Theorem sqmul

Description: Distribution of square over multiplication. (Contributed by NM, 21-Mar-2008)

	Ref	Expression
Assertion	sqmul	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) ↑ 2 ) = ( ( 𝐴 ↑ 2 ) · ( 𝐵 ↑ 2 ) ) )

Proof

Step	Hyp	Ref	Expression
1		2nn0	⊢ 2 ∈ ℕ0
2		mulexp	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 2 ∈ ℕ0 ) → ( ( 𝐴 · 𝐵 ) ↑ 2 ) = ( ( 𝐴 ↑ 2 ) · ( 𝐵 ↑ 2 ) ) )
3	1 2	mp3an3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) ↑ 2 ) = ( ( 𝐴 ↑ 2 ) · ( 𝐵 ↑ 2 ) ) )