Metamath Proof Explorer
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 ) ) ) |