Definition df-ur 13141

Description: Define the multiplicative identity, i.e., the monoid identity (df-0g 12706) of the multiplicative monoid (df-mgp 13129) of a ring-like structure. This multiplicative identity is also called "ring unity" or "unity element".

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 0_g element, that is an identity with respect to the operation which started out in the ._r slot.

See also dfur2g 13143, 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 13140	. 2 class 1r
2		c0g 12704	. . 3 class 0g
3		cmgp 13128	. . 3 class mulGrp
4	2, 3	ccom 4630	. 2 class (0g ∘ mulGrp)
5	1, 4	wceq 1353	1 wff 1r = (0g ∘ mulGrp)

This definition is referenced by:  ringidvalg  13142