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Mirrors > Home > MPE Home > Th. List > Mathboxes > df-bj-arg | Structured version Visualization version GIF version |
Description: Define the argument of a nonzero extended complex number. By convention, it has values in (-π, π]. Another convention chooses [0, 2π) but the present one simplifies formulas giving the argument as an arctangent. (Contributed by BJ, 22-Jun-2019.) |
Ref | Expression |
---|---|
df-bj-arg | ⊢ Arg = (𝑥 ∈ (ℂ̅ ∖ {0}) ↦ if(𝑥 ∈ ℂ, (ℑ‘(log‘𝑥)), (1st ‘𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | carg 33130 | . 2 class Arg | |
2 | vx | . . 3 setvar 𝑥 | |
3 | cccbar 33102 | . . . 4 class ℂ̅ | |
4 | cc0 9936 | . . . . 5 class 0 | |
5 | 4 | csn 4177 | . . . 4 class {0} |
6 | 3, 5 | cdif 3571 | . . 3 class (ℂ̅ ∖ {0}) |
7 | 2 | cv 1482 | . . . . 5 class 𝑥 |
8 | cc 9934 | . . . . 5 class ℂ | |
9 | 7, 8 | wcel 1990 | . . . 4 wff 𝑥 ∈ ℂ |
10 | clog 24301 | . . . . . 6 class log | |
11 | 7, 10 | cfv 5888 | . . . . 5 class (log‘𝑥) |
12 | cim 13838 | . . . . 5 class ℑ | |
13 | 11, 12 | cfv 5888 | . . . 4 class (ℑ‘(log‘𝑥)) |
14 | c1st 7166 | . . . . 5 class 1st | |
15 | 7, 14 | cfv 5888 | . . . 4 class (1st ‘𝑥) |
16 | 9, 13, 15 | cif 4086 | . . 3 class if(𝑥 ∈ ℂ, (ℑ‘(log‘𝑥)), (1st ‘𝑥)) |
17 | 2, 6, 16 | cmpt 4729 | . 2 class (𝑥 ∈ (ℂ̅ ∖ {0}) ↦ if(𝑥 ∈ ℂ, (ℑ‘(log‘𝑥)), (1st ‘𝑥))) |
18 | 1, 17 | wceq 1483 | 1 wff Arg = (𝑥 ∈ (ℂ̅ ∖ {0}) ↦ if(𝑥 ∈ ℂ, (ℑ‘(log‘𝑥)), (1st ‘𝑥))) |
Colors of variables: wff setvar class |
This definition is referenced by: (None) |
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