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Metamath Proof Explorer

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 ) )