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Metamath Proof Explorer

Theorem gimco

Description: The composition of group isomorphisms is a group isomorphism. (Contributed by Mario Carneiro, 21-Apr-2016)

Ref Expression
Assertion gimco 
|- ( ( F e. ( T GrpIso U ) /\ G e. ( S GrpIso T ) ) -> ( F o. G ) e. ( S GrpIso U ) )

Proof

Step Hyp Ref Expression
1 isgim2 
 |-  ( F e. ( T GrpIso U ) <-> ( F e. ( T GrpHom U ) /\ `' F e. ( U GrpHom T ) ) )
2 isgim2 
 |-  ( G e. ( S GrpIso T ) <-> ( G e. ( S GrpHom T ) /\ `' G e. ( T GrpHom S ) ) )
3 ghmco 
 |-  ( ( F e. ( T GrpHom U ) /\ G e. ( S GrpHom T ) ) -> ( F o. G ) e. ( S GrpHom U ) )
4 cnvco 
 |-  `' ( F o. G ) = ( `' G o. `' F )
5 ghmco 
 |-  ( ( `' G e. ( T GrpHom S ) /\ `' F e. ( U GrpHom T ) ) -> ( `' G o. `' F ) e. ( U GrpHom S ) )
6 5 ancoms 
 |-  ( ( `' F e. ( U GrpHom T ) /\ `' G e. ( T GrpHom S ) ) -> ( `' G o. `' F ) e. ( U GrpHom S ) )
7 4 6 eqeltrid 
 |-  ( ( `' F e. ( U GrpHom T ) /\ `' G e. ( T GrpHom S ) ) -> `' ( F o. G ) e. ( U GrpHom S ) )
8 3 7 anim12i 
 |-  ( ( ( F e. ( T GrpHom U ) /\ G e. ( S GrpHom T ) ) /\ ( `' F e. ( U GrpHom T ) /\ `' G e. ( T GrpHom S ) ) ) -> ( ( F o. G ) e. ( S GrpHom U ) /\ `' ( F o. G ) e. ( U GrpHom S ) ) )
9 8 an4s 
 |-  ( ( ( F e. ( T GrpHom U ) /\ `' F e. ( U GrpHom T ) ) /\ ( G e. ( S GrpHom T ) /\ `' G e. ( T GrpHom S ) ) ) -> ( ( F o. G ) e. ( S GrpHom U ) /\ `' ( F o. G ) e. ( U GrpHom S ) ) )
10 1 2 9 syl2anb 
 |-  ( ( F e. ( T GrpIso U ) /\ G e. ( S GrpIso T ) ) -> ( ( F o. G ) e. ( S GrpHom U ) /\ `' ( F o. G ) e. ( U GrpHom S ) ) )
11 isgim2 
 |-  ( ( F o. G ) e. ( S GrpIso U ) <-> ( ( F o. G ) e. ( S GrpHom U ) /\ `' ( F o. G ) e. ( U GrpHom S ) ) )
12 10 11 sylibr 
 |-  ( ( F e. ( T GrpIso U ) /\ G e. ( S GrpIso T ) ) -> ( F o. G ) e. ( S GrpIso U ) )