Theorem vscandxnplusgndx

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

	Ref	Expression
Assertion	vscandxnplusgndx	$$⊢{\cdot }_{\mathrm{ndx}}\ne {+}_{\mathrm{ndx}}$$

Proof

Step	Hyp	Ref	Expression
1		2re	$$⊢2\in ℝ$$
2		2lt6	$$⊢2<6$$
3	1 2	gtneii	$$⊢6\ne 2$$
4		vscandx	$$⊢{\cdot }_{\mathrm{ndx}}=6$$
5		plusgndx	$$⊢{+}_{\mathrm{ndx}}=2$$
6	4 5	neeq12i	$$⊢{\cdot }_{\mathrm{ndx}}\ne {+}_{\mathrm{ndx}}\leftrightarrow 6\ne 2$$
7	3 6	mpbir	$$⊢{\cdot }_{\mathrm{ndx}}\ne {+}_{\mathrm{ndx}}$$