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Metamath Proof Explorer

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 ) ) )