Step |
Hyp |
Ref |
Expression |
1 |
|
pire |
⊢ π ∈ ℝ |
2 |
1
|
a1i |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → π ∈ ℝ ) |
3 |
2
|
renegcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → - π ∈ ℝ ) |
4 |
|
logcl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( log ‘ 𝐴 ) ∈ ℂ ) |
5 |
4
|
imcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ℑ ‘ ( log ‘ 𝐴 ) ) ∈ ℝ ) |
6 |
|
logimcl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( - π < ( ℑ ‘ ( log ‘ 𝐴 ) ) ∧ ( ℑ ‘ ( log ‘ 𝐴 ) ) ≤ π ) ) |
7 |
6
|
simpld |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → - π < ( ℑ ‘ ( log ‘ 𝐴 ) ) ) |
8 |
3 5 7
|
ltled |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → - π ≤ ( ℑ ‘ ( log ‘ 𝐴 ) ) ) |
9 |
6
|
simprd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ℑ ‘ ( log ‘ 𝐴 ) ) ≤ π ) |
10 |
5 2
|
absled |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ( abs ‘ ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ≤ π ↔ ( - π ≤ ( ℑ ‘ ( log ‘ 𝐴 ) ) ∧ ( ℑ ‘ ( log ‘ 𝐴 ) ) ≤ π ) ) ) |
11 |
8 9 10
|
mpbir2and |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( abs ‘ ( ℑ ‘ ( log ‘ 𝐴 ) ) ) ≤ π ) |