This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.
Metamath Proof Explorer
Description: Square of value of absolute value function. (Contributed by Mario
Carneiro, 29-May-2016)
|
|
Ref |
Expression |
|
Hypothesis |
abscld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
|
Assertion |
absvalsq2d |
⊢ ( 𝜑 → ( ( abs ‘ 𝐴 ) ↑ 2 ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
abscld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
| 2 |
|
absvalsq2 |
⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ 𝐴 ) ↑ 2 ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) |
| 3 |
1 2
|
syl |
⊢ ( 𝜑 → ( ( abs ‘ 𝐴 ) ↑ 2 ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) |