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Metamath Proof Explorer

Theorem rimcnv

Description: The converse of a ring isomorphism is a ring isomorphism. (Contributed by SN, 10-Jan-2025)

Ref Expression
Assertion rimcnv 
|- ( F e. ( R RingIso S ) -> `' F e. ( S RingIso R ) )

Proof

Step Hyp Ref Expression
1 eqid 
 |-  ( Base ` R ) = ( Base ` R )
2 eqid 
 |-  ( Base ` S ) = ( Base ` S )
3 1 2 rhmf 
 |-  ( F e. ( R RingHom S ) -> F : ( Base ` R ) --> ( Base ` S ) )
4 frel 
 |-  ( F : ( Base ` R ) --> ( Base ` S ) -> Rel F )
5 dfrel2 
 |-  ( Rel F <-> `' `' F = F )
6 4 5 sylib 
 |-  ( F : ( Base ` R ) --> ( Base ` S ) -> `' `' F = F )
7 3 6 syl 
 |-  ( F e. ( R RingHom S ) -> `' `' F = F )
8 id 
 |-  ( F e. ( R RingHom S ) -> F e. ( R RingHom S ) )
9 7 8 eqeltrd 
 |-  ( F e. ( R RingHom S ) -> `' `' F e. ( R RingHom S ) )
10 9 anim1ci 
 |-  ( ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) -> ( `' F e. ( S RingHom R ) /\ `' `' F e. ( R RingHom S ) ) )
11 isrim0 
 |-  ( F e. ( R RingIso S ) <-> ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) )
12 isrim0 
 |-  ( `' F e. ( S RingIso R ) <-> ( `' F e. ( S RingHom R ) /\ `' `' F e. ( R RingHom S ) ) )
13 10 11 12 3imtr4i 
 |-  ( F e. ( R RingIso S ) -> `' F e. ( S RingIso R ) )