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Definition df-subg 14896
Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 14914), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 14909), contains the neutral element of the group (see subg0 14905) and contains the inverses for all of its elements (see subginvcl 14908). (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 14893 . 2  class SubGrp
2 vw . . 3  set  w
3 cgrp 14640 . . 3  class  Grp
42cv 1648 . . . . . 6  class  w
5 vs . . . . . . 7  set  s
65cv 1648 . . . . . 6  class  s
7 cress 13425 . . . . . 6  classs
84, 6, 7co 6040 . . . . 5  class  ( ws  s )
98, 3wcel 1721 . . . 4  wff  ( ws  s )  e.  Grp
10 cbs 13424 . . . . . 6  class  Base
114, 10cfv 5413 . . . . 5  class  ( Base `  w )
1211cpw 3759 . . . 4  class  ~P ( Base `  w )
139, 5, 12crab 2670 . . 3  class  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp }
142, 3, 13cmpt 4226 . 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  14899
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