Theorem ipndxnbasendx

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

	Ref	Expression
Assertion	ipndxnbasendx	$$⊢{\cdot }_{𝑖}\left(\mathrm{ndx}\right)\ne {\mathrm{Base}}_{\mathrm{ndx}}$$

Proof

Step	Hyp	Ref	Expression
1		1re	$$⊢1\in ℝ$$
2		1lt8	$$⊢1<8$$
3	1 2	gtneii	$$⊢8\ne 1$$
4		ipndx	$$⊢{\cdot }_{𝑖}\left(\mathrm{ndx}\right)=8$$
5		basendx	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}=1$$
6	4 5	neeq12i	$$⊢{\cdot }_{𝑖}\left(\mathrm{ndx}\right)\ne {\mathrm{Base}}_{\mathrm{ndx}}\leftrightarrow 8\ne 1$$
7	3 6	mpbir	$$⊢{\cdot }_{𝑖}\left(\mathrm{ndx}\right)\ne {\mathrm{Base}}_{\mathrm{ndx}}$$