Definition df-nm 24595

Description: Define the norm on a group or ring (when it makes sense) in terms of the distance to zero. (Contributed by Mario Carneiro, 2-Oct-2015.)

Assertion

Ref	Expression
df-nm	⊢ norm = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g‘𝑤))))

Distinct variable group:   𝑥,𝑤

Detailed syntax breakdown of Definition df-nm

Step	Hyp	Ref	Expression
1		cnm 24589	. 2 class norm
2		vw	. . 3 setvar 𝑤
3		cvv 3480	. . 3 class V
4		vx	. . . 4 setvar 𝑥
5	2	cv 1539	. . . . 5 class 𝑤
6		cbs 17247	. . . . 5 class Base
7	5, 6	cfv 6561	. . . 4 class (Base‘𝑤)
8	4	cv 1539	. . . . 5 class 𝑥
9		c0g 17484	. . . . . 6 class 0g
10	5, 9	cfv 6561	. . . . 5 class (0g‘𝑤)
11		cds 17306	. . . . . 6 class dist
12	5, 11	cfv 6561	. . . . 5 class (dist‘𝑤)
13	8, 10, 12	co 7431	. . . 4 class (𝑥(dist‘𝑤)(0g‘𝑤))
14	4, 7, 13	cmpt 5225	. . 3 class (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g‘𝑤)))
15	2, 3, 14	cmpt 5225	. 2 class (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g‘𝑤))))
16	1, 15	wceq 1540	1 wff norm = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g‘𝑤))))

This definition is referenced by:  nmfval  24601