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Definition df-mgp 15649
 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 15772 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 15670) or "the multiplicative identity" in terms of the identity of a monoid (df-1r 8940). (Contributed by Mario Carneiro, 21-Dec-2014.)
Assertion
Ref Expression
df-mgp mulGrp sSet

Detailed syntax breakdown of Definition df-mgp
StepHypRef Expression
1 cmgp 15648 . 2 mulGrp
2 vw . . 3
3 cvv 2956 . . 3
42cv 1651 . . . 4
5 cnx 13466 . . . . . 6
6 cplusg 13529 . . . . . 6
75, 6cfv 5454 . . . . 5
8 cmulr 13530 . . . . . 6
94, 8cfv 5454 . . . . 5
107, 9cop 3817 . . . 4
11 csts 13467 . . . 4 sSet
124, 10, 11co 6081 . . 3 sSet
132, 3, 12cmpt 4266 . 2 sSet
141, 13wceq 1652 1 mulGrp sSet
 Colors of variables: wff set class This definition is referenced by:  fnmgp  15650  mgpval  15651
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