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Definition df-mgp 13129
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 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" or "the multiplicative identity" in terms of the identity of a monoid (df-ur 13141). (Contributed by Mario Carneiro, 21-Dec-2014.)
Assertion
Ref Expression
df-mgp mulGrp = (𝑀 ∈ V ↦ (𝑀 sSet ⟨(+gβ€˜ndx), (.rβ€˜π‘€)⟩))

Detailed syntax breakdown of Definition df-mgp
StepHypRef Expression
1 cmgp 13128 . 2 class mulGrp
2 vw . . 3 setvar 𝑀
3 cvv 2737 . . 3 class V
42cv 1352 . . . 4 class 𝑀
5 cnx 12458 . . . . . 6 class ndx
6 cplusg 12535 . . . . . 6 class +g
75, 6cfv 5216 . . . . 5 class (+gβ€˜ndx)
8 cmulr 12536 . . . . . 6 class .r
94, 8cfv 5216 . . . . 5 class (.rβ€˜π‘€)
107, 9cop 3595 . . . 4 class ⟨(+gβ€˜ndx), (.rβ€˜π‘€)⟩
11 csts 12459 . . . 4 class sSet
124, 10, 11co 5874 . . 3 class (𝑀 sSet ⟨(+gβ€˜ndx), (.rβ€˜π‘€)⟩)
132, 3, 12cmpt 4064 . 2 class (𝑀 ∈ V ↦ (𝑀 sSet ⟨(+gβ€˜ndx), (.rβ€˜π‘€)⟩))
141, 13wceq 1353 1 wff mulGrp = (𝑀 ∈ V ↦ (𝑀 sSet ⟨(+gβ€˜ndx), (.rβ€˜π‘€)⟩))
Colors of variables: wff set class
This definition is referenced by:  fnmgp  13130  mgpvalg  13131
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