Theorem slotstnscsi

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

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

Proof

Step	Hyp	Ref	Expression
1		5re	$$⊢5\in ℝ$$
2		5lt9	$$⊢5<9$$
3	1 2	gtneii	$$⊢9\ne 5$$
4		tsetndx	$$⊢\mathrm{TopSet}\left(\mathrm{ndx}\right)=9$$
5		scandx	$$⊢\mathrm{Scalar}\left(\mathrm{ndx}\right)=5$$
6	4 5	neeq12i	$$⊢\mathrm{TopSet}\left(\mathrm{ndx}\right)\ne \mathrm{Scalar}\left(\mathrm{ndx}\right)\leftrightarrow 9\ne 5$$
7	3 6	mpbir	$$⊢\mathrm{TopSet}\left(\mathrm{ndx}\right)\ne \mathrm{Scalar}\left(\mathrm{ndx}\right)$$
8		6re	$$⊢6\in ℝ$$
9		6lt9	$$⊢6<9$$
10	8 9	gtneii	$$⊢9\ne 6$$
11		vscandx	$$⊢{\cdot }_{\mathrm{ndx}}=6$$
12	4 11	neeq12i	$$⊢\mathrm{TopSet}\left(\mathrm{ndx}\right)\ne {\cdot }_{\mathrm{ndx}}\leftrightarrow 9\ne 6$$
13	10 12	mpbir	$$⊢\mathrm{TopSet}\left(\mathrm{ndx}\right)\ne {\cdot }_{\mathrm{ndx}}$$
14		8re	$$⊢8\in ℝ$$
15		8lt9	$$⊢8<9$$
16	14 15	gtneii	$$⊢9\ne 8$$
17		ipndx	$$⊢{\cdot }_{𝑖}\left(\mathrm{ndx}\right)=8$$
18	4 17	neeq12i	$$⊢\mathrm{TopSet}\left(\mathrm{ndx}\right)\ne {\cdot }_{𝑖}\left(\mathrm{ndx}\right)\leftrightarrow 9\ne 8$$
19	16 18	mpbir	$$⊢\mathrm{TopSet}\left(\mathrm{ndx}\right)\ne {\cdot }_{𝑖}\left(\mathrm{ndx}\right)$$
20	7 13 19	3pm3.2i	$$⊢\left(\mathrm{TopSet}\left(\mathrm{ndx}\right)\ne \mathrm{Scalar}\left(\mathrm{ndx}\right)\wedge \mathrm{TopSet}\left(\mathrm{ndx}\right)\ne {\cdot }_{\mathrm{ndx}}\wedge \mathrm{TopSet}\left(\mathrm{ndx}\right)\ne {\cdot }_{𝑖}\left(\mathrm{ndx}\right)\right)$$