Theorem vscandxnbasendx

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

	Ref	Expression
Assertion	vscandxnbasendx	$$⊢{\cdot }_{\mathrm{ndx}}\ne {\mathrm{Base}}_{\mathrm{ndx}}$$

Proof

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