Definition df-ur 19471

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 19459). See also dfur2 19473, 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 19470	. 2 class 1r
2		c0g 16898	. . 3 class 0g
3		cmgp 19458	. . 3 class mulGrp
4	2, 3	ccom 5540	. 2 class (0g ∘ mulGrp)
5	1, 4	wceq 1543	1 wff 1r = (0g ∘ mulGrp)

This definition is referenced by:  ringidval  19472  prds1  19586  pws1  19588