Theorem expdivd

Description: Nonnegative integer exponentiation of a quotient. (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 )
	expdivd.3	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
Assertion	expdivd	⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) ↑ 𝑁 ) = ( ( 𝐴 ↑ 𝑁 ) / ( 𝐵 ↑ 𝑁 ) ) )

Proof

Step	Hyp	Ref	Expression
1		expcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		mulexpd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		sqdivd.3	⊢ ( 𝜑 → 𝐵 ≠ 0 )
4		expdivd.3	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
5		expdiv	⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝑁 ∈ ℕ0 ) → ( ( 𝐴 / 𝐵 ) ↑ 𝑁 ) = ( ( 𝐴 ↑ 𝑁 ) / ( 𝐵 ↑ 𝑁 ) ) )
6	1 2 3 4 5	syl121anc	⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) ↑ 𝑁 ) = ( ( 𝐴 ↑ 𝑁 ) / ( 𝐵 ↑ 𝑁 ) ) )