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Definition df-subg 14861
Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 14879), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 14874), contains the neutral element of the group (see subg0 14870) and contains the inverses for all of its elements (see subginvcl 14873). (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 14858 . 2  class SubGrp
2 vw . . 3  set  w
3 cgrp 14605 . . 3  class  Grp
42cv 1648 . . . . . 6  class  w
5 vs . . . . . . 7  set  s
65cv 1648 . . . . . 6  class  s
7 cress 13390 . . . . . 6  classs
84, 6, 7co 6013 . . . . 5  class  ( ws  s )
98, 3wcel 1717 . . . 4  wff  ( ws  s )  e.  Grp
10 cbs 13389 . . . . . 6  class  Base
114, 10cfv 5387 . . . . 5  class  ( Base `  w )
1211cpw 3735 . . . 4  class  ~P ( Base `  w )
139, 5, 12crab 2646 . . 3  class  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp }
142, 3, 13cmpt 4200 . 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  14864
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