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Metamath Proof Explorer

Theorem ghmpropd

Description: Group homomorphism depends only on the group attributes of structures. (Contributed by Mario Carneiro, 12-Jun-2015)

Ref Expression
Hypotheses ghmpropd.a 
|- ( ph -> B = ( Base ` J ) )
ghmpropd.b 
|- ( ph -> C = ( Base ` K ) )
ghmpropd.c 
|- ( ph -> B = ( Base ` L ) )
ghmpropd.d 
|- ( ph -> C = ( Base ` M ) )
ghmpropd.e 
|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` J ) y ) = ( x ( +g ` L ) y ) )
ghmpropd.f 
|- ( ( ph /\ ( x e. C /\ y e. C ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` M ) y ) )
Assertion ghmpropd 
|- ( ph -> ( J GrpHom K ) = ( L GrpHom M ) )

Proof

Step Hyp Ref Expression
1 ghmpropd.a 
 |-  ( ph -> B = ( Base ` J ) )
2 ghmpropd.b 
 |-  ( ph -> C = ( Base ` K ) )
3 ghmpropd.c 
 |-  ( ph -> B = ( Base ` L ) )
4 ghmpropd.d 
 |-  ( ph -> C = ( Base ` M ) )
5 ghmpropd.e 
 |-  ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` J ) y ) = ( x ( +g ` L ) y ) )
6 ghmpropd.f 
 |-  ( ( ph /\ ( x e. C /\ y e. C ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` M ) y ) )
7 1 3 5 grppropd 
 |-  ( ph -> ( J e. Grp <-> L e. Grp ) )
8 2 4 6 grppropd 
 |-  ( ph -> ( K e. Grp <-> M e. Grp ) )
9 7 8 anbi12d 
 |-  ( ph -> ( ( J e. Grp /\ K e. Grp ) <-> ( L e. Grp /\ M e. Grp ) ) )
10 1 2 3 4 5 6 mhmpropd 
 |-  ( ph -> ( J MndHom K ) = ( L MndHom M ) )
11 10 eleq2d 
 |-  ( ph -> ( f e. ( J MndHom K ) <-> f e. ( L MndHom M ) ) )
12 9 11 anbi12d 
 |-  ( ph -> ( ( ( J e. Grp /\ K e. Grp ) /\ f e. ( J MndHom K ) ) <-> ( ( L e. Grp /\ M e. Grp ) /\ f e. ( L MndHom M ) ) ) )
13 ghmgrp1 
 |-  ( f e. ( J GrpHom K ) -> J e. Grp )
14 ghmgrp2 
 |-  ( f e. ( J GrpHom K ) -> K e. Grp )
15 13 14 jca 
 |-  ( f e. ( J GrpHom K ) -> ( J e. Grp /\ K e. Grp ) )
16 ghmmhmb 
 |-  ( ( J e. Grp /\ K e. Grp ) -> ( J GrpHom K ) = ( J MndHom K ) )
17 16 eleq2d 
 |-  ( ( J e. Grp /\ K e. Grp ) -> ( f e. ( J GrpHom K ) <-> f e. ( J MndHom K ) ) )
18 15 17 biadanii 
 |-  ( f e. ( J GrpHom K ) <-> ( ( J e. Grp /\ K e. Grp ) /\ f e. ( J MndHom K ) ) )
19 ghmgrp1 
 |-  ( f e. ( L GrpHom M ) -> L e. Grp )
20 ghmgrp2 
 |-  ( f e. ( L GrpHom M ) -> M e. Grp )
21 19 20 jca 
 |-  ( f e. ( L GrpHom M ) -> ( L e. Grp /\ M e. Grp ) )
22 ghmmhmb 
 |-  ( ( L e. Grp /\ M e. Grp ) -> ( L GrpHom M ) = ( L MndHom M ) )
23 22 eleq2d 
 |-  ( ( L e. Grp /\ M e. Grp ) -> ( f e. ( L GrpHom M ) <-> f e. ( L MndHom M ) ) )
24 21 23 biadanii 
 |-  ( f e. ( L GrpHom M ) <-> ( ( L e. Grp /\ M e. Grp ) /\ f e. ( L MndHom M ) ) )
25 12 18 24 3bitr4g 
 |-  ( ph -> ( f e. ( J GrpHom K ) <-> f e. ( L GrpHom M ) ) )
26 25 eqrdv 
 |-  ( ph -> ( J GrpHom K ) = ( L GrpHom M ) )