Theorem scandxnbasendx

Description: The slot for the scalar is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024)

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

Proof

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