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Metamath Proof Explorer

Theorem rpexpcld

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

Ref Expression
Hypotheses rpexpcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
rpexpcld.2 ⊢ ( 𝜑 → 𝑁 ∈ ℤ )
Assertion rpexpcld ( 𝜑 → ( 𝐴 ↑ 𝑁 ) ∈ ℝ+ )

Proof

Step Hyp Ref Expression
1 rpexpcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
2 rpexpcld.2 ⊢ ( 𝜑 → 𝑁 ∈ ℤ )
3 rpexpcl ⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ 𝑁 ) ∈ ℝ+ )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( 𝐴 ↑ 𝑁 ) ∈ ℝ+ )