Theorem ipndxnmulrndx

Description: The slot for the inner product is not the slot for the ring (multiplication) operation in an extensible structure. Formerly part of proof for mgpsca . (Contributed by AV, 29-Oct-2024)

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

Proof

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