Theorem sqabsaddi

Description: Square of absolute value of sum. Proposition 10-3.7(g) of Gleason p. 133. (Contributed by NM, 2-Oct-1999)

	Ref	Expression
Hypotheses	absvalsqi.1	⊢ 𝐴 ∈ ℂ
	abssub.2	⊢ 𝐵 ∈ ℂ
Assertion	sqabsaddi	⊢ ( ( abs ‘ ( 𝐴 + 𝐵 ) ) ↑ 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		sqabsadd	⊢ ( ( 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 ) ) ) ) )