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Definition df-subg 16814
Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 16832), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 16827), contains the neutral element of the group (see subg0 16823) and contains the inverses for all of its elements (see subginvcl 16826). (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 16811 . 2  class SubGrp
2 vw . . 3  setvar  w
3 cgrp 16669 . . 3  class  Grp
42cv 1443 . . . . . 6  class  w
5 vs . . . . . . 7  setvar  s
65cv 1443 . . . . . 6  class  s
7 cress 15122 . . . . . 6  classs
84, 6, 7co 6290 . . . . 5  class  ( ws  s )
98, 3wcel 1887 . . . 4  wff  ( ws  s )  e.  Grp
10 cbs 15121 . . . . . 6  class  Base
114, 10cfv 5582 . . . . 5  class  ( Base `  w )
1211cpw 3951 . . . 4  class  ~P ( Base `  w )
139, 5, 12crab 2741 . . 3  class  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp }
142, 3, 13cmpt 4461 . 2  class  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp } )
151, 14wceq 1444 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  16817
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