MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-subg Unicode version

Definition df-subg 14869
Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 14887), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 14882), contains the neutral element of the group (see subg0 14878) and contains the inverses for all of its elements (see subginvcl 14881). (Contributed by Mario Carneiro, 2-Dec-2014.)
Assertion
Ref Expression
df-subg  |- SubGrp  =  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e. 
Grp } )
Distinct variable group:    w, s

Detailed syntax breakdown of Definition df-subg
StepHypRef Expression
1 csubg 14866 . 2  class SubGrp
2 vw . . 3  set  w
3 cgrp 14613 . . 3  class  Grp
42cv 1648 . . . . . 6  class  w
5 vs . . . . . . 7  set  s
65cv 1648 . . . . . 6  class  s
7 cress 13398 . . . . . 6  classs
84, 6, 7co 6021 . . . . 5  class  ( ws  s )
98, 3wcel 1717 . . . 4  wff  ( ws  s )  e.  Grp
10 cbs 13397 . . . . . 6  class  Base
114, 10cfv 5395 . . . . 5  class  ( Base `  w )
1211cpw 3743 . . . 4  class  ~P ( Base `  w )
139, 5, 12crab 2654 . . 3  class  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp }
142, 3, 13cmpt 4208 . 2  class  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp } )
151, 14wceq 1649 1  wff SubGrp  =  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e. 
Grp } )
Colors of variables: wff set class
This definition is referenced by:  issubg  14872
  Copyright terms: Public domain W3C validator