Theorem plendxnplusgndx

Description: The slot for the "less than or equal to" ordering is not the slot for the group operation in an extensible structure. Formerly part of proof for oppgle . (Contributed by AV, 18-Oct-2024)

	Ref	Expression
Assertion	plendxnplusgndx	$$⊢{\le }_{\mathrm{ndx}}\ne {+}_{\mathrm{ndx}}$$

Proof

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