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Metamath Proof Explorer

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 ) )