Step |
Hyp |
Ref |
Expression |
1 |
|
rnghommul.1 |
|- G = ( 1st ` R ) |
2 |
|
rnghommul.2 |
|- X = ran G |
3 |
|
rnghommul.3 |
|- H = ( 2nd ` R ) |
4 |
|
rnghommul.4 |
|- K = ( 2nd ` S ) |
5 |
|
eqid |
|- ( GId ` H ) = ( GId ` H ) |
6 |
|
eqid |
|- ( 1st ` S ) = ( 1st ` S ) |
7 |
|
eqid |
|- ran ( 1st ` S ) = ran ( 1st ` S ) |
8 |
|
eqid |
|- ( GId ` K ) = ( GId ` K ) |
9 |
1 3 2 5 6 4 7 8
|
isrngohom |
|- ( ( R e. RingOps /\ S e. RingOps ) -> ( F e. ( R RngHom S ) <-> ( F : X --> ran ( 1st ` S ) /\ ( F ` ( GId ` H ) ) = ( GId ` K ) /\ A. x e. X A. y e. X ( ( F ` ( x G y ) ) = ( ( F ` x ) ( 1st ` S ) ( F ` y ) ) /\ ( F ` ( x H y ) ) = ( ( F ` x ) K ( F ` y ) ) ) ) ) ) |
10 |
9
|
biimpa |
|- ( ( ( R e. RingOps /\ S e. RingOps ) /\ F e. ( R RngHom S ) ) -> ( F : X --> ran ( 1st ` S ) /\ ( F ` ( GId ` H ) ) = ( GId ` K ) /\ A. x e. X A. y e. X ( ( F ` ( x G y ) ) = ( ( F ` x ) ( 1st ` S ) ( F ` y ) ) /\ ( F ` ( x H y ) ) = ( ( F ` x ) K ( F ` y ) ) ) ) ) |
11 |
10
|
simp3d |
|- ( ( ( R e. RingOps /\ S e. RingOps ) /\ F e. ( R RngHom S ) ) -> A. x e. X A. y e. X ( ( F ` ( x G y ) ) = ( ( F ` x ) ( 1st ` S ) ( F ` y ) ) /\ ( F ` ( x H y ) ) = ( ( F ` x ) K ( F ` y ) ) ) ) |
12 |
11
|
3impa |
|- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> A. x e. X A. y e. X ( ( F ` ( x G y ) ) = ( ( F ` x ) ( 1st ` S ) ( F ` y ) ) /\ ( F ` ( x H y ) ) = ( ( F ` x ) K ( F ` y ) ) ) ) |
13 |
|
simpr |
|- ( ( ( F ` ( x G y ) ) = ( ( F ` x ) ( 1st ` S ) ( F ` y ) ) /\ ( F ` ( x H y ) ) = ( ( F ` x ) K ( F ` y ) ) ) -> ( F ` ( x H y ) ) = ( ( F ` x ) K ( F ` y ) ) ) |
14 |
13
|
2ralimi |
|- ( A. x e. X A. y e. X ( ( F ` ( x G y ) ) = ( ( F ` x ) ( 1st ` S ) ( F ` y ) ) /\ ( F ` ( x H y ) ) = ( ( F ` x ) K ( F ` y ) ) ) -> A. x e. X A. y e. X ( F ` ( x H y ) ) = ( ( F ` x ) K ( F ` y ) ) ) |
15 |
12 14
|
syl |
|- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> A. x e. X A. y e. X ( F ` ( x H y ) ) = ( ( F ` x ) K ( F ` y ) ) ) |
16 |
|
fvoveq1 |
|- ( x = A -> ( F ` ( x H y ) ) = ( F ` ( A H y ) ) ) |
17 |
|
fveq2 |
|- ( x = A -> ( F ` x ) = ( F ` A ) ) |
18 |
17
|
oveq1d |
|- ( x = A -> ( ( F ` x ) K ( F ` y ) ) = ( ( F ` A ) K ( F ` y ) ) ) |
19 |
16 18
|
eqeq12d |
|- ( x = A -> ( ( F ` ( x H y ) ) = ( ( F ` x ) K ( F ` y ) ) <-> ( F ` ( A H y ) ) = ( ( F ` A ) K ( F ` y ) ) ) ) |
20 |
|
oveq2 |
|- ( y = B -> ( A H y ) = ( A H B ) ) |
21 |
20
|
fveq2d |
|- ( y = B -> ( F ` ( A H y ) ) = ( F ` ( A H B ) ) ) |
22 |
|
fveq2 |
|- ( y = B -> ( F ` y ) = ( F ` B ) ) |
23 |
22
|
oveq2d |
|- ( y = B -> ( ( F ` A ) K ( F ` y ) ) = ( ( F ` A ) K ( F ` B ) ) ) |
24 |
21 23
|
eqeq12d |
|- ( y = B -> ( ( F ` ( A H y ) ) = ( ( F ` A ) K ( F ` y ) ) <-> ( F ` ( A H B ) ) = ( ( F ` A ) K ( F ` B ) ) ) ) |
25 |
19 24
|
rspc2v |
|- ( ( A e. X /\ B e. X ) -> ( A. x e. X A. y e. X ( F ` ( x H y ) ) = ( ( F ` x ) K ( F ` y ) ) -> ( F ` ( A H B ) ) = ( ( F ` A ) K ( F ` B ) ) ) ) |
26 |
15 25
|
mpan9 |
|- ( ( ( 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 ) ) ) |