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Definition df-subg 14620
Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 14638), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 14633), contains the neutral element of the group (see subg0 14629) and contains the inverses for all of its elements (see subginvcl 14632). (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 14617 . 2  class SubGrp
2 vw . . 3  set  w
3 cgrp 14364 . . 3  class  Grp
42cv 1624 . . . . . 6  class  w
5 vs . . . . . . 7  set  s
65cv 1624 . . . . . 6  class  s
7 cress 13151 . . . . . 6  classs
84, 6, 7co 5860 . . . . 5  class  ( ws  s )
98, 3wcel 1686 . . . 4  wff  ( ws  s )  e.  Grp
10 cbs 13150 . . . . . 6  class  Base
114, 10cfv 5257 . . . . 5  class  ( Base `  w )
1211cpw 3627 . . . 4  class  ~P ( Base `  w )
139, 5, 12crab 2549 . . 3  class  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp }
142, 3, 13cmpt 4079 . 2  class  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp } )
151, 14wceq 1625 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  14623
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