Theorem cxpmuld

Description: Product of exponents law for complex exponentiation. Proposition 10-4.2(b) of Gleason p. 135. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	rpcxpcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
	rpcxpcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	cxpmuld.4	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
Assertion	cxpmuld	⊢ ( 𝜑 → ( 𝐴 ↑𝑐 ( 𝐵 · 𝐶 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) ↑𝑐 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		rpcxpcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
2		rpcxpcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		cxpmuld.4	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		cxpmul	⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 ↑𝑐 ( 𝐵 · 𝐶 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) ↑𝑐 𝐶 ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( 𝐴 ↑𝑐 ( 𝐵 · 𝐶 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) ↑𝑐 𝐶 ) )