MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-ur Structured version   Visualization version   GIF version

Definition df-ur 19183
Description: Define the multiplicative neutral element of a ring. This definition works by extracting the 0g element, i.e. the neutral element in a group or monoid, and transferring it to the multiplicative monoid via the mulGrp function (df-mgp 19171). See also dfur2 19185, 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 ∘ mulGrp)

Detailed syntax breakdown of Definition df-ur
StepHypRef Expression
1 cur 19182 . 2 class 1r
2 c0g 16703 . . 3 class 0g
3 cmgp 19170 . . 3 class mulGrp
42, 3ccom 5553 . 2 class (0g ∘ mulGrp)
51, 4wceq 1528 1 wff 1r = (0g ∘ mulGrp)
Colors of variables: wff setvar class
This definition is referenced by:  ringidval  19184  prds1  19295  pws1  19297
  Copyright terms: Public domain W3C validator