MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-cring Unicode version

Definition df-cring 15551
Description: Define class of all commutative rings. (Contributed by Mario Carneiro, 7-Jan-2015.)
Assertion
Ref Expression
df-cring  |-  CRing  =  {
f  e.  Ring  |  (mulGrp `  f )  e. CMnd }

Detailed syntax breakdown of Definition df-cring
StepHypRef Expression
1 ccrg 15548 . 2  class  CRing
2 vf . . . . . 6  set  f
32cv 1646 . . . . 5  class  f
4 cmgp 15535 . . . . 5  class mulGrp
53, 4cfv 5358 . . . 4  class  (mulGrp `  f )
6 ccmn 15299 . . . 4  class CMnd
75, 6wcel 1715 . . 3  wff  (mulGrp `  f )  e. CMnd
8 crg 15547 . . 3  class  Ring
97, 2, 8crab 2632 . 2  class  { f  e.  Ring  |  (mulGrp `  f )  e. CMnd }
101, 9wceq 1647 1  wff  CRing  =  {
f  e.  Ring  |  (mulGrp `  f )  e. CMnd }
Colors of variables: wff set class
This definition is referenced by:  iscrng  15558
  Copyright terms: Public domain W3C validator