Definition df-nm 22387

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 e. _V |-> ( x e. ( 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 22381	. 2 class norm
2		vw	. . 3 setvar w
3		cvv 3200	. . 3 class _V
4		vx	. . . 4 setvar x
5	2	cv 1482	. . . . 5 class w
6		cbs 15857	. . . . 5 class Base
7	5, 6	cfv 5888	. . . 4 class ( Base ` w )
8	4	cv 1482	. . . . 5 class x
9		c0g 16100	. . . . . 6 class 0g
10	5, 9	cfv 5888	. . . . 5 class ( 0g ` w )
11		cds 15950	. . . . . 6 class dist
12	5, 11	cfv 5888	. . . . 5 class ( dist ` w )
13	8, 10, 12	co 6650	. . . 4 class ( x ( dist ` w ) ( 0g ` w ) )
14	4, 7, 13	cmpt 4729	. . 3 class ( x e. ( Base ` w ) |-> ( x ( dist ` w ) ( 0g ` w ) ) )
15	2, 3, 14	cmpt 4729	. 2 class ( w e. _V |-> ( x e. ( Base ` w ) |-> ( x ( dist ` w ) ( 0g ` w ) ) ) )
16	1, 15	wceq 1483	1 wff norm = ( w e. _V |-> ( x e. ( Base ` w ) |-> ( x ( dist ` w ) ( 0g ` w ) ) ) )

This definition is referenced by:  nmfval  22393