Definition df-ascl 21875

Description: Every unital algebra contains a canonical homomorphic image of its ring of scalars as scalar multiples of the unity element. This names the homomorphism. (Contributed by Mario Carneiro, 8-Mar-2015.)

Assertion

Ref	Expression
df-ascl	⊢ algSc = (𝑤 ∈ V ↦ (𝑥 ∈ (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 21872	. 2 class algSc
2		vw	. . 3 setvar 𝑤
3		cvv 3480	. . 3 class V
4		vx	. . . 4 setvar 𝑥
5	2	cv 1539	. . . . . 6 class 𝑤
6		csca 17300	. . . . . 6 class Scalar
7	5, 6	cfv 6561	. . . . 5 class (Scalar‘𝑤)
8		cbs 17247	. . . . 5 class Base
9	7, 8	cfv 6561	. . . 4 class (Base‘(Scalar‘𝑤))
10	4	cv 1539	. . . . 5 class 𝑥
11		cur 20178	. . . . . 6 class 1r
12	5, 11	cfv 6561	. . . . 5 class (1r‘𝑤)
13		cvsca 17301	. . . . . 6 class ·𝑠
14	5, 13	cfv 6561	. . . . 5 class ( ·𝑠 ‘𝑤)
15	10, 12, 14	co 7431	. . . 4 class (𝑥( ·𝑠 ‘𝑤)(1r‘𝑤))
16	4, 9, 15	cmpt 5225	. . 3 class (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·𝑠 ‘𝑤)(1r‘𝑤)))
17	2, 3, 16	cmpt 5225	. 2 class (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·𝑠 ‘𝑤)(1r‘𝑤))))
18	1, 17	wceq 1540	1 wff algSc = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·𝑠 ‘𝑤)(1r‘𝑤))))

This definition is referenced by:  asclfval  21899