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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