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Definition df-subg 15660
Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 15678), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 15673), contains the neutral element of the group (see subg0 15669) and contains the inverses for all of its elements (see subginvcl 15672). (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 15657 . 2  class SubGrp
2 vw . . 3  setvar  w
3 cgrp 15395 . . 3  class  Grp
42cv 1363 . . . . . 6  class  w
5 vs . . . . . . 7  setvar  s
65cv 1363 . . . . . 6  class  s
7 cress 14160 . . . . . 6  classs
84, 6, 7co 6082 . . . . 5  class  ( ws  s )
98, 3wcel 1757 . . . 4  wff  ( ws  s )  e.  Grp
10 cbs 14159 . . . . . 6  class  Base
114, 10cfv 5408 . . . . 5  class  ( Base `  w )
1211cpw 3850 . . . 4  class  ~P ( Base `  w )
139, 5, 12crab 2711 . . 3  class  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp }
142, 3, 13cmpt 4340 . 2  class  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp } )
151, 14wceq 1364 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  15663
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