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Mirrors > Home > MPE Home > Th. List > df-mgp | Structured version Visualization version 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 unitgrp 20334 shows that we get a group if we restrict to the elements that have inverses. This allows to formalize such notions as "the multiplication operation of a ring is a monoid" (ringmgp 20191) or "the multiplicative identity" in terms of the identity of a monoid (df-ur 20134). (Contributed by Mario Carneiro, 21-Dec-2014.) |
Ref | Expression |
---|---|
df-mgp | ⊢ mulGrp = (𝑤 ∈ V ↦ (𝑤 sSet 〈(+g‘ndx), (.r‘𝑤)〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cmgp 20086 | . 2 class mulGrp | |
2 | vw | . . 3 setvar 𝑤 | |
3 | cvv 3461 | . . 3 class V | |
4 | 2 | cv 1532 | . . . 4 class 𝑤 |
5 | cnx 17165 | . . . . . 6 class ndx | |
6 | cplusg 17236 | . . . . . 6 class +g | |
7 | 5, 6 | cfv 6549 | . . . . 5 class (+g‘ndx) |
8 | cmulr 17237 | . . . . . 6 class .r | |
9 | 4, 8 | cfv 6549 | . . . . 5 class (.r‘𝑤) |
10 | 7, 9 | cop 4636 | . . . 4 class 〈(+g‘ndx), (.r‘𝑤)〉 |
11 | csts 17135 | . . . 4 class sSet | |
12 | 4, 10, 11 | co 7419 | . . 3 class (𝑤 sSet 〈(+g‘ndx), (.r‘𝑤)〉) |
13 | 2, 3, 12 | cmpt 5232 | . 2 class (𝑤 ∈ V ↦ (𝑤 sSet 〈(+g‘ndx), (.r‘𝑤)〉)) |
14 | 1, 13 | wceq 1533 | 1 wff mulGrp = (𝑤 ∈ V ↦ (𝑤 sSet 〈(+g‘ndx), (.r‘𝑤)〉)) |
Colors of variables: wff setvar class |
This definition is referenced by: fnmgp 20088 mgpval 20089 |
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