Theorem vscandxnscandx

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

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

Proof

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