Theorem exp0

Description: Value of a complex number raised to the zeroth power. Under our definition, 0 ^ 0 = 1 ( 0exp0e1 ), following standard convention, for instance 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 ∈ ℂ → ( 𝐴 ↑ 0 ) = 1 )

Proof

Step	Hyp	Ref	Expression
1		0z	⊢ 0 ∈ ℤ
2		expval	⊢ ( ( 𝐴 ∈ ℂ ∧ 0 ∈ ℤ ) → ( 𝐴 ↑ 0 ) = if ( 0 = 0 , 1 , if ( 0 < 0 , ( seq 1 ( · , ( ℕ × { 𝐴 } ) ) ‘ 0 ) , ( 1 / ( seq 1 ( · , ( ℕ × { 𝐴 } ) ) ‘ - 0 ) ) ) ) )
3	1 2	mpan2	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ↑ 0 ) = if ( 0 = 0 , 1 , if ( 0 < 0 , ( seq 1 ( · , ( ℕ × { 𝐴 } ) ) ‘ 0 ) , ( 1 / ( seq 1 ( · , ( ℕ × { 𝐴 } ) ) ‘ - 0 ) ) ) ) )
4		eqid	⊢ 0 = 0
5	4	iftruei	⊢ if ( 0 = 0 , 1 , if ( 0 < 0 , ( seq 1 ( · , ( ℕ × { 𝐴 } ) ) ‘ 0 ) , ( 1 / ( seq 1 ( · , ( ℕ × { 𝐴 } ) ) ‘ - 0 ) ) ) ) = 1
6	3 5	eqtrdi	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ↑ 0 ) = 1 )