Theorem 1cxp

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

	Ref	Expression
Assertion	1cxp	$$⊢A\in ℂ\to 1^A=1$$

Proof

Step	Hyp	Ref	Expression
1		ax-1cn	$$⊢1\in ℂ$$
2		ax-1ne0	$$⊢1\ne 0$$
3		cxpef	$$⊢\left(1\in ℂ\wedge 1\ne 0\wedge A\in ℂ\right)\to 1^A={\mathrm{e}}^{A\log 1}$$
4	1 2 3	mp3an12	$$⊢A\in ℂ\to 1^A={\mathrm{e}}^{A\log 1}$$
5		log1	$$⊢\log 1=0$$
6	5	oveq2i	$$⊢A\log 1=A\cdot 0$$
7		mul01	$$⊢A\in ℂ\to A\cdot 0=0$$
8	6 7	eqtrid	$$⊢A\in ℂ\to A\log 1=0$$
9	8	fveq2d	$$⊢A\in ℂ\to {\mathrm{e}}^{A\log 1}={\mathrm{e}}^0$$
10		ef0	$$⊢{\mathrm{e}}^0=1$$
11	9 10	eqtrdi	$$⊢A\in ℂ\to {\mathrm{e}}^{A\log 1}=1$$
12	4 11	eqtrd	$$⊢A\in ℂ\to 1^A=1$$