expgt0

Description: A positive real raised to an integer power is positive. (Contributed by NM, 16-Dec-2005) (Revised by Mario Carneiro, 4-Jun-2014)

		Ref	Expression
	Assertion	expgt0	$⊢ (A \in ℝ \land N \in ℤ \land 0 < A) \to 0 < A^{N}$

Step	Hyp	Ref	Expression
1		elrp	$⊢ A \in ℝ^{+} \leftrightarrow (A \in ℝ \land 0 < A)$
2		rpexpcl	$⊢ (A \in ℝ^{+} \land N \in ℤ) \to A^{N} \in ℝ^{+}$
3	2	rpgt0d	$⊢ (A \in ℝ^{+} \land N \in ℤ) \to 0 < A^{N}$
4	1 3	sylanbr	$⊢ ((A \in ℝ \land 0 < A) \land N \in ℤ) \to 0 < A^{N}$
5	4	3impa	$⊢ (A \in ℝ \land 0 < A \land N \in ℤ) \to 0 < A^{N}$
6	5	3com23	$⊢ (A \in ℝ \land N \in ℤ \land 0 < A) \to 0 < A^{N}$

Metamath Proof Explorer