Theorem divcxpd

Description: Complex exponentiation of a quotient. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	recxpcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	recxpcld.2	⊢ ( 𝜑 → 0 ≤ 𝐴 )
	divcxpd.4	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
	divcxpd.5	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
Assertion	divcxpd	⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) ↑𝑐 𝐶 ) = ( ( 𝐴 ↑𝑐 𝐶 ) / ( 𝐵 ↑𝑐 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		recxpcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		recxpcld.2	⊢ ( 𝜑 → 0 ≤ 𝐴 )
3		divcxpd.4	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
4		divcxpd.5	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
5		divcxp	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 𝐵 ∈ ℝ+ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 / 𝐵 ) ↑𝑐 𝐶 ) = ( ( 𝐴 ↑𝑐 𝐶 ) / ( 𝐵 ↑𝑐 𝐶 ) ) )
6	1 2 3 4 5	syl211anc	⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) ↑𝑐 𝐶 ) = ( ( 𝐴 ↑𝑐 𝐶 ) / ( 𝐵 ↑𝑐 𝐶 ) ) )