Theorem slotsdnscsi

Description: The slots Scalar , .s and .i are different from the slot dist . Formerly part of sralem and proofs using it. (Contributed by AV, 29-Oct-2024)

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

Proof

Step	Hyp	Ref	Expression
1		5re	$$⊢5\in ℝ$$
2		1nn	$$⊢1\in \mathbb{N}$$
3		2nn0	$$⊢2\in {\mathbb{N}}_0$$
4		5nn0	$$⊢5\in {\mathbb{N}}_0$$
5		5lt10	$$⊢5<10$$
6	2 3 4 5	declti	$$⊢5<12$$
7	1 6	gtneii	$$⊢12\ne 5$$
8		dsndx	$$⊢\mathrm{dist}\left(\mathrm{ndx}\right)=12$$
9		scandx	$$⊢\mathrm{Scalar}\left(\mathrm{ndx}\right)=5$$
10	8 9	neeq12i	$$⊢\mathrm{dist}\left(\mathrm{ndx}\right)\ne \mathrm{Scalar}\left(\mathrm{ndx}\right)\leftrightarrow 12\ne 5$$
11	7 10	mpbir	$$⊢\mathrm{dist}\left(\mathrm{ndx}\right)\ne \mathrm{Scalar}\left(\mathrm{ndx}\right)$$
12		6re	$$⊢6\in ℝ$$
13		6nn0	$$⊢6\in {\mathbb{N}}_0$$
14		6lt10	$$⊢6<10$$
15	2 3 13 14	declti	$$⊢6<12$$
16	12 15	gtneii	$$⊢12\ne 6$$
17		vscandx	$$⊢{\cdot }_{\mathrm{ndx}}=6$$
18	8 17	neeq12i	$$⊢\mathrm{dist}\left(\mathrm{ndx}\right)\ne {\cdot }_{\mathrm{ndx}}\leftrightarrow 12\ne 6$$
19	16 18	mpbir	$$⊢\mathrm{dist}\left(\mathrm{ndx}\right)\ne {\cdot }_{\mathrm{ndx}}$$
20		8re	$$⊢8\in ℝ$$
21		8nn0	$$⊢8\in {\mathbb{N}}_0$$
22		8lt10	$$⊢8<10$$
23	2 3 21 22	declti	$$⊢8<12$$
24	20 23	gtneii	$$⊢12\ne 8$$
25		ipndx	$$⊢{\cdot }_{𝑖}\left(\mathrm{ndx}\right)=8$$
26	8 25	neeq12i	$$⊢\mathrm{dist}\left(\mathrm{ndx}\right)\ne {\cdot }_{𝑖}\left(\mathrm{ndx}\right)\leftrightarrow 12\ne 8$$
27	24 26	mpbir	$$⊢\mathrm{dist}\left(\mathrm{ndx}\right)\ne {\cdot }_{𝑖}\left(\mathrm{ndx}\right)$$
28	11 19 27	3pm3.2i	$$⊢\left(\mathrm{dist}\left(\mathrm{ndx}\right)\ne \mathrm{Scalar}\left(\mathrm{ndx}\right)\wedge \mathrm{dist}\left(\mathrm{ndx}\right)\ne {\cdot }_{\mathrm{ndx}}\wedge \mathrm{dist}\left(\mathrm{ndx}\right)\ne {\cdot }_{𝑖}\left(\mathrm{ndx}\right)\right)$$