Theorem efiargd

Description: The exponential of the "arg" function Im o. log , deduction version. (Contributed by Thierry Arnoux, 5-Nov-2025)

	Ref	Expression
Hypotheses	efiargd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	efiargd.2	⊢ ( 𝜑 → 𝐴 ≠ 0 )
Assertion	efiargd	⊢ ( 𝜑 → ( exp ‘ ( i · ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ) = ( 𝐴 / ( abs ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		efiargd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		efiargd.2	⊢ ( 𝜑 → 𝐴 ≠ 0 )
3		efiarg	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( exp ‘ ( i · ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ) = ( 𝐴 / ( abs ‘ 𝐴 ) ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( exp ‘ ( i · ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ) = ( 𝐴 / ( abs ‘ 𝐴 ) ) )