Theorem dsndxnmulrndx

Description: The slot for the distance function is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024)

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

Proof

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