Metamath Proof Explorer

Theorem rimgim

Description: An isomorphism of rings is an isomorphism of their additive groups. (Contributed by AV, 24-Dec-2019)

Ref Expression
Assertion rimgim 
|- ( F e. ( R RingIso S ) -> F e. ( R GrpIso S ) )

Proof

Step Hyp Ref Expression
1 eqid 
 |-  ( Base ` R ) = ( Base ` R )
2 eqid 
 |-  ( Base ` S ) = ( Base ` S )
3 1 2 rimrhm 
 |-  ( F e. ( R RingIso S ) -> F e. ( R RingHom S ) )
4 rhmghm 
 |-  ( F e. ( R RingHom S ) -> F e. ( R GrpHom S ) )
5 3 4 syl 
 |-  ( F e. ( R RingIso S ) -> F e. ( R GrpHom S ) )
6 1 2 rimf1o 
 |-  ( F e. ( R RingIso S ) -> F : ( Base ` R ) -1-1-onto-> ( Base ` S ) )
7 1 2 isgim 
 |-  ( F e. ( R GrpIso S ) <-> ( F e. ( R GrpHom S ) /\ F : ( Base ` R ) -1-1-onto-> ( Base ` S ) ) )
8 5 6 7 sylanbrc 
 |-  ( F e. ( R RingIso S ) -> F e. ( R GrpIso S ) )