Step |
Hyp |
Ref |
Expression |
1 |
|
logrncl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( log ‘ 𝐴 ) ∈ ran log ) |
2 |
|
ellogrn |
⊢ ( ( log ‘ 𝐴 ) ∈ ran log ↔ ( ( log ‘ 𝐴 ) ∈ ℂ ∧ - π < ( ℑ ‘ ( log ‘ 𝐴 ) ) ∧ ( ℑ ‘ ( log ‘ 𝐴 ) ) ≤ π ) ) |
3 |
1 2
|
sylib |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ( log ‘ 𝐴 ) ∈ ℂ ∧ - π < ( ℑ ‘ ( log ‘ 𝐴 ) ) ∧ ( ℑ ‘ ( log ‘ 𝐴 ) ) ≤ π ) ) |
4 |
|
3simpc |
⊢ ( ( ( log ‘ 𝐴 ) ∈ ℂ ∧ - π < ( ℑ ‘ ( log ‘ 𝐴 ) ) ∧ ( ℑ ‘ ( log ‘ 𝐴 ) ) ≤ π ) → ( - π < ( ℑ ‘ ( log ‘ 𝐴 ) ) ∧ ( ℑ ‘ ( log ‘ 𝐴 ) ) ≤ π ) ) |
5 |
3 4
|
syl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( - π < ( ℑ ‘ ( log ‘ 𝐴 ) ) ∧ ( ℑ ‘ ( log ‘ 𝐴 ) ) ≤ π ) ) |