Step |
Hyp |
Ref |
Expression |
1 |
|
recl |
⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ 𝐴 ) ∈ ℝ ) |
2 |
1
|
recnd |
⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ 𝐴 ) ∈ ℂ ) |
3 |
2
|
sqvald |
⊢ ( 𝐴 ∈ ℂ → ( ( ℜ ‘ 𝐴 ) ↑ 2 ) = ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐴 ) ) ) |
4 |
|
imcl |
⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ 𝐴 ) ∈ ℝ ) |
5 |
4
|
recnd |
⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ 𝐴 ) ∈ ℂ ) |
6 |
5
|
sqvald |
⊢ ( 𝐴 ∈ ℂ → ( ( ℑ ‘ 𝐴 ) ↑ 2 ) = ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) |
7 |
3 6
|
oveq12d |
⊢ ( 𝐴 ∈ ℂ → ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐴 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) ) |
8 |
|
ipcnval |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐴 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) ) |
9 |
8
|
anidms |
⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℜ ‘ 𝐴 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℑ ‘ 𝐴 ) ) ) ) |
10 |
|
cjmulrcl |
⊢ ( 𝐴 ∈ ℂ → ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ∈ ℝ ) |
11 |
|
rere |
⊢ ( ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ∈ ℝ → ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) = ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) |
12 |
10 11
|
syl |
⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) = ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) |
13 |
7 9 12
|
3eqtr2rd |
⊢ ( 𝐴 ∈ ℂ → ( 𝐴 · ( ∗ ‘ 𝐴 ) ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) |