Description: Reverse closure for the subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014)
Ref | Expression | ||
---|---|---|---|
Assertion | subgrcl | |- ( S e. ( SubGrp ` G ) -> G e. Grp ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid | |- ( Base ` G ) = ( Base ` G ) |
|
2 | 1 | issubg | |- ( S e. ( SubGrp ` G ) <-> ( G e. Grp /\ S C_ ( Base ` G ) /\ ( G |`s S ) e. Grp ) ) |
3 | 2 | simp1bi | |- ( S e. ( SubGrp ` G ) -> G e. Grp ) |