ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-ur Unicode version

Definition df-ur 12936
Description: Define the multiplicative identity, i.e., the monoid identity (df-0g 12628) of the multiplicative monoid (df-mgp 12926) of a ring-like structure. This definition works by transferring the multiplicative operation from the  .r slot to the  +g slot and then looking at the element which is then the  0g element, that is an identity with respect to the operation which started out in the  .r slot.

See also dfur2g 12938, which derives the "traditional" definition as the unique element of a ring which is left- and right-neutral under multiplication. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)

Assertion
Ref Expression
df-ur  |-  1r  =  ( 0g  o. mulGrp )

Detailed syntax breakdown of Definition df-ur
StepHypRef Expression
1 cur 12935 . 2  class  1r
2 c0g 12626 . . 3  class  0g
3 cmgp 12925 . . 3  class mulGrp
42, 3ccom 4624 . 2  class  ( 0g  o. mulGrp )
51, 4wceq 1353 1  wff  1r  =  ( 0g  o. mulGrp )
Colors of variables: wff set class
This definition is referenced by:  ringidvalg  12937
  Copyright terms: Public domain W3C validator