Theorem ltexp1d

Description: Elevating to a positive power does not affect inequalities. Similar to ltmul1d for exponentiation of positive reals. (Contributed by Steven Nguyen, 22-Aug-2023)

	Ref	Expression
Hypotheses	ltexp1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
	ltexp1d.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
	ltexp1d.3	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
Assertion	ltexp1d	⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ ( 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		rpexpmord	⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐴 < 𝐵 ↔ ( 𝐴 ↑ 𝑁 ) < ( 𝐵 ↑ 𝑁 ) ) )
5	3 1 2 4	syl3anc	⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ ( 𝐴 ↑ 𝑁 ) < ( 𝐵 ↑ 𝑁 ) ) )