Definition df-bj-arg 35415

Description: Define the argument of a nonzero extended complex number. By convention, it has values in (-π, π]. Another convention chooses values in [0, 2π) but the present convention simplifies formulas giving the argument as an arctangent. (Contributed by BJ, 22-Jun-2019.) The "else" case of the second conditional operator, corresponding to infinite extended complex numbers other than -∞, gives a definition depending on the specific definition chosen for these numbers (df-bj-inftyexpitau 35370), and therefore should not be relied upon. (New usage is discouraged.)

Assertion

Ref	Expression
df-bj-arg	⊢ Arg = (𝑥 ∈ (ℂ̅ ∖ {0}) ↦ if(𝑥 ∈ ℂ, (ℑ‘(log‘𝑥)), if(𝑥<ℝ̅0, π, (((1st ‘𝑥) / (2 · π)) − π))))

Detailed syntax breakdown of Definition df-bj-arg

Step	Hyp	Ref	Expression
1		carg 35414	. 2 class Arg
2		vx	. . 3 setvar 𝑥
3		cccbar 35386	. . . 4 class ℂ̅
4		cc0 10871	. . . . 5 class 0
5	4	csn 4561	. . . 4 class {0}
6	3, 5	cdif 3884	. . 3 class (ℂ̅ ∖ {0})
7	2	cv 1538	. . . . 5 class 𝑥
8		cc 10869	. . . . 5 class ℂ
9	7, 8	wcel 2106	. . . 4 wff 𝑥 ∈ ℂ
10		clog 25710	. . . . . 6 class log
11	7, 10	cfv 6433	. . . . 5 class (log‘𝑥)
12		cim 14809	. . . . 5 class ℑ
13	11, 12	cfv 6433	. . . 4 class (ℑ‘(log‘𝑥))
14		cltxr 35412	. . . . . 6 class <ℝ̅
15	7, 4, 14	wbr 5074	. . . . 5 wff 𝑥<ℝ̅0
16		cpi 15776	. . . . 5 class π
17		c1st 7829	. . . . . . . 8 class 1st
18	7, 17	cfv 6433	. . . . . . 7 class (1st ‘𝑥)
19		c2 12028	. . . . . . . 8 class 2
20		cmul 10876	. . . . . . . 8 class ·
21	19, 16, 20	co 7275	. . . . . . 7 class (2 · π)
22		cdiv 11632	. . . . . . 7 class /
23	18, 21, 22	co 7275	. . . . . 6 class ((1st ‘𝑥) / (2 · π))
24		cmin 11205	. . . . . 6 class −
25	23, 16, 24	co 7275	. . . . 5 class (((1st ‘𝑥) / (2 · π)) − π)
26	15, 16, 25	cif 4459	. . . 4 class if(𝑥<ℝ̅0, π, (((1st ‘𝑥) / (2 · π)) − π))
27	9, 13, 26	cif 4459	. . 3 class if(𝑥 ∈ ℂ, (ℑ‘(log‘𝑥)), if(𝑥<ℝ̅0, π, (((1st ‘𝑥) / (2 · π)) − π)))
28	2, 6, 27	cmpt 5157	. 2 class (𝑥 ∈ (ℂ̅ ∖ {0}) ↦ if(𝑥 ∈ ℂ, (ℑ‘(log‘𝑥)), if(𝑥<ℝ̅0, π, (((1st ‘𝑥) / (2 · π)) − π))))
29	1, 28	wceq 1539	1 wff Arg = (𝑥 ∈ (ℂ̅ ∖ {0}) ↦ if(𝑥 ∈ ℂ, (ℑ‘(log‘𝑥)), if(𝑥<ℝ̅0, π, (((1st ‘𝑥) / (2 · π)) − π))))

This definition is referenced by: (None)