Theorem scandxnplusgndx

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

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

Proof

Step	Hyp	Ref	Expression
1		2re	$$⊢2\in ℝ$$
2		2lt5	$$⊢2<5$$
3	1 2	gtneii	$$⊢5\ne 2$$
4		scandx	$$⊢\mathrm{Scalar}\left(\mathrm{ndx}\right)=5$$
5		plusgndx	$$⊢{+}_{\mathrm{ndx}}=2$$
6	4 5	neeq12i	$$⊢\mathrm{Scalar}\left(\mathrm{ndx}\right)\ne {+}_{\mathrm{ndx}}\leftrightarrow 5\ne 2$$
7	3 6	mpbir	$$⊢\mathrm{Scalar}\left(\mathrm{ndx}\right)\ne {+}_{\mathrm{ndx}}$$