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Metamath Proof Explorer

Theorem reexpclzd

Description: Closure of exponentiation of reals. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses rpexpclzd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
rpexpclzd.2 ⊢ ( 𝜑 → 𝐴 ≠ 0 )
rpexpclzd.3 ⊢ ( 𝜑 → 𝑁 ∈ ℤ )
Assertion reexpclzd ( 𝜑 → ( 𝐴 ↑ 𝑁 ) ∈ ℝ )

Proof

Step Hyp Ref Expression
1 rpexpclzd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 rpexpclzd.2 ⊢ ( 𝜑 → 𝐴 ≠ 0 )
3 rpexpclzd.3 ⊢ ( 𝜑 → 𝑁 ∈ ℤ )
4 reexpclz ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ 𝑁 ) ∈ ℝ )
5 1 2 3 4 syl3anc ⊢ ( 𝜑 → ( 𝐴 ↑ 𝑁 ) ∈ ℝ )