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Metamath Proof Explorer

Theorem divcxpd

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

Ref Expression
Hypotheses recxpcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
recxpcld.2 ⊢ ( 𝜑 → 0 ≤ 𝐴 )
divcxpd.4 ⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
divcxpd.5 ⊢ ( 𝜑 → 𝐶 ∈ ℂ )
Assertion divcxpd ( 𝜑 → ( ( 𝐴 / 𝐵 ) ↑𝑐 𝐶 ) = ( ( 𝐴 ↑𝑐 𝐶 ) / ( 𝐵 ↑𝑐 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 recxpcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 recxpcld.2 ⊢ ( 𝜑 → 0 ≤ 𝐴 )
3 divcxpd.4 ⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
4 divcxpd.5 ⊢ ( 𝜑 → 𝐶 ∈ ℂ )
5 divcxp ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 𝐵 ∈ ℝ+ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 / 𝐵 ) ↑𝑐 𝐶 ) = ( ( 𝐴 ↑𝑐 𝐶 ) / ( 𝐵 ↑𝑐 𝐶 ) ) )
6 1 2 3 4 5 syl211anc ⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) ↑𝑐 𝐶 ) = ( ( 𝐴 ↑𝑐 𝐶 ) / ( 𝐵 ↑𝑐 𝐶 ) ) )