# Metamath Proof Explorer

## 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 e. CC -> ( A ^ 0 ) = 1 )`

### Proof

Step Hyp Ref Expression
1 0z
` |-  0 e. ZZ`
2 expval
` |-  ( ( A e. CC /\ 0 e. ZZ ) -> ( A ^ 0 ) = if ( 0 = 0 , 1 , if ( 0 < 0 , ( seq 1 ( x. , ( NN X. { A } ) ) ` 0 ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u 0 ) ) ) ) )`
3 1 2 mpan2
` |-  ( A e. CC -> ( A ^ 0 ) = if ( 0 = 0 , 1 , if ( 0 < 0 , ( seq 1 ( x. , ( NN X. { A } ) ) ` 0 ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u 0 ) ) ) ) )`
4 eqid
` |-  0 = 0`
5 4 iftruei
` |-  if ( 0 = 0 , 1 , if ( 0 < 0 , ( seq 1 ( x. , ( NN X. { A } ) ) ` 0 ) , ( 1 / ( seq 1 ( x. , ( NN X. { A } ) ) ` -u 0 ) ) ) ) = 1`
6 3 5 eqtrdi
` |-  ( A e. CC -> ( A ^ 0 ) = 1 )`