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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