Theorem slotsdifipndx

Description: The slot for the scalar is not the index of other slots. Formerly part of proof for srasca and sravsca . (Contributed by AV, 12-Nov-2024)

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

Proof

Step	Hyp	Ref	Expression
1		6re	$$⊢6\in ℝ$$
2		6lt8	$$⊢6<8$$
3	1 2	ltneii	$$⊢6\ne 8$$
4		vscandx	$$⊢{\cdot }_{\mathrm{ndx}}=6$$
5		ipndx	$$⊢{\cdot }_{𝑖}\left(\mathrm{ndx}\right)=8$$
6	4 5	neeq12i	$$⊢{\cdot }_{\mathrm{ndx}}\ne {\cdot }_{𝑖}\left(\mathrm{ndx}\right)\leftrightarrow 6\ne 8$$
7	3 6	mpbir	$$⊢{\cdot }_{\mathrm{ndx}}\ne {\cdot }_{𝑖}\left(\mathrm{ndx}\right)$$
8		5re	$$⊢5\in ℝ$$
9		5lt8	$$⊢5<8$$
10	8 9	ltneii	$$⊢5\ne 8$$
11		scandx	$$⊢\mathrm{Scalar}\left(\mathrm{ndx}\right)=5$$
12	11 5	neeq12i	$$⊢\mathrm{Scalar}\left(\mathrm{ndx}\right)\ne {\cdot }_{𝑖}\left(\mathrm{ndx}\right)\leftrightarrow 5\ne 8$$
13	10 12	mpbir	$$⊢\mathrm{Scalar}\left(\mathrm{ndx}\right)\ne {\cdot }_{𝑖}\left(\mathrm{ndx}\right)$$
14	7 13	pm3.2i	$$⊢\left({\cdot }_{\mathrm{ndx}}\ne {\cdot }_{𝑖}\left(\mathrm{ndx}\right)\wedge \mathrm{Scalar}\left(\mathrm{ndx}\right)\ne {\cdot }_{𝑖}\left(\mathrm{ndx}\right)\right)$$