Theorem gicrcl

Description: Isomorphism implies the right side is a group. (Contributed by Mario Carneiro, 6-May-2015)

	Ref	Expression
Assertion	gicrcl	|- ( R ~=g S -> S e. Grp )

Proof

Step	Hyp	Ref	Expression
1		brgic	|- ( R ~=g S <-> ( R GrpIso S ) =/= (/) )
2		n0	|- ( ( R GrpIso S ) =/= (/) <-> E. f f e. ( R GrpIso S ) )
3	1 2	bitri	|- ( R ~=g S <-> E. f f e. ( R GrpIso S ) )
4		gimghm	|- ( f e. ( R GrpIso S ) -> f e. ( R GrpHom S ) )
5		ghmgrp2	|- ( f e. ( R GrpHom S ) -> S e. Grp )
6	4 5	syl	|- ( f e. ( R GrpIso S ) -> S e. Grp )
7	6	exlimiv	|- ( E. f f e. ( R GrpIso S ) -> S e. Grp )
8	3 7	sylbi	|- ( R ~=g S -> S e. Grp )