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Mirrors > Home > ILE Home > Th. List > df-mgp | GIF version |
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.) |
Ref | Expression |
---|---|
df-mgp | β’ mulGrp = (π€ β V β¦ (π€ sSet β¨(+gβndx), (.rβπ€)β©)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cmgp 13128 | . 2 class mulGrp | |
2 | vw | . . 3 setvar π€ | |
3 | cvv 2737 | . . 3 class V | |
4 | 2 | cv 1352 | . . . 4 class π€ |
5 | cnx 12458 | . . . . . 6 class ndx | |
6 | cplusg 12535 | . . . . . 6 class +g | |
7 | 5, 6 | cfv 5216 | . . . . 5 class (+gβndx) |
8 | cmulr 12536 | . . . . . 6 class .r | |
9 | 4, 8 | cfv 5216 | . . . . 5 class (.rβπ€) |
10 | 7, 9 | cop 3595 | . . . 4 class β¨(+gβndx), (.rβπ€)β© |
11 | csts 12459 | . . . 4 class sSet | |
12 | 4, 10, 11 | co 5874 | . . 3 class (π€ sSet β¨(+gβndx), (.rβπ€)β©) |
13 | 2, 3, 12 | cmpt 4064 | . 2 class (π€ β V β¦ (π€ sSet β¨(+gβndx), (.rβπ€)β©)) |
14 | 1, 13 | wceq 1353 | 1 wff mulGrp = (π€ β V β¦ (π€ sSet β¨(+gβndx), (.rβπ€)β©)) |
Colors of variables: wff set class |
This definition is referenced by: fnmgp 13130 mgpvalg 13131 |
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