Definition df-mgp 15604

Description: Define a structure that puts the multiplication operation of a ring in the addition slot. Note that this will not actually be a group for the average ring, or even for a field, but it will be a monoid, and unitgrp 15727 shows that we get a group if we restrict to the elements that have inverses. This allows us to formalize such notions as "the multiplication operation of a ring is a monoid" (rngmgp 15625) or "the multiplicative identity" in terms of the identity of a monoid (df-1r 8896). (Contributed by Mario Carneiro, 21-Dec-2014.)

Assertion

Ref	Expression
df-mgp	|- mulGrp = ( w e. _V |-> ( w sSet <. ( +g ` ndx ) , ( .r ` w ) >. ) )

Detailed syntax breakdown of Definition df-mgp

Step	Hyp	Ref	Expression
1		cmgp 15603	. 2 class mulGrp
2		vw	. . 3 set w
3		cvv 2916	. . 3 class _V
4	2	cv 1648	. . . 4 class w
5		cnx 13421	. . . . . 6 class ndx
6		cplusg 13484	. . . . . 6 class +g
7	5, 6	cfv 5413	. . . . 5 class ( +g ` ndx )
8		cmulr 13485	. . . . . 6 class .r
9	4, 8	cfv 5413	. . . . 5 class ( .r ` w )
10	7, 9	cop 3777	. . . 4 class <. ( +g ` ndx ) , ( .r ` w ) >.
11		csts 13422	. . . 4 class sSet
12	4, 10, 11	co 6040	. . 3 class ( w sSet <. ( +g ` ndx ) , ( .r ` w ) >. )
13	2, 3, 12	cmpt 4226	. 2 class ( w e. _V |-> ( w sSet <. ( +g ` ndx ) , ( .r ` w ) >. ) )
14	1, 13	wceq 1649	1 wff mulGrp = ( w e. _V |-> ( w sSet <. ( +g ` ndx ) , ( .r ` w ) >. ) )

This definition is referenced by:  fnmgp  15605  mgpval  15606