Theorem cxpmul2d

Description: Product of exponents law for complex exponentiation. Variation on cxpmul with more general conditions on A and B when C is an integer. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	cxp0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	cxpcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	cxpmul2d.4	⊢ ( 𝜑 → 𝐶 ∈ ℕ0 )
Assertion	cxpmul2d	⊢ ( 𝜑 → ( 𝐴 ↑𝑐 ( 𝐵 · 𝐶 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) ↑ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		cxp0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		cxpcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		cxpmul2d.4	⊢ ( 𝜑 → 𝐶 ∈ ℕ0 )
4		cxpmul2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℕ0 ) → ( 𝐴 ↑𝑐 ( 𝐵 · 𝐶 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) ↑ 𝐶 ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( 𝐴 ↑𝑐 ( 𝐵 · 𝐶 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) ↑ 𝐶 ) )