Metamath Proof Explorer

Theorem lmimcnv

Description: The converse of a bijective module homomorphism is a bijective module homomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015) (Revised by Mario Carneiro, 6-May-2015)

Ref Expression
Assertion lmimcnv 
|- ( F e. ( S LMIso T ) -> `' F e. ( T LMIso S ) )

Proof

Step Hyp Ref Expression
1 eqid 
 |-  ( Base ` S ) = ( Base ` S )
2 eqid 
 |-  ( Base ` T ) = ( Base ` T )
3 1 2 lmhmf 
 |-  ( F e. ( S LMHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
4 frel 
 |-  ( F : ( Base ` S ) --> ( Base ` T ) -> Rel F )
5 3 4 syl 
 |-  ( F e. ( S LMHom T ) -> Rel F )
6 dfrel2 
 |-  ( Rel F <-> `' `' F = F )
7 5 6 sylib 
 |-  ( F e. ( S LMHom T ) -> `' `' F = F )
8 id 
 |-  ( F e. ( S LMHom T ) -> F e. ( S LMHom T ) )
9 7 8 eqeltrd 
 |-  ( F e. ( S LMHom T ) -> `' `' F e. ( S LMHom T ) )
10 9 anim1ci 
 |-  ( ( F e. ( S LMHom T ) /\ `' F e. ( T LMHom S ) ) -> ( `' F e. ( T LMHom S ) /\ `' `' F e. ( S LMHom T ) ) )
11 islmim2 
 |-  ( F e. ( S LMIso T ) <-> ( F e. ( S LMHom T ) /\ `' F e. ( T LMHom S ) ) )
12 islmim2 
 |-  ( `' F e. ( T LMIso S ) <-> ( `' F e. ( T LMHom S ) /\ `' `' F e. ( S LMHom T ) ) )
13 10 11 12 3imtr4i 
 |-  ( F e. ( S LMIso T ) -> `' F e. ( T LMIso S ) )