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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 15434 for its closure and absval 15277 or absval2i 15436 for its value. For example, (abs‘-2) = 2 (ex-abs 30474). (Contributed by NM, 27-Jul-1999.) |
| Ref | Expression |
|---|---|
| df-abs | ⊢ abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cabs 15273 | . 2 class abs | |
| 2 | vx | . . 3 setvar 𝑥 | |
| 3 | cc 11153 | . . 3 class ℂ | |
| 4 | 2 | cv 1539 | . . . . 5 class 𝑥 |
| 5 | ccj 15135 | . . . . . 6 class ∗ | |
| 6 | 4, 5 | cfv 6561 | . . . . 5 class (∗‘𝑥) |
| 7 | cmul 11160 | . . . . 5 class · | |
| 8 | 4, 6, 7 | co 7431 | . . . 4 class (𝑥 · (∗‘𝑥)) |
| 9 | csqrt 15272 | . . . 4 class √ | |
| 10 | 8, 9 | cfv 6561 | . . 3 class (√‘(𝑥 · (∗‘𝑥))) |
| 11 | 2, 3, 10 | cmpt 5225 | . 2 class (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥)))) |
| 12 | 1, 11 | wceq 1540 | 1 wff abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥)))) |
| Colors of variables: wff setvar class |
| This definition is referenced by: absval 15277 absf 15376 absfico 45223 |
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