Theorem ltexp1dd

Description: Raising both sides of 'less than' to the same positive integer preserves ordering. (Contributed by Steven Nguyen, 24-Aug-2023)

	Ref	Expression
Hypotheses	ltexp1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
	ltexp1d.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
	ltexp1d.3	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
	ltexp1dd.4	⊢ ( 𝜑 → 𝐴 < 𝐵 )
Assertion	ltexp1dd	⊢ ( 𝜑 → ( 𝐴 ↑ 𝑁 ) < ( 𝐵 ↑ 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		ltexp1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
2		ltexp1d.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
3		ltexp1d.3	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
4		ltexp1dd.4	⊢ ( 𝜑 → 𝐴 < 𝐵 )
5	1 2 3	ltexp1d	⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ ( 𝐴 ↑ 𝑁 ) < ( 𝐵 ↑ 𝑁 ) ) )
6	4 5	mpbid	⊢ ( 𝜑 → ( 𝐴 ↑ 𝑁 ) < ( 𝐵 ↑ 𝑁 ) )