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Definition df-mgp 18255
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 18432 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" (ringmgp 18318) or "the multiplicative identity" in terms of the identity of a monoid (df-1r 9735). (Contributed by Mario Carneiro, 21-Dec-2014.)
Assertion
Ref Expression
df-mgp mulGrp = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+g‘ndx), (.r𝑤)⟩))

Detailed syntax breakdown of Definition df-mgp
StepHypRef Expression
1 cmgp 18254 . 2 class mulGrp
2 vw . . 3 setvar 𝑤
3 cvv 3168 . . 3 class V
42cv 1473 . . . 4 class 𝑤
5 cnx 15634 . . . . . 6 class ndx
6 cplusg 15710 . . . . . 6 class +g
75, 6cfv 5786 . . . . 5 class (+g‘ndx)
8 cmulr 15711 . . . . . 6 class .r
94, 8cfv 5786 . . . . 5 class (.r𝑤)
107, 9cop 4126 . . . 4 class ⟨(+g‘ndx), (.r𝑤)⟩
11 csts 15635 . . . 4 class sSet
124, 10, 11co 6523 . . 3 class (𝑤 sSet ⟨(+g‘ndx), (.r𝑤)⟩)
132, 3, 12cmpt 4633 . 2 class (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+g‘ndx), (.r𝑤)⟩))
141, 13wceq 1474 1 wff mulGrp = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+g‘ndx), (.r𝑤)⟩))
Colors of variables: wff setvar class
This definition is referenced by:  fnmgp  18256  mgpval  18257
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