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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 → 𝐴 ∈ ℝ )
	reexpcld.2	⊢ ( 𝜑 → 𝑁 ∈ ℕ₀ )
	expge1d.3	⊢ ( 𝜑 → 1 ≤ 𝐴 )
Assertion	expge1d	⊢ ( 𝜑 → 1 ≤ ( 𝐴 ↑ 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		reexpcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		reexpcld.2	⊢ ( 𝜑 → 𝑁 ∈ ℕ₀ )
3		expge1d.3	⊢ ( 𝜑 → 1 ≤ 𝐴 )
4		expge1	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 1 ≤ 𝐴 ) → 1 ≤ ( 𝐴 ↑ 𝑁 ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → 1 ≤ ( 𝐴 ↑ 𝑁 ) )