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Definition df-rpcxp 13008
 Description: Define the power function on complex numbers. Because df-relog 13007 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.)
Assertion
Ref Expression
df-rpcxp 𝑐 = (𝑥 ∈ ℝ+, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥))))
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-rpcxp
StepHypRef Expression
1 ccxp 13006 . 2 class 𝑐
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 crp 9490 . . 3 class +
5 cc 7662 . . 3 class
63cv 1331 . . . . 5 class 𝑦
72cv 1331 . . . . . 6 class 𝑥
8 clog 13005 . . . . . 6 class log
97, 8cfv 5132 . . . . 5 class (log‘𝑥)
10 cmul 7669 . . . . 5 class ·
116, 9, 10co 5783 . . . 4 class (𝑦 · (log‘𝑥))
12 ce 11405 . . . 4 class exp
1311, 12cfv 5132 . . 3 class (exp‘(𝑦 · (log‘𝑥)))
142, 3, 4, 5, 13cmpo 5785 . 2 class (𝑥 ∈ ℝ+, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥))))
151, 14wceq 1332 1 wff 𝑐 = (𝑥 ∈ ℝ+, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥))))
 Colors of variables: wff set class This definition is referenced by:  rpcxpef  13043
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