Definition df-abs 15193

Description: Define the function for the absolute value (modulus) of a complex number. See abscli 15353 for its closure and absval 15195 or absval2i 15355 for its value. For example, (abs‘-2) = 2 (ex-abs 30545). (Contributed by NM, 27-Jul-1999.)

Assertion

Ref	Expression
df-abs	⊢ abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥))))

Detailed syntax breakdown of Definition df-abs

Step	Hyp	Ref	Expression
1		cabs 15191	. 2 class abs
2		vx	. . 3 setvar 𝑥
3		cc 11032	. . 3 class ℂ
4	2	cv 1547	. . . . 5 class 𝑥
5		ccj 15053	. . . . . 6 class ∗
6	4, 5	cfv 6488	. . . . 5 class (∗‘𝑥)
7		cmul 11039	. . . . 5 class ·
8	4, 6, 7	co 7359	. . . 4 class (𝑥 · (∗‘𝑥))
9		csqrt 15190	. . . 4 class √
10	8, 9	cfv 6488	. . 3 class (√‘(𝑥 · (∗‘𝑥)))
11	2, 3, 10	cmpt 5155	. 2 class (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥))))
12	1, 11	wceq 1548	1 wff abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥))))

This definition is referenced by:  absval  15195  absf  15295  absfico  45675