Step |
Hyp |
Ref |
Expression |
1 |
|
logcl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( log ‘ 𝐴 ) ∈ ℂ ) |
2 |
1
|
recld |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ℜ ‘ ( log ‘ 𝐴 ) ) ∈ ℝ ) |
3 |
|
relogef |
⊢ ( ( ℜ ‘ ( log ‘ 𝐴 ) ) ∈ ℝ → ( log ‘ ( exp ‘ ( ℜ ‘ ( log ‘ 𝐴 ) ) ) ) = ( ℜ ‘ ( log ‘ 𝐴 ) ) ) |
4 |
2 3
|
syl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( log ‘ ( exp ‘ ( ℜ ‘ ( log ‘ 𝐴 ) ) ) ) = ( ℜ ‘ ( log ‘ 𝐴 ) ) ) |
5 |
|
absef |
⊢ ( ( log ‘ 𝐴 ) ∈ ℂ → ( abs ‘ ( exp ‘ ( log ‘ 𝐴 ) ) ) = ( exp ‘ ( ℜ ‘ ( log ‘ 𝐴 ) ) ) ) |
6 |
1 5
|
syl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( abs ‘ ( exp ‘ ( log ‘ 𝐴 ) ) ) = ( exp ‘ ( ℜ ‘ ( log ‘ 𝐴 ) ) ) ) |
7 |
|
eflog |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( exp ‘ ( log ‘ 𝐴 ) ) = 𝐴 ) |
8 |
7
|
fveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( abs ‘ ( exp ‘ ( log ‘ 𝐴 ) ) ) = ( abs ‘ 𝐴 ) ) |
9 |
6 8
|
eqtr3d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( exp ‘ ( ℜ ‘ ( log ‘ 𝐴 ) ) ) = ( abs ‘ 𝐴 ) ) |
10 |
9
|
fveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( log ‘ ( exp ‘ ( ℜ ‘ ( log ‘ 𝐴 ) ) ) ) = ( log ‘ ( abs ‘ 𝐴 ) ) ) |
11 |
4 10
|
eqtr3d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ℜ ‘ ( log ‘ 𝐴 ) ) = ( log ‘ ( abs ‘ 𝐴 ) ) ) |