Theorem ltexp2rd

Description: The power of a positive number smaller than 1 decreases as its exponent increases. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	rpexpcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
	rpexpcld.2	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
	ltexp2rd.3	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
	ltexp2rd.4	⊢ ( 𝜑 → 𝐴 < 1 )
Assertion	ltexp2rd	⊢ ( 𝜑 → ( 𝑀 < 𝑁 ↔ ( 𝐴 ↑ 𝑁 ) < ( 𝐴 ↑ 𝑀 ) ) )

Proof

Step	Hyp	Ref	Expression
1		rpexpcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
2		rpexpcld.2	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
3		ltexp2rd.3	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
4		ltexp2rd.4	⊢ ( 𝜑 → 𝐴 < 1 )
5		ltexp2r	⊢ ( ( ( 𝐴 ∈ ℝ+ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ 𝐴 < 1 ) → ( 𝑀 < 𝑁 ↔ ( 𝐴 ↑ 𝑁 ) < ( 𝐴 ↑ 𝑀 ) ) )
6	1 3 2 4 5	syl31anc	⊢ ( 𝜑 → ( 𝑀 < 𝑁 ↔ ( 𝐴 ↑ 𝑁 ) < ( 𝐴 ↑ 𝑀 ) ) )