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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 14845 for its closure and absval 14687 or absval2i 14847 for its value. For example, (abs‘-2) = 2 (ex-abs 28392). (Contributed by NM, 27-Jul-1999.) |
Ref | Expression |
---|---|
df-abs | ⊢ abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cabs 14683 | . 2 class abs | |
2 | vx | . . 3 setvar 𝑥 | |
3 | cc 10613 | . . 3 class ℂ | |
4 | 2 | cv 1541 | . . . . 5 class 𝑥 |
5 | ccj 14545 | . . . . . 6 class ∗ | |
6 | 4, 5 | cfv 6339 | . . . . 5 class (∗‘𝑥) |
7 | cmul 10620 | . . . . 5 class · | |
8 | 4, 6, 7 | co 7170 | . . . 4 class (𝑥 · (∗‘𝑥)) |
9 | csqrt 14682 | . . . 4 class √ | |
10 | 8, 9 | cfv 6339 | . . 3 class (√‘(𝑥 · (∗‘𝑥))) |
11 | 2, 3, 10 | cmpt 5110 | . 2 class (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥)))) |
12 | 1, 11 | wceq 1542 | 1 wff abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥)))) |
Colors of variables: wff setvar class |
This definition is referenced by: absval 14687 absf 14787 absfico 42296 |
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