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Definition df-subg 15800
Description: Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 15818), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 15813), contains the neutral element of the group (see subg0 15809) and contains the inverses for all of its elements (see subginvcl 15812). (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 15797 . 2  class SubGrp
2 vw . . 3  setvar  w
3 cgrp 15532 . . 3  class  Grp
42cv 1369 . . . . . 6  class  w
5 vs . . . . . . 7  setvar  s
65cv 1369 . . . . . 6  class  s
7 cress 14296 . . . . . 6  classs
84, 6, 7co 6203 . . . . 5  class  ( ws  s )
98, 3wcel 1758 . . . 4  wff  ( ws  s )  e.  Grp
10 cbs 14295 . . . . . 6  class  Base
114, 10cfv 5529 . . . . 5  class  ( Base `  w )
1211cpw 3971 . . . 4  class  ~P ( Base `  w )
139, 5, 12crab 2803 . . 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 1370 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  15803
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