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Definition df-subg 16000
Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 16018), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 16013), contains the neutral element of the group (see subg0 16009) and contains the inverses for all of its elements (see subginvcl 16012). (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 15997 . 2  class SubGrp
2 vw . . 3  setvar  w
3 cgrp 15726 . . 3  class  Grp
42cv 1378 . . . . . 6  class  w
5 vs . . . . . . 7  setvar  s
65cv 1378 . . . . . 6  class  s
7 cress 14490 . . . . . 6  classs
84, 6, 7co 6283 . . . . 5  class  ( ws  s )
98, 3wcel 1767 . . . 4  wff  ( ws  s )  e.  Grp
10 cbs 14489 . . . . . 6  class  Base
114, 10cfv 5587 . . . . 5  class  ( Base `  w )
1211cpw 4010 . . . 4  class  ~P ( Base `  w )
139, 5, 12crab 2818 . . 3  class  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp }
142, 3, 13cmpt 4505 . 2  class  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp } )
151, 14wceq 1379 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  16003
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