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		eqid	$$⊢{+}_R={+}_R$$
4	3 1	oppgval	$$⊢O=R\mathrm{sSet}⟨{+}_{\mathrm{ndx}},tpos{+}_R⟩$$
5		pleid	$$⊢\mathrm{le}=\mathrm{Slot}{\le }_{\mathrm{ndx}}$$
6		plendxnplusgndx	$$⊢{\le }_{\mathrm{ndx}}\ne {+}_{\mathrm{ndx}}$$
7	4 5 6	setsplusg	$$⊢{\le }_R={\le }_O$$
8	2 7	eqtri	$$⊢\underset{\dot{}}{\le }={\le }_O$$