Theorem nnge2recfl0

Description: The floor of the reciprocal of an integer greater than 1 is 0. (Contributed by AV, 10-Apr-2026)

	Ref	Expression
Assertion	nnge2recfl0	$$⊢N\in {\mathbb{Z}}_{\ge 2}\to ⌊\frac{1}{N}⌋=0$$

Proof

Step	Hyp	Ref	Expression
1		nnge2recico01	$$⊢N\in {\mathbb{Z}}_{\ge 2}\to \frac{1}{N}\in \left[0,1\right)$$
2		ico01fl0	$$⊢\frac{1}{N}\in \left[0,1\right)\to ⌊\frac{1}{N}⌋=0$$
3	1 2	syl	$$⊢N\in {\mathbb{Z}}_{\ge 2}\to ⌊\frac{1}{N}⌋=0$$