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Metamath Proof Explorer

Theorem ltexp2rd

Description: The power of a positive number less 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 ) ) )