Metamath Proof Explorer


Theorem lmimlmhm

Description: An isomorphism of modules is a homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015)

Ref Expression
Assertion lmimlmhm
|- ( F e. ( R LMIso S ) -> F e. ( R LMHom S ) )

Proof

Step Hyp Ref Expression
1 eqid
 |-  ( Base ` R ) = ( Base ` R )
2 eqid
 |-  ( Base ` S ) = ( Base ` S )
3 1 2 islmim
 |-  ( F e. ( R LMIso S ) <-> ( F e. ( R LMHom S ) /\ F : ( Base ` R ) -1-1-onto-> ( Base ` S ) ) )
4 3 simplbi
 |-  ( F e. ( R LMIso S ) -> F e. ( R LMHom S ) )