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Definition df-ascl 16055
Description: Every unital algebra contains a canonical homomorphic image of its ring of scalars as scalar multiples of the unit. This names the homomorphism. (Contributed by Mario Carneiro, 8-Mar-2015.)
Assertion
Ref Expression
df-ascl  |- algSc  =  ( w  e.  _V  |->  ( x  e.  ( Base `  (Scalar `  w )
)  |->  ( x ( .s `  w ) ( 1r `  w
) ) ) )
Distinct variable group:    x, w

Detailed syntax breakdown of Definition df-ascl
StepHypRef Expression
1 cascl 16052 . 2  class algSc
2 vw . . 3  set  w
3 cvv 2788 . . 3  class  _V
4 vx . . . 4  set  x
52cv 1622 . . . . . 6  class  w
6 csca 13211 . . . . . 6  class Scalar
75, 6cfv 5255 . . . . 5  class  (Scalar `  w )
8 cbs 13148 . . . . 5  class  Base
97, 8cfv 5255 . . . 4  class  ( Base `  (Scalar `  w )
)
104cv 1622 . . . . 5  class  x
11 cur 15339 . . . . . 6  class  1r
125, 11cfv 5255 . . . . 5  class  ( 1r
`  w )
13 cvsca 13212 . . . . . 6  class  .s
145, 13cfv 5255 . . . . 5  class  ( .s
`  w )
1510, 12, 14co 5858 . . . 4  class  ( x ( .s `  w
) ( 1r `  w ) )
164, 9, 15cmpt 4077 . . 3  class  ( x  e.  ( Base `  (Scalar `  w ) )  |->  ( x ( .s `  w ) ( 1r
`  w ) ) )
172, 3, 16cmpt 4077 . 2  class  ( w  e.  _V  |->  ( x  e.  ( Base `  (Scalar `  w ) )  |->  ( x ( .s `  w ) ( 1r
`  w ) ) ) )
181, 17wceq 1623 1  wff algSc  =  ( w  e.  _V  |->  ( x  e.  ( Base `  (Scalar `  w )
)  |->  ( x ( .s `  w ) ( 1r `  w
) ) ) )
Colors of variables: wff set class
This definition is referenced by:  asclfval  16074
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