This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem expcld

Description: Closure law for nonnegative integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses expcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
expcld.2 ⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
Assertion expcld ( 𝜑 → ( 𝐴 ↑ 𝑁 ) ∈ ℂ )

Proof

Step Hyp Ref Expression
1 expcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2 expcld.2 ⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
3 expcl ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑁 ) ∈ ℂ )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( 𝐴 ↑ 𝑁 ) ∈ ℂ )