Theorem plendxnmulrndx

Description: The slot for the "less than or equal to" ordering is not the slot for the ring multiplication operation in an extensible structure. Formerly part of proof for opsrmulr . (Contributed by AV, 1-Nov-2024)

	Ref	Expression
Assertion	plendxnmulrndx	$$⊢{\le }_{\mathrm{ndx}}\ne {\cdot }_{\mathrm{ndx}}$$

Proof

Step	Hyp	Ref	Expression
1		3re	$$⊢3\in ℝ$$
2		3lt10	$$⊢3<10$$
3	1 2	gtneii	$$⊢10\ne 3$$
4		plendx	$$⊢{\le }_{\mathrm{ndx}}=10$$
5		mulrndx	$$⊢{\cdot }_{\mathrm{ndx}}=3$$
6	4 5	neeq12i	$$⊢{\le }_{\mathrm{ndx}}\ne {\cdot }_{\mathrm{ndx}}\leftrightarrow 10\ne 3$$
7	3 6	mpbir	$$⊢{\le }_{\mathrm{ndx}}\ne {\cdot }_{\mathrm{ndx}}$$