Step |
Hyp |
Ref |
Expression |
1 |
|
rnghomsub.1 |
|- G = ( 1st ` R ) |
2 |
|
rnghomsub.2 |
|- X = ran G |
3 |
|
rnghomsub.3 |
|- H = ( /g ` G ) |
4 |
|
rnghomsub.4 |
|- J = ( 1st ` S ) |
5 |
|
rnghomsub.5 |
|- K = ( /g ` J ) |
6 |
1
|
rngogrpo |
|- ( R e. RingOps -> G e. GrpOp ) |
7 |
6
|
3ad2ant1 |
|- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> G e. GrpOp ) |
8 |
4
|
rngogrpo |
|- ( S e. RingOps -> J e. GrpOp ) |
9 |
8
|
3ad2ant2 |
|- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> J e. GrpOp ) |
10 |
1 4
|
rngogrphom |
|- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> F e. ( G GrpOpHom J ) ) |
11 |
7 9 10
|
3jca |
|- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( G e. GrpOp /\ J e. GrpOp /\ F e. ( G GrpOpHom J ) ) ) |
12 |
2 3 5
|
ghomdiv |
|- ( ( ( G e. GrpOp /\ J e. GrpOp /\ F e. ( G GrpOpHom J ) ) /\ ( A e. X /\ B e. X ) ) -> ( F ` ( A H B ) ) = ( ( F ` A ) K ( F ` B ) ) ) |
13 |
11 12
|
sylan |
|- ( ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) /\ ( A e. X /\ B e. X ) ) -> ( F ` ( A H B ) ) = ( ( F ` A ) K ( F ` B ) ) ) |