Metamath Proof Explorer


Theorem sqabssubi

Description: Square of absolute value of difference. (Contributed by Steve Rodriguez, 20-Jan-2007)

Ref Expression
Hypotheses absvalsqi.1 𝐴 ∈ ℂ
abssub.2 𝐵 ∈ ℂ
Assertion sqabssubi ( ( abs ‘ ( 𝐴𝐵 ) ) ↑ 2 ) = ( ( ( ( abs ‘ 𝐴 ) ↑ 2 ) + ( ( abs ‘ 𝐵 ) ↑ 2 ) ) − ( 2 · ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐵 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 absvalsqi.1 𝐴 ∈ ℂ
2 abssub.2 𝐵 ∈ ℂ
3 sqabssub ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( abs ‘ ( 𝐴𝐵 ) ) ↑ 2 ) = ( ( ( ( abs ‘ 𝐴 ) ↑ 2 ) + ( ( abs ‘ 𝐵 ) ↑ 2 ) ) − ( 2 · ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐵 ) ) ) ) ) )
4 1 2 3 mp2an ( ( abs ‘ ( 𝐴𝐵 ) ) ↑ 2 ) = ( ( ( ( abs ‘ 𝐴 ) ↑ 2 ) + ( ( abs ‘ 𝐵 ) ↑ 2 ) ) − ( 2 · ( ℜ ‘ ( 𝐴 · ( ∗ ‘ 𝐵 ) ) ) ) )