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Definition df-mgp 15342
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 15465 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 15363) or "the multiplicative identity" in terms of the identity of a monoid (df-1r 8703). (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
StepHypRef Expression
1 cmgp 15341 . 2  class mulGrp
2 vw . . 3  set  w
3 cvv 2801 . . 3  class  _V
42cv 1631 . . . 4  class  w
5 cnx 13161 . . . . . 6  class  ndx
6 cplusg 13224 . . . . . 6  class  +g
75, 6cfv 5271 . . . . 5  class  ( +g  ` 
ndx )
8 cmulr 13225 . . . . . 6  class  .r
94, 8cfv 5271 . . . . 5  class  ( .r
`  w )
107, 9cop 3656 . . . 4  class  <. ( +g  `  ndx ) ,  ( .r `  w
) >.
11 csts 13162 . . . 4  class sSet
124, 10, 11co 5874 . . 3  class  ( w sSet  <. ( +g  `  ndx ) ,  ( .r `  w ) >. )
132, 3, 12cmpt 4093 . 2  class  ( w  e.  _V  |->  ( w sSet  <. ( +g  `  ndx ) ,  ( .r `  w ) >. )
)
141, 13wceq 1632 1  wff mulGrp  =  ( w  e.  _V  |->  ( w sSet  <. ( +g  `  ndx ) ,  ( .r `  w ) >. )
)
Colors of variables: wff set class
This definition is referenced by:  fnmgp  15343  mgpval  15344
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