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Mirrors > Home > ILE Home > Th. List > df-rpcxp | GIF version |
Description: Define the power function on complex numbers. Because df-relog 13118 is only defined on positive reals, this definition only allows for a base which is a positive real. (Contributed by Jim Kingdon, 12-Jun-2024.) |
Ref | Expression |
---|---|
df-rpcxp | ⊢ ↑𝑐 = (𝑥 ∈ ℝ+, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ccxp 13117 | . 2 class ↑𝑐 | |
2 | vx | . . 3 setvar 𝑥 | |
3 | vy | . . 3 setvar 𝑦 | |
4 | crp 9538 | . . 3 class ℝ+ | |
5 | cc 7709 | . . 3 class ℂ | |
6 | 3 | cv 1331 | . . . . 5 class 𝑦 |
7 | 2 | cv 1331 | . . . . . 6 class 𝑥 |
8 | clog 13116 | . . . . . 6 class log | |
9 | 7, 8 | cfv 5163 | . . . . 5 class (log‘𝑥) |
10 | cmul 7716 | . . . . 5 class · | |
11 | 6, 9, 10 | co 5814 | . . . 4 class (𝑦 · (log‘𝑥)) |
12 | ce 11516 | . . . 4 class exp | |
13 | 11, 12 | cfv 5163 | . . 3 class (exp‘(𝑦 · (log‘𝑥))) |
14 | 2, 3, 4, 5, 13 | cmpo 5816 | . 2 class (𝑥 ∈ ℝ+, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥)))) |
15 | 1, 14 | wceq 1332 | 1 wff ↑𝑐 = (𝑥 ∈ ℝ+, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥)))) |
Colors of variables: wff set class |
This definition is referenced by: rpcxpef 13154 |
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