Theorem sqdivd

Description: Distribution of square over division. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	expcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	mulexpd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	sqdivd.3	⊢ ( 𝜑 → 𝐵 ≠ 0 )
Assertion	sqdivd	⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) ↑ 2 ) = ( ( 𝐴 ↑ 2 ) / ( 𝐵 ↑ 2 ) ) )

Proof

Step	Hyp	Ref	Expression
1		expcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		mulexpd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		sqdivd.3	⊢ ( 𝜑 → 𝐵 ≠ 0 )
4		sqdiv	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 / 𝐵 ) ↑ 2 ) = ( ( 𝐴 ↑ 2 ) / ( 𝐵 ↑ 2 ) ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) ↑ 2 ) = ( ( 𝐴 ↑ 2 ) / ( 𝐵 ↑ 2 ) ) )