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Definition df-subg 14929
Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 14947), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 14942), contains the neutral element of the group (see subg0 14938) and contains the inverses for all of its elements (see subginvcl 14941). (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 14926 . 2  class SubGrp
2 vw . . 3  set  w
3 cgrp 14673 . . 3  class  Grp
42cv 1651 . . . . . 6  class  w
5 vs . . . . . . 7  set  s
65cv 1651 . . . . . 6  class  s
7 cress 13458 . . . . . 6  classs
84, 6, 7co 6072 . . . . 5  class  ( ws  s )
98, 3wcel 1725 . . . 4  wff  ( ws  s )  e.  Grp
10 cbs 13457 . . . . . 6  class  Base
114, 10cfv 5445 . . . . 5  class  ( Base `  w )
1211cpw 3791 . . . 4  class  ~P ( Base `  w )
139, 5, 12crab 2701 . . 3  class  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp }
142, 3, 13cmpt 4258 . 2  class  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp } )
151, 14wceq 1652 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  14932
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