Description: Square of value of absolute value function. (Contributed by NM, 1-Feb-2007)
Ref | Expression | ||
---|---|---|---|
Assertion | absvalsq2 | ⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ 𝐴 ) ↑ 2 ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | absvalsq | ⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ 𝐴 ) ↑ 2 ) = ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) | |
2 | cjmulval | ⊢ ( 𝐴 ∈ ℂ → ( 𝐴 · ( ∗ ‘ 𝐴 ) ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) | |
3 | 1 2 | eqtrd | ⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ 𝐴 ) ↑ 2 ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) ) |