Theorem oppggic

Description: Every group is (naturally) isomorphic to its opposite. (Contributed by Stefan O'Rear, 26-Aug-2015)

	Ref	Expression
Hypothesis	oppggic.o	⊢ 𝑂 = ( oppg ‘ 𝐺 )
Assertion	oppggic	⊢ ( 𝐺 ∈ Grp → 𝐺 ≃𝑔 𝑂 )

Proof

Step	Hyp	Ref	Expression
1		oppggic.o	⊢ 𝑂 = ( oppg ‘ 𝐺 )
2		eqid	⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 )
3	1 2	invoppggim	⊢ ( 𝐺 ∈ Grp → ( invg ‘ 𝐺 ) ∈ ( 𝐺 GrpIso 𝑂 ) )
4		brgici	⊢ ( ( invg ‘ 𝐺 ) ∈ ( 𝐺 GrpIso 𝑂 ) → 𝐺 ≃𝑔 𝑂 )
5	3 4	syl	⊢ ( 𝐺 ∈ Grp → 𝐺 ≃𝑔 𝑂 )