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Metamath Proof Explorer

Theorem rpexpcl

Description: Closure law for integer exponentiation of positive reals. (Contributed by NM, 24-Feb-2008) (Revised by Mario Carneiro, 9-Sep-2014)

Ref Expression
Assertion rpexpcl ( ( 𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ 𝑁 ) ∈ ℝ+ )

Proof

Step Hyp Ref Expression
1 simpl ⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ ) → 𝐴 ∈ ℝ+ )
2 rpne0 ⊢ ( 𝐴 ∈ ℝ+ → 𝐴 ≠ 0 )
3 2 adantr ⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ ) → 𝐴 ≠ 0 )
4 simpr ⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ ) → 𝑁 ∈ ℤ )
5 rpssre ⊢ ℝ+ ⊆ ℝ
6 ax-resscn ⊢ ℝ ⊆ ℂ
7 5 6 sstri ⊢ ℝ+ ⊆ ℂ
8 rpmulcl ⊢ ( ( 𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+ ) → ( 𝑥 · 𝑦 ) ∈ ℝ+ )
9 1rp ⊢ 1 ∈ ℝ+
10 rpreccl ⊢ ( 𝑥 ∈ ℝ+ → ( 1 / 𝑥 ) ∈ ℝ+ )
11 10 adantr ⊢ ( ( 𝑥 ∈ ℝ+ ∧ 𝑥 ≠ 0 ) → ( 1 / 𝑥 ) ∈ ℝ+ )
12 7 8 9 11 expcl2lem ⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ 𝑁 ) ∈ ℝ+ )
13 1 3 4 12 syl3anc ⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ 𝑁 ) ∈ ℝ+ )