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Definition df-subg 16176
Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 16194), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 16189), contains the neutral element of the group (see subg0 16185) and contains the inverses for all of its elements (see subginvcl 16188). (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 16173 . 2  class SubGrp
2 vw . . 3  setvar  w
3 cgrp 16031 . . 3  class  Grp
42cv 1382 . . . . . 6  class  w
5 vs . . . . . . 7  setvar  s
65cv 1382 . . . . . 6  class  s
7 cress 14614 . . . . . 6  classs
84, 6, 7co 6281 . . . . 5  class  ( ws  s )
98, 3wcel 1804 . . . 4  wff  ( ws  s )  e.  Grp
10 cbs 14613 . . . . . 6  class  Base
114, 10cfv 5578 . . . . 5  class  ( Base `  w )
1211cpw 3997 . . . 4  class  ~P ( Base `  w )
139, 5, 12crab 2797 . . 3  class  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp }
142, 3, 13cmpt 4495 . 2  class  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp } )
151, 14wceq 1383 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  16179
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