Theorem oppglt

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

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

Proof

Step	Hyp	Ref	Expression
1		oppglt.1	$$⊢O={opp}_{𝑔}\left(R\right)$$
2		oppglt.2	$$⊢\underset{\dot{}}{<}={<}_R$$
3		eqid	$$⊢{\le }_R={\le }_R$$
4	3 2	pltfval	$$⊢R\in V\to \underset{\dot{}}{<}={\le }_R\setminus \mathrm{I}$$
5	1	fvexi	$$⊢O\in \mathrm{V}$$
6	1 3	oppgle	$$⊢{\le }_R={\le }_O$$
7		eqid	$$⊢{<}_O={<}_O$$
8	6 7	pltfval	$$⊢O\in \mathrm{V}\to {<}_O={\le }_R\setminus \mathrm{I}$$
9	5 8	ax-mp	$$⊢{<}_O={\le }_R\setminus \mathrm{I}$$
10	4 9	eqtr4di	$$⊢R\in V\to \underset{\dot{}}{<}={<}_O$$