Theorem dsndxnn

Description: The index of the slot for the distance in an extensible structure is a positive integer. Formerly part of proof for tmslem . (Contributed by AV, 28-Oct-2024)

	Ref	Expression
Assertion	dsndxnn	$$⊢\mathrm{dist}\left(\mathrm{ndx}\right)\in \mathbb{N}$$

Proof

Step	Hyp	Ref	Expression
1		dsndx	$$⊢\mathrm{dist}\left(\mathrm{ndx}\right)=12$$
2		1nn0	$$⊢1\in {\mathbb{N}}_0$$
3		2nn	$$⊢2\in \mathbb{N}$$
4	2 3	decnncl	$$⊢12\in \mathbb{N}$$
5	1 4	eqeltri	$$⊢\mathrm{dist}\left(\mathrm{ndx}\right)\in \mathbb{N}$$