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Metamath Proof Explorer

Theorem expge1d

Description: A real greater than or equal to 1 raised to a nonnegative integer is greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	reexpcld.1	$⊢ φ \to A \in ℝ$
	reexpcld.2	$⊢ φ \to N \in ℕ_{0}$
	expge1d.3	$⊢ φ \to 1 \leq A$
Assertion	expge1d	$⊢ φ \to 1 \leq A^{N}$

Proof

Step	Hyp	Ref	Expression
1		reexpcld.1	$⊢ φ \to A \in ℝ$
2		reexpcld.2	$⊢ φ \to N \in ℕ_{0}$
3		expge1d.3	$⊢ φ \to 1 \leq A$
4		expge1	$⊢ (A \in ℝ \land N \in ℕ_{0} \land 1 \leq A) \to 1 \leq A^{N}$
5	1 2 3 4	syl3anc	$⊢ φ \to 1 \leq A^{N}$