Theorem cxpef

Description: Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014)

	Ref	Expression
Assertion	cxpef	$$⊢\left(A\in ℂ\wedge A\ne 0\wedge B\in ℂ\right)\to A^B={\mathrm{e}}^{B\log A}$$

Proof

Step	Hyp	Ref	Expression
1		cxpval	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to A^B=if\left(A=0,if\left(B=0,1,0\right),{\mathrm{e}}^{B\log A}\right)$$
2	1	3adant2	$$⊢\left(A\in ℂ\wedge A\ne 0\wedge B\in ℂ\right)\to A^B=if\left(A=0,if\left(B=0,1,0\right),{\mathrm{e}}^{B\log A}\right)$$
3		simp2	$$⊢\left(A\in ℂ\wedge A\ne 0\wedge B\in ℂ\right)\to A\ne 0$$
4	3	neneqd	$$⊢\left(A\in ℂ\wedge A\ne 0\wedge B\in ℂ\right)\to \neg A=0$$
5	4	iffalsed	$$⊢\left(A\in ℂ\wedge A\ne 0\wedge B\in ℂ\right)\to if\left(A=0,if\left(B=0,1,0\right),{\mathrm{e}}^{B\log A}\right)={\mathrm{e}}^{B\log A}$$
6	2 5	eqtrd	$$⊢\left(A\in ℂ\wedge A\ne 0\wedge B\in ℂ\right)\to A^B={\mathrm{e}}^{B\log A}$$