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	$$⊢\phi \to A\in {ℝ}^{+}$$
	ltexp1d.2	$$⊢\phi \to B\in {ℝ}^{+}$$
	ltexp1d.3	$$⊢\phi \to N\in \mathbb{N}$$
Assertion	ltexp1d	$$⊢\phi \to \left(A<B\leftrightarrow A^N<B^N\right)$$

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		rpexpmord	$$⊢\left(N\in \mathbb{N}\wedge A\in {ℝ}^{+}\wedge B\in {ℝ}^{+}\right)\to \left(A<B\leftrightarrow A^N<B^N\right)$$
5	3 1 2 4	syl3anc	$$⊢\phi \to \left(A<B\leftrightarrow A^N<B^N\right)$$