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Theorem isidom 16045
Description: An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
Assertion
Ref Expression
isidom  |-  ( R  e. IDomn 
<->  ( R  e.  CRing  /\  R  e. Domn ) )

Proof of Theorem isidom
StepHypRef Expression
1 df-idom 16026 . 2  |- IDomn  =  (
CRing  i^i Domn )
21elin2 3359 1  |-  ( R  e. IDomn 
<->  ( R  e.  CRing  /\  R  e. Domn ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1684   CRingccrg 15338  Domncdomn 16021  IDomncidom 16022
This theorem is referenced by:  fldidom  16046  fiidomfld  16049  znfld  16514  znidomb  16515  ply1idom  19510  fta1glem1  19551  fta1glem2  19552  fta1g  19553  fta1b  19555  lgsqrlem1  20580  lgsqrlem2  20581  lgsqrlem3  20582  lgsqrlem4  20583  idomrootle  27511  idomodle  27512  proot1mul  27515  proot1hash  27519
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-idom 16026
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