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 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 )