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Definition df-subg 16522
Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 16540), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 16535), contains the neutral element of the group (see subg0 16531) and contains the inverses for all of its elements (see subginvcl 16534). (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 16519 . 2  class SubGrp
2 vw . . 3  setvar  w
3 cgrp 16377 . . 3  class  Grp
42cv 1404 . . . . . 6  class  w
5 vs . . . . . . 7  setvar  s
65cv 1404 . . . . . 6  class  s
7 cress 14842 . . . . . 6  classs
84, 6, 7co 6278 . . . . 5  class  ( ws  s )
98, 3wcel 1842 . . . 4  wff  ( ws  s )  e.  Grp
10 cbs 14841 . . . . . 6  class  Base
114, 10cfv 5569 . . . . 5  class  ( Base `  w )
1211cpw 3955 . . . 4  class  ~P ( Base `  w )
139, 5, 12crab 2758 . . 3  class  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp }
142, 3, 13cmpt 4453 . 2  class  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp } )
151, 14wceq 1405 1  wff SubGrp  =  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e. 
Grp } )
Colors of variables: wff setvar class
This definition is referenced by:  issubg  16525
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