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 ) |
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 ) |