This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )