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Metamath Proof Explorer

Theorem cxpmul2d

Description: Product of exponents law for complex exponentiation. Variation on cxpmul with more general conditions on A and B when C is a nonnegative 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 ) )