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Metamath Proof Explorer

Theorem cxpgt0d

Description: A positive real raised to a real power is positive. (Contributed by SN, 6-Apr-2023)

Ref Expression
Hypotheses cxpgt0d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
cxpgt0d.2 ⊢ ( 𝜑 → 𝑁 ∈ ℝ )
Assertion cxpgt0d ( 𝜑 → 0 < ( 𝐴 ↑𝑐 𝑁 ) )

Proof

Step Hyp Ref Expression
1 cxpgt0d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
2 cxpgt0d.2 ⊢ ( 𝜑 → 𝑁 ∈ ℝ )
3 1 2 rpcxpcld ⊢ ( 𝜑 → ( 𝐴 ↑𝑐 𝑁 ) ∈ ℝ+ )
4 3 rpgt0d ⊢ ( 𝜑 → 0 < ( 𝐴 ↑𝑐 𝑁 ) )