Theorem dsndxntsetndx

Description: The slot for the distance function is not the slot for the topology in an extensible structure. Formerly part of proof for tngds . (Contributed by AV, 29-Oct-2024)

	Ref	Expression
Assertion	dsndxntsetndx	$$⊢\mathrm{dist}\left(\mathrm{ndx}\right)\ne \mathrm{TopSet}\left(\mathrm{ndx}\right)$$

Proof

Step	Hyp	Ref	Expression
1		9re	$$⊢9\in ℝ$$
2		1nn	$$⊢1\in \mathbb{N}$$
3		2nn0	$$⊢2\in {\mathbb{N}}_0$$
4		9nn0	$$⊢9\in {\mathbb{N}}_0$$
5		9lt10	$$⊢9<10$$
6	2 3 4 5	declti	$$⊢9<12$$
7	1 6	gtneii	$$⊢12\ne 9$$
8		dsndx	$$⊢\mathrm{dist}\left(\mathrm{ndx}\right)=12$$
9		tsetndx	$$⊢\mathrm{TopSet}\left(\mathrm{ndx}\right)=9$$
10	8 9	neeq12i	$$⊢\mathrm{dist}\left(\mathrm{ndx}\right)\ne \mathrm{TopSet}\left(\mathrm{ndx}\right)\leftrightarrow 12\ne 9$$
11	7 10	mpbir	$$⊢\mathrm{dist}\left(\mathrm{ndx}\right)\ne \mathrm{TopSet}\left(\mathrm{ndx}\right)$$