Metamath Proof Explorer


Theorem exp0

Description: Value of a complex number raised to the 0th power. Note that under our definition, 0 ^ 0 = 1 ( 0exp0e1 ) , 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 A 0 = 1

Proof

Step Hyp Ref Expression
1 0z 0
2 expval A 0 A 0 = if 0 = 0 1 if 0 < 0 seq 1 × × A 0 1 seq 1 × × A 0
3 1 2 mpan2 A A 0 = if 0 = 0 1 if 0 < 0 seq 1 × × A 0 1 seq 1 × × A 0
4 eqid 0 = 0
5 4 iftruei if 0 = 0 1 if 0 < 0 seq 1 × × A 0 1 seq 1 × × A 0 = 1
6 3 5 eqtrdi A A 0 = 1