 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-grp Structured version   Visualization version   GIF version

Definition df-grp 17419
 Description: Define class of all groups. A group is a monoid (df-mnd 17289) whose internal operation is such that every element admits a left inverse (which can be proven to be a two-sided inverse). Thus, a group 𝐺 is an algebraic structure formed from a base set of elements (notated (Base‘𝐺) per df-base 15857) and an internal group operation (notated (+g‘𝐺) per df-plusg 15948). The operation combines any two elements of the group base set and must satisfy the 4 group axioms: closure (the result of the group operation must always be a member of the base set, see grpcl 17424), associativity (so ((𝑎+g𝑏)+g𝑐) = (𝑎+g(𝑏+g𝑐)) for any a, b, c, see grpass 17425), identity (there must be an element 𝑒 = (0g‘𝐺) such that 𝑒+g𝑎 = 𝑎+g𝑒 = 𝑎 for any a), and inverse (for each element a in the base set, there must be an element 𝑏 = invg𝑎 in the base set such that 𝑎+g𝑏 = 𝑏+g𝑎 = 𝑒). It can be proven that the identity element is unique (grpideu 17427). Groups need not be commutative; a commutative group is an Abelian group (see df-abl 18190). Subgroups can often be formed from groups, see df-subg 17585. An example of an (Abelian) group is the set of complex numbers ℂ over the group operation + (addition), as proven in cnaddablx 18265; an Abelian group is a group as proven in ablgrp 18192. Other structures include groups, including unital rings (df-ring 18543) and fields (df-field 18744). (Contributed by NM, 17-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
Assertion
Ref Expression
df-grp Grp = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g𝑔)𝑎) = (0g𝑔)}
Distinct variable group:   𝑔,𝑎,𝑚

Detailed syntax breakdown of Definition df-grp
StepHypRef Expression
1 cgrp 17416 . 2 class Grp
2 vm . . . . . . . 8 setvar 𝑚
32cv 1481 . . . . . . 7 class 𝑚
4 va . . . . . . . 8 setvar 𝑎
54cv 1481 . . . . . . 7 class 𝑎
6 vg . . . . . . . . 9 setvar 𝑔
76cv 1481 . . . . . . . 8 class 𝑔
8 cplusg 15935 . . . . . . . 8 class +g
97, 8cfv 5886 . . . . . . 7 class (+g𝑔)
103, 5, 9co 6647 . . . . . 6 class (𝑚(+g𝑔)𝑎)
11 c0g 16094 . . . . . . 7 class 0g
127, 11cfv 5886 . . . . . 6 class (0g𝑔)
1310, 12wceq 1482 . . . . 5 wff (𝑚(+g𝑔)𝑎) = (0g𝑔)
14 cbs 15851 . . . . . 6 class Base
157, 14cfv 5886 . . . . 5 class (Base‘𝑔)
1613, 2, 15wrex 2912 . . . 4 wff 𝑚 ∈ (Base‘𝑔)(𝑚(+g𝑔)𝑎) = (0g𝑔)
1716, 4, 15wral 2911 . . 3 wff 𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g𝑔)𝑎) = (0g𝑔)
18 cmnd 17288 . . 3 class Mnd
1917, 6, 18crab 2915 . 2 class {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g𝑔)𝑎) = (0g𝑔)}
201, 19wceq 1482 1 wff Grp = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g𝑔)𝑎) = (0g𝑔)}
 Colors of variables: wff setvar class This definition is referenced by:  isgrp  17422
 Copyright terms: Public domain W3C validator