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Metamath Proof Explorer

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 ) )