Theorem sqmuli

Description: Distribution of squaring over multiplication. (Contributed by NM, 3-Sep-1999)

	Ref	Expression
Hypotheses	sqval.1	⊢ 𝐴 ∈ ℂ
	sqmul.2	⊢ 𝐵 ∈ ℂ
Assertion	sqmuli	⊢ ( ( 𝐴 · 𝐵 ) ↑ 2 ) = ( ( 𝐴 ↑ 2 ) · ( 𝐵 ↑ 2 ) )

Proof

Step	Hyp	Ref	Expression
1		sqval.1	⊢ 𝐴 ∈ ℂ
2		sqmul.2	⊢ 𝐵 ∈ ℂ
3		sqmul	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) ↑ 2 ) = ( ( 𝐴 ↑ 2 ) · ( 𝐵 ↑ 2 ) ) )
4	1 2 3	mp2an	⊢ ( ( 𝐴 · 𝐵 ) ↑ 2 ) = ( ( 𝐴 ↑ 2 ) · ( 𝐵 ↑ 2 ) )