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Metamath Proof Explorer
Description: Square of value of absolute value function. (Contributed by NM, 2-Oct-1999)
|
|
Ref |
Expression |
|
Hypothesis |
absvalsqi.1 |
⊢ 𝐴 ∈ ℂ |
|
Assertion |
absvalsq2i |
⊢ ( ( abs ‘ 𝐴 ) ↑ 2 ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
absvalsqi.1 |
⊢ 𝐴 ∈ ℂ |
| 2 |
|
absvalsq2 |
⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ 𝐴 ) ↑ 2 ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) |
| 3 |
1 2
|
ax-mp |
⊢ ( ( abs ‘ 𝐴 ) ↑ 2 ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) |