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Mirrors > Home > MPE Home > Th. List > df-mgp | Structured version Visualization version Unicode 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 18667 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" (ringmgp 18553) or "the multiplicative identity" in terms of the identity of a monoid (df-1r 9883). (Contributed by Mario Carneiro, 21-Dec-2014.) |
Ref | Expression |
---|---|
df-mgp | mulGrp sSet |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cmgp 18489 | . 2 mulGrp | |
2 | vw | . . 3 | |
3 | cvv 3200 | . . 3 | |
4 | 2 | cv 1482 | . . . 4 |
5 | cnx 15854 | . . . . . 6 | |
6 | cplusg 15941 | . . . . . 6 | |
7 | 5, 6 | cfv 5888 | . . . . 5 |
8 | cmulr 15942 | . . . . . 6 | |
9 | 4, 8 | cfv 5888 | . . . . 5 |
10 | 7, 9 | cop 4183 | . . . 4 |
11 | csts 15855 | . . . 4 sSet | |
12 | 4, 10, 11 | co 6650 | . . 3 sSet |
13 | 2, 3, 12 | cmpt 4729 | . 2 sSet |
14 | 1, 13 | wceq 1483 | 1 mulGrp sSet |
Colors of variables: wff setvar class |
This definition is referenced by: fnmgp 18491 mgpval 18492 |
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