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Definition df-subg 14634
Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 14652), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 14647), contains the neutral element of the group (see subg0 14643) and contains the inverses for all of its elements (see subginvcl 14646). (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 14631 . 2  class SubGrp
2 vw . . 3  set  w
3 cgrp 14378 . . 3  class  Grp
42cv 1631 . . . . . 6  class  w
5 vs . . . . . . 7  set  s
65cv 1631 . . . . . 6  class  s
7 cress 13165 . . . . . 6  classs
84, 6, 7co 5874 . . . . 5  class  ( ws  s )
98, 3wcel 1696 . . . 4  wff  ( ws  s )  e.  Grp
10 cbs 13164 . . . . . 6  class  Base
114, 10cfv 5271 . . . . 5  class  ( Base `  w )
1211cpw 3638 . . . 4  class  ~P ( Base `  w )
139, 5, 12crab 2560 . . 3  class  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp }
142, 3, 13cmpt 4093 . 2  class  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp } )
151, 14wceq 1632 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  14637
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