Theorem ipndxnplusgndx

Description: The slot for the inner product is not the slot for the group operation in an extensible structure. (Contributed by AV, 29-Oct-2024)

	Ref	Expression
Assertion	ipndxnplusgndx	$$⊢{\cdot }_{𝑖}\left(\mathrm{ndx}\right)\ne {+}_{\mathrm{ndx}}$$

Proof

Step	Hyp	Ref	Expression
1		2re	$$⊢2\in ℝ$$
2		2lt8	$$⊢2<8$$
3	1 2	gtneii	$$⊢8\ne 2$$
4		ipndx	$$⊢{\cdot }_{𝑖}\left(\mathrm{ndx}\right)=8$$
5		plusgndx	$$⊢{+}_{\mathrm{ndx}}=2$$
6	4 5	neeq12i	$$⊢{\cdot }_{𝑖}\left(\mathrm{ndx}\right)\ne {+}_{\mathrm{ndx}}\leftrightarrow 8\ne 2$$
7	3 6	mpbir	$$⊢{\cdot }_{𝑖}\left(\mathrm{ndx}\right)\ne {+}_{\mathrm{ndx}}$$