Definition df-abs 15145

Description: Define the function for the absolute value (modulus) of a complex number. See abscli 15305 for its closure and absval 15147 or absval2i 15307 for its value. For example, (abs‘-2) = 2 (ex-abs 30437). (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 15143	. 2 class abs
2		vx	. . 3 setvar 𝑥
3		cc 11011	. . 3 class ℂ
4	2	cv 1540	. . . . 5 class 𝑥
5		ccj 15005	. . . . . 6 class ∗
6	4, 5	cfv 6486	. . . . 5 class (∗‘𝑥)
7		cmul 11018	. . . . 5 class ·
8	4, 6, 7	co 7352	. . . 4 class (𝑥 · (∗‘𝑥))
9		csqrt 15142	. . . 4 class √
10	8, 9	cfv 6486	. . 3 class (√‘(𝑥 · (∗‘𝑥)))
11	2, 3, 10	cmpt 5174	. 2 class (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥))))
12	1, 11	wceq 1541	1 wff abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥))))

This definition is referenced by:  absval  15147  absf  15247  absfico  45339