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Definition df-subg 15669
Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 15687), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 15682), contains the neutral element of the group (see subg0 15678) and contains the inverses for all of its elements (see subginvcl 15681). (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 15666 . 2  class SubGrp
2 vw . . 3  setvar  w
3 cgrp 15402 . . 3  class  Grp
42cv 1368 . . . . . 6  class  w
5 vs . . . . . . 7  setvar  s
65cv 1368 . . . . . 6  class  s
7 cress 14167 . . . . . 6  classs
84, 6, 7co 6086 . . . . 5  class  ( ws  s )
98, 3wcel 1756 . . . 4  wff  ( ws  s )  e.  Grp
10 cbs 14166 . . . . . 6  class  Base
114, 10cfv 5413 . . . . 5  class  ( Base `  w )
1211cpw 3855 . . . 4  class  ~P ( Base `  w )
139, 5, 12crab 2714 . . 3  class  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp }
142, 3, 13cmpt 4345 . 2  class  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp } )
151, 14wceq 1369 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  15672
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