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Definition df-subg 16807
Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 16825), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 16820), contains the neutral element of the group (see subg0 16816) and contains the inverses for all of its elements (see subginvcl 16819). (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 16804 . 2  class SubGrp
2 vw . . 3  setvar  w
3 cgrp 16662 . . 3  class  Grp
42cv 1437 . . . . . 6  class  w
5 vs . . . . . . 7  setvar  s
65cv 1437 . . . . . 6  class  s
7 cress 15115 . . . . . 6  classs
84, 6, 7co 6303 . . . . 5  class  ( ws  s )
98, 3wcel 1869 . . . 4  wff  ( ws  s )  e.  Grp
10 cbs 15114 . . . . . 6  class  Base
114, 10cfv 5599 . . . . 5  class  ( Base `  w )
1211cpw 3980 . . . 4  class  ~P ( Base `  w )
139, 5, 12crab 2780 . . 3  class  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp }
142, 3, 13cmpt 4480 . 2  class  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp } )
151, 14wceq 1438 1  wff SubGrp  =  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e. 
Grp } )
Colors of variables: wff setvar class
This definition is referenced by:  issubg  16810
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