Theorem denexp

Description: Elevating to a nonnegative power commutes with canonical denominator. Similar to densq , extended to nonnegative exponents. (Contributed by Steven Nguyen, 5-Apr-2023)

	Ref	Expression
Assertion	denexp	$$⊢\left(A\in \mathbb{Q}\wedge N\in {\mathbb{N}}_0\right)\to \mathrm{denom}\left(A^N\right)={\mathrm{denom}\left(A\right)}^N$$

Proof

Step	Hyp	Ref	Expression
1		numdenexp	$$⊢\left(A\in \mathbb{Q}\wedge N\in {\mathbb{N}}_0\right)\to \left(\mathrm{numer}\left(A^N\right)={\mathrm{numer}\left(A\right)}^N\wedge \mathrm{denom}\left(A^N\right)={\mathrm{denom}\left(A\right)}^N\right)$$
2	1	simprd	$$⊢\left(A\in \mathbb{Q}\wedge N\in {\mathbb{N}}_0\right)\to \mathrm{denom}\left(A^N\right)={\mathrm{denom}\left(A\right)}^N$$