Theorem 0exp

Description: Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004)

	Ref	Expression
Assertion	0exp	⊢ ( 𝑁 ∈ ℕ → ( 0 ↑ 𝑁 ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ 0 = 0
2		0cn	⊢ 0 ∈ ℂ
3		expeq0	⊢ ( ( 0 ∈ ℂ ∧ 𝑁 ∈ ℕ ) → ( ( 0 ↑ 𝑁 ) = 0 ↔ 0 = 0 ) )
4	2 3	mpan	⊢ ( 𝑁 ∈ ℕ → ( ( 0 ↑ 𝑁 ) = 0 ↔ 0 = 0 ) )
5	1 4	mpbiri	⊢ ( 𝑁 ∈ ℕ → ( 0 ↑ 𝑁 ) = 0 )