Definition df-ascl 19314

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

Step	Hyp	Ref	Expression
1		cascl 19311	. 2 class algSc
2		vw	. . 3 setvar w
3		cvv 3200	. . 3 class _V
4		vx	. . . 4 setvar x
5	2	cv 1482	. . . . . 6 class w
6		csca 15944	. . . . . 6 class Scalar
7	5, 6	cfv 5888	. . . . 5 class (Scalar ` w )
8		cbs 15857	. . . . 5 class Base
9	7, 8	cfv 5888	. . . 4 class ( Base ` (Scalar ` w ) )
10	4	cv 1482	. . . . 5 class x
11		cur 18501	. . . . . 6 class 1r
12	5, 11	cfv 5888	. . . . 5 class ( 1r ` w )
13		cvsca 15945	. . . . . 6 class .s
14	5, 13	cfv 5888	. . . . 5 class ( .s ` w )
15	10, 12, 14	co 6650	. . . 4 class ( x ( .s ` w ) ( 1r ` w ) )
16	4, 9, 15	cmpt 4729	. . 3 class ( x e. ( Base ` (Scalar ` w ) ) |-> ( x ( .s ` w ) ( 1r ` w ) ) )
17	2, 3, 16	cmpt 4729	. 2 class ( w e. _V |-> ( x e. ( Base ` (Scalar ` w ) ) |-> ( x ( .s ` w ) ( 1r ` w ) ) ) )
18	1, 17	wceq 1483	1 wff algSc = ( w e. _V |-> ( x e. ( Base ` (Scalar ` w ) ) |-> ( x ( .s ` w ) ( 1r ` w ) ) ) )

This definition is referenced by:  asclfval  19334