Theorem cxprecd

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

	Ref	Expression
Hypotheses	rpcxpcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
	cxprecd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
Assertion	cxprecd	⊢ ( 𝜑 → ( ( 1 / 𝐴 ) ↑𝑐 𝐵 ) = ( 1 / ( 𝐴 ↑𝑐 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		rpcxpcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
2		cxprecd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		cxprec	⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ ) → ( ( 1 / 𝐴 ) ↑𝑐 𝐵 ) = ( 1 / ( 𝐴 ↑𝑐 𝐵 ) ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( ( 1 / 𝐴 ) ↑𝑐 𝐵 ) = ( 1 / ( 𝐴 ↑𝑐 𝐵 ) ) )