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Mirrors > Home > MPE Home > Th. List > df-abs | Structured version Visualization version GIF version |
Description: Define the function for the absolute value (modulus) of a complex number. See abscli 15339 for its closure and absval 15182 or absval2i 15341 for its value. For example, (abs‘-2) = 2 (ex-abs 30177). (Contributed by NM, 27-Jul-1999.) |
Ref | Expression |
---|---|
df-abs | ⊢ abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cabs 15178 | . 2 class abs | |
2 | vx | . . 3 setvar 𝑥 | |
3 | cc 11104 | . . 3 class ℂ | |
4 | 2 | cv 1532 | . . . . 5 class 𝑥 |
5 | ccj 15040 | . . . . . 6 class ∗ | |
6 | 4, 5 | cfv 6533 | . . . . 5 class (∗‘𝑥) |
7 | cmul 11111 | . . . . 5 class · | |
8 | 4, 6, 7 | co 7401 | . . . 4 class (𝑥 · (∗‘𝑥)) |
9 | csqrt 15177 | . . . 4 class √ | |
10 | 8, 9 | cfv 6533 | . . 3 class (√‘(𝑥 · (∗‘𝑥))) |
11 | 2, 3, 10 | cmpt 5221 | . 2 class (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥)))) |
12 | 1, 11 | wceq 1533 | 1 wff abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥)))) |
Colors of variables: wff setvar class |
This definition is referenced by: absval 15182 absf 15281 absfico 44402 |
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