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Definition df-ur 19914
Description: Define the multiplicative identity, i.e., the monoid identity (df-0g 17323) of the multiplicative monoid (df-mgp 19897) 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 0g element, that is an identity with respect to the operation which started out in the .r slot.

See also dfur2 19916, 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 19913 . 2 class 1r
2 c0g 17321 . . 3 class 0g
3 cmgp 19896 . . 3 class mulGrp
42, 3ccom 5637 . 2 class (0g ∘ mulGrp)
51, 4wceq 1541 1 wff 1r = (0g ∘ mulGrp)
Colors of variables: wff setvar class
This definition is referenced by:  ringidval  19915  prds1  20038  pws1  20040
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