Theorem exp0

Description: Value of a complex number raised to the 0th power. Note that under our definition, 0 ^ 0 = 1 , following the convention used by Gleason. Part of Definition 10-4.1 of Gleason p. 134. (Contributed by NM, 20-May-2004) (Revised by Mario Carneiro, 4-Jun-2014)

	Ref	Expression
Assertion	exp0	$$⊢A\in ℂ\to A^0=1$$

Proof

Step	Hyp	Ref	Expression
1		0z	$$⊢0\in \mathbb{Z}$$
2		expval	$$⊢\left(A\in ℂ\wedge 0\in \mathbb{Z}\right)\to A^0=if\left(0=0,1,if\left(0<0,seq1\left(\times ,\left(\mathbb{N}\times \left\{A\right\}\right)\right)\left(0\right),\frac{1}{seq1\left(\times ,\left(\mathbb{N}\times \left\{A\right\}\right)\right)\left(-0\right)}\right)\right)$$
3	1 2	mpan2	$$⊢A\in ℂ\to A^0=if\left(0=0,1,if\left(0<0,seq1\left(\times ,\left(\mathbb{N}\times \left\{A\right\}\right)\right)\left(0\right),\frac{1}{seq1\left(\times ,\left(\mathbb{N}\times \left\{A\right\}\right)\right)\left(-0\right)}\right)\right)$$
4		eqid	$$⊢0=0$$
5	4	iftruei	$$⊢if\left(0=0,1,if\left(0<0,seq1\left(\times ,\left(\mathbb{N}\times \left\{A\right\}\right)\right)\left(0\right),\frac{1}{seq1\left(\times ,\left(\mathbb{N}\times \left\{A\right\}\right)\right)\left(-0\right)}\right)\right)=1$$
6	3 5	syl6eq	$$⊢A\in ℂ\to A^0=1$$