Definition df-mgp 20162

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 20409 shows that we get a group if we restrict to the elements that have inverses. This allows to formalize such notions as "the multiplication operation of a ring is a monoid" (ringmgp 20266) or "the multiplicative identity" in terms of the identity of a monoid (df-ur 20209). (Contributed by Mario Carneiro, 21-Dec-2014.)

Assertion

Ref	Expression
df-mgp	⊢ mulGrp = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+g‘ndx), (.r‘𝑤)⟩))

Detailed syntax breakdown of Definition df-mgp

Step	Hyp	Ref	Expression
1		cmgp 20161	. 2 class mulGrp
2		vw	. . 3 setvar 𝑤
3		cvv 3488	. . 3 class V
4	2	cv 1536	. . . 4 class 𝑤
5		cnx 17240	. . . . . 6 class ndx
6		cplusg 17311	. . . . . 6 class +g
7	5, 6	cfv 6573	. . . . 5 class (+g‘ndx)
8		cmulr 17312	. . . . . 6 class .r
9	4, 8	cfv 6573	. . . . 5 class (.r‘𝑤)
10	7, 9	cop 4654	. . . 4 class ⟨(+g‘ndx), (.r‘𝑤)⟩
11		csts 17210	. . . 4 class sSet
12	4, 10, 11	co 7448	. . 3 class (𝑤 sSet ⟨(+g‘ndx), (.r‘𝑤)⟩)
13	2, 3, 12	cmpt 5249	. 2 class (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+g‘ndx), (.r‘𝑤)⟩))
14	1, 13	wceq 1537	1 wff mulGrp = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+g‘ndx), (.r‘𝑤)⟩))

This definition is referenced by:  fnmgp  20163  mgpval  20164