Theorem leexp2rd

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

	Ref	Expression
Hypotheses	sqgt0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	leexp2rd.2	⊢ ( 𝜑 → 𝑀 ∈ ℕ0 )
	leexp2rd.3	⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
	leexp2rd.4	⊢ ( 𝜑 → 0 ≤ 𝐴 )
	leexp2rd.5	⊢ ( 𝜑 → 𝐴 ≤ 1 )
Assertion	leexp2rd	⊢ ( 𝜑 → ( 𝐴 ↑ 𝑁 ) ≤ ( 𝐴 ↑ 𝑀 ) )

Proof

Step	Hyp	Ref	Expression
1		sqgt0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		leexp2rd.2	⊢ ( 𝜑 → 𝑀 ∈ ℕ0 )
3		leexp2rd.3	⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
4		leexp2rd.4	⊢ ( 𝜑 → 0 ≤ 𝐴 )
5		leexp2rd.5	⊢ ( 𝜑 → 𝐴 ≤ 1 )
6		leexp2r	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) → ( 𝐴 ↑ 𝑁 ) ≤ ( 𝐴 ↑ 𝑀 ) )
7	1 2 3 4 5 6	syl32anc	⊢ ( 𝜑 → ( 𝐴 ↑ 𝑁 ) ≤ ( 𝐴 ↑ 𝑀 ) )