Definition df-ur 19719

Description: Define the multiplicative neutral element of a ring. This definition works by extracting the 0_g element, i.e. the neutral element in a group or monoid, and transferring it to the multiplicative monoid via the mulGrp function (df-mgp 19702). See also dfur2 19721, 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

Step	Hyp	Ref	Expression
1		cur 19718	. 2 class 1r
2		c0g 17131	. . 3 class 0g
3		cmgp 19701	. . 3 class mulGrp
4	2, 3	ccom 5592	. 2 class (0g ∘ mulGrp)
5	1, 4	wceq 1541	1 wff 1r = (0g ∘ mulGrp)

This definition is referenced by:  ringidval  19720  prds1  19834  pws1  19836