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	$$⊢\phi \to A\in {ℝ}^{+}$$
	ltexp1d.2	$$⊢\phi \to B\in {ℝ}^{+}$$
	ltexp1d.3	$$⊢\phi \to N\in \mathbb{N}$$
	ltexp1dd.4	$$⊢\phi \to A<B$$
Assertion	ltexp1dd	$$⊢\phi \to A^N<B^N$$

Proof

Step	Hyp	Ref	Expression
1		ltexp1d.1	$$⊢\phi \to A\in {ℝ}^{+}$$
2		ltexp1d.2	$$⊢\phi \to B\in {ℝ}^{+}$$
3		ltexp1d.3	$$⊢\phi \to N\in \mathbb{N}$$
4		ltexp1dd.4	$$⊢\phi \to A<B$$
5	1 2 3	ltexp1d	$$⊢\phi \to \left(A<B\leftrightarrow A^N<B^N\right)$$
6	4 5	mpbid	$$⊢\phi \to A^N<B^N$$