Step |
Hyp |
Ref |
Expression |
1 |
|
ghmgrp1 |
|- ( F e. ( S GrpHom T ) -> S e. Grp ) |
2 |
|
grpmnd |
|- ( S e. Grp -> S e. Mnd ) |
3 |
1 2
|
syl |
|- ( F e. ( S GrpHom T ) -> S e. Mnd ) |
4 |
|
ghmgrp2 |
|- ( F e. ( S GrpHom T ) -> T e. Grp ) |
5 |
|
grpmnd |
|- ( T e. Grp -> T e. Mnd ) |
6 |
4 5
|
syl |
|- ( F e. ( S GrpHom T ) -> T e. Mnd ) |
7 |
|
eqid |
|- ( Base ` S ) = ( Base ` S ) |
8 |
|
eqid |
|- ( Base ` T ) = ( Base ` T ) |
9 |
7 8
|
ghmf |
|- ( F e. ( S GrpHom T ) -> F : ( Base ` S ) --> ( Base ` T ) ) |
10 |
|
eqid |
|- ( +g ` S ) = ( +g ` S ) |
11 |
|
eqid |
|- ( +g ` T ) = ( +g ` T ) |
12 |
7 10 11
|
ghmlin |
|- ( ( F e. ( S GrpHom T ) /\ x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) ) |
13 |
12
|
3expb |
|- ( ( F e. ( S GrpHom T ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) ) -> ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) ) |
14 |
13
|
ralrimivva |
|- ( F e. ( S GrpHom T ) -> A. x e. ( Base ` S ) A. y e. ( Base ` S ) ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) ) |
15 |
|
eqid |
|- ( 0g ` S ) = ( 0g ` S ) |
16 |
|
eqid |
|- ( 0g ` T ) = ( 0g ` T ) |
17 |
15 16
|
ghmid |
|- ( F e. ( S GrpHom T ) -> ( F ` ( 0g ` S ) ) = ( 0g ` T ) ) |
18 |
9 14 17
|
3jca |
|- ( F e. ( S GrpHom T ) -> ( F : ( Base ` S ) --> ( Base ` T ) /\ A. x e. ( Base ` S ) A. y e. ( Base ` S ) ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) /\ ( F ` ( 0g ` S ) ) = ( 0g ` T ) ) ) |
19 |
7 8 10 11 15 16
|
ismhm |
|- ( F e. ( S MndHom T ) <-> ( ( S e. Mnd /\ T e. Mnd ) /\ ( F : ( Base ` S ) --> ( Base ` T ) /\ A. x e. ( Base ` S ) A. y e. ( Base ` S ) ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) /\ ( F ` ( 0g ` S ) ) = ( 0g ` T ) ) ) ) |
20 |
3 6 18 19
|
syl21anbrc |
|- ( F e. ( S GrpHom T ) -> F e. ( S MndHom T ) ) |