Theorem dsndxnplusgndx

Description: The slot for the distance function is not the slot for the group operation in an extensible structure. Formerly part of proof for mgpds . (Contributed by AV, 18-Oct-2024)

	Ref	Expression
Assertion	dsndxnplusgndx	$$⊢\mathrm{dist}\left(\mathrm{ndx}\right)\ne {+}_{\mathrm{ndx}}$$

Proof

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