R  e.  Dmn  <->  ( R  e.  PrRing  /\  R  e.  Com2 ) )
2 prrngorngo 26345 . . . 4  |-  ( R  e.  PrRing  ->  R  e.  RingOps )
3 iscrngo 26291 . . . . 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 1717   RingOpscrngo 21804   Com2ccm2 21839  CRingOpsccring 26289   PrRingcprrng 26340   Dmncdmn 26341
This theorem is referenced by:  dmncrng  26350  flddmn  26352  isdmn3  26368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-iota 5351  df-fv 5395  df-crngo 26290  df-prrngo 26342  df-dmn 26343
  Copyright terms: Public domain W3C validator