Theorem tsetndxnmulrndx

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

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

Proof

Step	Hyp	Ref	Expression
1		3re	$$⊢3\in ℝ$$
2		3lt9	$$⊢3<9$$
3	1 2	gtneii	$$⊢9\ne 3$$
4		tsetndx	$$⊢\mathrm{TopSet}\left(\mathrm{ndx}\right)=9$$
5		mulrndx	$$⊢{\cdot }_{\mathrm{ndx}}=3$$
6	4 5	neeq12i	$$⊢\mathrm{TopSet}\left(\mathrm{ndx}\right)\ne {\cdot }_{\mathrm{ndx}}\leftrightarrow 9\ne 3$$
7	3 6	mpbir	$$⊢\mathrm{TopSet}\left(\mathrm{ndx}\right)\ne {\cdot }_{\mathrm{ndx}}$$