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Definition df-cntz 14788
Description: Define the centralizer of a subset of a magma, which is the set of elements each of which commutes with each element of the given subset. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Assertion
Ref Expression
df-cntz  |- Cntz  =  ( m  e.  _V  |->  ( s  e.  ~P ( Base `  m )  |->  { x  e.  ( Base `  m )  |  A. y  e.  s  (
x ( +g  `  m
) y )  =  ( y ( +g  `  m ) x ) } ) )
Distinct variable group:    m, s, x, y

Detailed syntax breakdown of Definition df-cntz
StepHypRef Expression
1 ccntz 14786 . 2  class Cntz
2 vm . . 3  set  m
3 cvv 2790 . . 3  class  _V
4 vs . . . 4  set  s
52cv 1623 . . . . . 6  class  m
6 cbs 13143 . . . . . 6  class  Base
75, 6cfv 5222 . . . . 5  class  ( Base `  m )
87cpw 3627 . . . 4  class  ~P ( Base `  m )
9 vx . . . . . . . . 9  set  x
109cv 1623 . . . . . . . 8  class  x
11 vy . . . . . . . . 9  set  y
1211cv 1623 . . . . . . . 8  class  y
13 cplusg 13203 . . . . . . . . 9  class  +g
145, 13cfv 5222 . . . . . . . 8  class  ( +g  `  m )
1510, 12, 14co 5820 . . . . . . 7  class  ( x ( +g  `  m
) y )
1612, 10, 14co 5820 . . . . . . 7  class  ( y ( +g  `  m
) x )
1715, 16wceq 1624 . . . . . 6  wff  ( x ( +g  `  m
) y )  =  ( y ( +g  `  m ) x )
184cv 1623 . . . . . 6  class  s
1917, 11, 18wral 2545 . . . . 5  wff  A. y  e.  s  ( x
( +g  `  m ) y )  =  ( y ( +g  `  m
) x )
2019, 9, 7crab 2549 . . . 4  class  { x  e.  ( Base `  m
)  |  A. y  e.  s  ( x
( +g  `  m ) y )  =  ( y ( +g  `  m
) x ) }
214, 8, 20cmpt 4079 . . 3  class  ( s  e.  ~P ( Base `  m )  |->  { x  e.  ( Base `  m
)  |  A. y  e.  s  ( x
( +g  `  m ) y )  =  ( y ( +g  `  m
) x ) } )
222, 3, 21cmpt 4079 . 2  class  ( m  e.  _V  |->  ( s  e.  ~P ( Base `  m )  |->  { x  e.  ( Base `  m
)  |  A. y  e.  s  ( x
( +g  `  m ) y )  =  ( y ( +g  `  m
) x ) } ) )
231, 22wceq 1624 1  wff Cntz  =  ( m  e.  _V  |->  ( s  e.  ~P ( Base `  m )  |->  { x  e.  ( Base `  m )  |  A. y  e.  s  (
x ( +g  `  m
) y )  =  ( y ( +g  `  m ) x ) } ) )
Colors of variables: wff set class
This definition is referenced by:  cntzfval  14791
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