Theorem leexp2ad

Description: Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	sqgt0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	leexp2ad.2	⊢ ( 𝜑 → 1 ≤ 𝐴 )
	leexp2ad.3	⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
Assertion	leexp2ad	⊢ ( 𝜑 → ( 𝐴 ↑ 𝑀 ) ≤ ( 𝐴 ↑ 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		sqgt0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		leexp2ad.2	⊢ ( 𝜑 → 1 ≤ 𝐴 )
3		leexp2ad.3	⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
4		leexp2a	⊢ ( ( 𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( 𝐴 ↑ 𝑀 ) ≤ ( 𝐴 ↑ 𝑁 ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( 𝐴 ↑ 𝑀 ) ≤ ( 𝐴 ↑ 𝑁 ) )