Theorem oppglt

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

	Ref	Expression
Hypotheses	oppglt.1	⊢ 𝑂 = ( oppg ‘ 𝑅 )
	oppglt.2	⊢ < = ( lt ‘ 𝑅 )
Assertion	oppglt	⊢ ( 𝑅 ∈ 𝑉 → < = ( lt ‘ 𝑂 ) )

Proof

Step	Hyp	Ref	Expression
1		oppglt.1	⊢ 𝑂 = ( oppg ‘ 𝑅 )
2		oppglt.2	⊢ < = ( lt ‘ 𝑅 )
3		eqid	⊢ ( le ‘ 𝑅 ) = ( le ‘ 𝑅 )
4	3 2	pltfval	⊢ ( 𝑅 ∈ 𝑉 → < = ( ( le ‘ 𝑅 ) ∖ I ) )
5	1	fvexi	⊢ 𝑂 ∈ V
6	1 3	oppgle	⊢ ( le ‘ 𝑅 ) = ( le ‘ 𝑂 )
7		eqid	⊢ ( lt ‘ 𝑂 ) = ( lt ‘ 𝑂 )
8	6 7	pltfval	⊢ ( 𝑂 ∈ V → ( lt ‘ 𝑂 ) = ( ( le ‘ 𝑅 ) ∖ I ) )
9	5 8	ax-mp	⊢ ( lt ‘ 𝑂 ) = ( ( le ‘ 𝑅 ) ∖ I )
10	4 9	eqtr4di	⊢ ( 𝑅 ∈ 𝑉 → < = ( lt ‘ 𝑂 ) )