Definition df-ur 20160

Description: Define the multiplicative identity, i.e., the monoid identity (df-0g 17401) of the multiplicative monoid (df-mgp 20119) 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 dfur2 20162, 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 20159	. 2 class 1r
2		c0g 17399	. . 3 class 0g
3		cmgp 20118	. . 3 class mulGrp
4	2, 3	ccom 5632	. 2 class (0g ∘ mulGrp)
5	1, 4	wceq 1542	1 wff 1r = (0g ∘ mulGrp)

This definition is referenced by:  ringidval  20161  prds1  20299  pws1  20301