Theorem qgt0numnn

Description: A rational is positive iff its canonical numerator is a positive integer. (Contributed by Stefan O'Rear, 15-Sep-2014)

	Ref	Expression
Assertion	qgt0numnn	$$⊢\left(A\in \mathbb{Q}\wedge 0<A\right)\to \mathrm{numer}\left(A\right)\in \mathbb{N}$$

Proof

Step	Hyp	Ref	Expression
1		qnumcl	$$⊢A\in \mathbb{Q}\to \mathrm{numer}\left(A\right)\in \mathbb{Z}$$
2	1	adantr	$$⊢\left(A\in \mathbb{Q}\wedge 0<A\right)\to \mathrm{numer}\left(A\right)\in \mathbb{Z}$$
3		qnumgt0	$$⊢A\in \mathbb{Q}\to \left(0<A\leftrightarrow 0<\mathrm{numer}\left(A\right)\right)$$
4	3	biimpa	$$⊢\left(A\in \mathbb{Q}\wedge 0<A\right)\to 0<\mathrm{numer}\left(A\right)$$
5		elnnz	$$⊢\mathrm{numer}\left(A\right)\in \mathbb{N}\leftrightarrow \left(\mathrm{numer}\left(A\right)\in \mathbb{Z}\wedge 0<\mathrm{numer}\left(A\right)\right)$$
6	2 4 5	sylanbrc	$$⊢\left(A\in \mathbb{Q}\wedge 0<A\right)\to \mathrm{numer}\left(A\right)\in \mathbb{N}$$