Definition df-abs 13976

Description: Define the function for the absolute value (modulus) of a complex number. See abscli 14134 for its closure and absval 13978 or absval2i 14136 for its value. For example, ( abs ` -u 2 ) = 2 (ex-abs 27312). (Contributed by NM, 27-Jul-1999.)

Assertion

Ref	Expression
df-abs	|- abs = ( x e. CC |-> ( sqr ` ( x x. ( * ` x ) ) ) )

Detailed syntax breakdown of Definition df-abs

Step	Hyp	Ref	Expression
1		cabs 13974	. 2 class abs
2		vx	. . 3 setvar x
3		cc 9934	. . 3 class CC
4	2	cv 1482	. . . . 5 class x
5		ccj 13836	. . . . . 6 class *
6	4, 5	cfv 5888	. . . . 5 class ( * ` x )
7		cmul 9941	. . . . 5 class x.
8	4, 6, 7	co 6650	. . . 4 class ( x x. ( * ` x ) )
9		csqrt 13973	. . . 4 class sqr
10	8, 9	cfv 5888	. . 3 class ( sqr ` ( x x. ( * ` x ) ) )
11	2, 3, 10	cmpt 4729	. 2 class ( x e. CC |-> ( sqr ` ( x x. ( * ` x ) ) ) )
12	1, 11	wceq 1483	1 wff abs = ( x e. CC |-> ( sqr ` ( x x. ( * ` x ) ) ) )

This definition is referenced by:  absval  13978  absf  14077  absfico  39410