Theorem oppgle

Description: less-than relation of an opposite group. (Contributed by Thierry Arnoux, 13-Apr-2018)

	Ref	Expression
Hypotheses	oppglt.1	$$⊢O={opp}_{𝑔}\left(R\right)$$
	oppgle.2	$$⊢\underset{\dot{}}{\le }={\le }_R$$
Assertion	oppgle	$$⊢\underset{\dot{}}{\le }={\le }_O$$

Proof

Step	Hyp	Ref	Expression
1		oppglt.1	$$⊢O={opp}_{𝑔}\left(R\right)$$
2		oppgle.2	$$⊢\underset{\dot{}}{\le }={\le }_R$$
3		df-ple	$$⊢\mathrm{le}=\mathrm{Slot}10$$
4		10nn	$$⊢10\in \mathbb{N}$$
5		2re	$$⊢2\in ℝ$$
6		2lt10	$$⊢2<10$$
7	5 6	gtneii	$$⊢10\ne 2$$
8	1 3 4 7	oppglem	$$⊢{\le }_R={\le }_O$$
9	2 8	eqtri	$$⊢\underset{\dot{}}{\le }={\le }_O$$