Theorem densq

Description: Square commutes with canonical denominator. (Contributed by Stefan O'Rear, 15-Sep-2014)

	Ref	Expression
Assertion	densq	$$⊢A\in \mathbb{Q}\to \mathrm{denom}\left(A^2\right)={\mathrm{denom}\left(A\right)}^2$$

Proof

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