Theorem oppgle

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

	Ref	Expression
Hypotheses	oppglt.1	⊢ 𝑂 = ( oppg ‘ 𝑅 )
	oppgle.2	⊢ ≤ = ( le ‘ 𝑅 )
Assertion	oppgle	⊢ ≤ = ( le ‘ 𝑂 )

Proof

Step	Hyp	Ref	Expression
1		oppglt.1	⊢ 𝑂 = ( oppg ‘ 𝑅 )
2		oppgle.2	⊢ ≤ = ( le ‘ 𝑅 )
3		eqid	⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 )
4	3 1	oppgval	⊢ 𝑂 = ( 𝑅 sSet ⟨ ( +g ‘ ndx ) , tpos ( +g ‘ 𝑅 ) ⟩ )
5		pleid	⊢ le = Slot ( le ‘ ndx )
6		plendxnplusgndx	⊢ ( le ‘ ndx ) ≠ ( +g ‘ ndx )
7	4 5 6	setsplusg	⊢ ( le ‘ 𝑅 ) = ( le ‘ 𝑂 )
8	2 7	eqtri	⊢ ≤ = ( le ‘ 𝑂 )