MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-abs Structured version   Visualization version   GIF version

Definition df-abs 14206
Description: Define the function for the absolute value (modulus) of a complex number. See abscli 14364 for its closure and absval 14208 or absval2i 14366 for its value. For example, (abs‘-2) = 2 (ex-abs 27653). (Contributed by NM, 27-Jul-1999.)
Assertion
Ref Expression
df-abs abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥))))

Detailed syntax breakdown of Definition df-abs
StepHypRef Expression
1 cabs 14204 . 2 class abs
2 vx . . 3 setvar 𝑥
3 cc 10226 . . 3 class
42cv 1636 . . . . 5 class 𝑥
5 ccj 14066 . . . . . 6 class
64, 5cfv 6108 . . . . 5 class (∗‘𝑥)
7 cmul 10233 . . . . 5 class ·
84, 6, 7co 6881 . . . 4 class (𝑥 · (∗‘𝑥))
9 csqrt 14203 . . . 4 class
108, 9cfv 6108 . . 3 class (√‘(𝑥 · (∗‘𝑥)))
112, 3, 10cmpt 4934 . 2 class (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥))))
121, 11wceq 1637 1 wff abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥))))
Colors of variables: wff setvar class
This definition is referenced by:  absval  14208  absf  14307  absfico  39902
  Copyright terms: Public domain W3C validator