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