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 ) ) |