MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-ascl Structured version   Unicode version

Definition df-ascl 17308
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 17305 . 2  class algSc
2 vw . . 3  setvar  w
3 cvv 2962 . . 3  class  _V
4 vx . . . 4  setvar  x
52cv 1361 . . . . . 6  class  w
6 csca 14224 . . . . . 6  class Scalar
75, 6cfv 5406 . . . . 5  class  (Scalar `  w )
8 cbs 14157 . . . . 5  class  Base
97, 8cfv 5406 . . . 4  class  ( Base `  (Scalar `  w )
)
104cv 1361 . . . . 5  class  x
11 cur 16579 . . . . . 6  class  1r
125, 11cfv 5406 . . . . 5  class  ( 1r
`  w )
13 cvsca 14225 . . . . . 6  class  .s
145, 13cfv 5406 . . . . 5  class  ( .s
`  w )
1510, 12, 14co 6080 . . . 4  class  ( x ( .s `  w
) ( 1r `  w ) )
164, 9, 15cmpt 4338 . . 3  class  ( x  e.  ( Base `  (Scalar `  w ) )  |->  ( x ( .s `  w ) ( 1r
`  w ) ) )
172, 3, 16cmpt 4338 . 2  class  ( w  e.  _V  |->  ( x  e.  ( Base `  (Scalar `  w ) )  |->  ( x ( .s `  w ) ( 1r
`  w ) ) ) )
181, 17wceq 1362 1  wff algSc  =  ( w  e.  _V  |->  ( x  e.  ( Base `  (Scalar `  w )
)  |->  ( x ( .s `  w ) ( 1r `  w
) ) ) )
Colors of variables: wff setvar class
This definition is referenced by:  asclfval  17327
  Copyright terms: Public domain W3C validator