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Metamath Proof Explorer

Theorem leexp2d

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

Ref Expression
Hypotheses sqgt0d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
ltexp2d.2 ⊢ ( 𝜑 → 𝑀 ∈ ℤ )
ltexp2d.3 ⊢ ( 𝜑 → 𝑁 ∈ ℤ )
ltexp2d.4 ⊢ ( 𝜑 → 1 < 𝐴 )
Assertion leexp2d ( 𝜑 → ( 𝑀 ≤ 𝑁 ↔ ( 𝐴 ↑ 𝑀 ) ≤ ( 𝐴 ↑ 𝑁 ) ) )

Proof

Step Hyp Ref Expression
1 sqgt0d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 ltexp2d.2 ⊢ ( 𝜑 → 𝑀 ∈ ℤ )
3 ltexp2d.3 ⊢ ( 𝜑 → 𝑁 ∈ ℤ )
4 ltexp2d.4 ⊢ ( 𝜑 → 1 < 𝐴 )
5 leexp2 ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ 1 < 𝐴 ) → ( 𝑀 ≤ 𝑁 ↔ ( 𝐴 ↑ 𝑀 ) ≤ ( 𝐴 ↑ 𝑁 ) ) )
6 1 2 3 4 5 syl31anc ⊢ ( 𝜑 → ( 𝑀 ≤ 𝑁 ↔ ( 𝐴 ↑ 𝑀 ) ≤ ( 𝐴 ↑ 𝑁 ) ) )