Metamath Proof Explorer
Description: Square of value of absolute value function. (Contributed by NM, 2Oct1999)


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

axmp 
⊢ ( ( abs ‘ 𝐴 ) ↑ 2 ) = ( ( ( ℜ ‘ 𝐴 ) ↑ 2 ) + ( ( ℑ ‘ 𝐴 ) ↑ 2 ) ) 