Step |
Hyp |
Ref |
Expression |
1 |
|
simpl |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> F e. ( R RingHom S ) ) |
2 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
3 |
|
eqid |
|- ( Unit ` R ) = ( Unit ` R ) |
4 |
2 3
|
unitss |
|- ( Unit ` R ) C_ ( Base ` R ) |
5 |
|
simpr |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> A e. ( Unit ` R ) ) |
6 |
4 5
|
sselid |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> A e. ( Base ` R ) ) |
7 |
|
rhmrcl1 |
|- ( F e. ( R RingHom S ) -> R e. Ring ) |
8 |
|
eqid |
|- ( 1r ` R ) = ( 1r ` R ) |
9 |
2 8
|
ringidcl |
|- ( R e. Ring -> ( 1r ` R ) e. ( Base ` R ) ) |
10 |
1 7 9
|
3syl |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> ( 1r ` R ) e. ( Base ` R ) ) |
11 |
|
eqid |
|- ( ||r ` R ) = ( ||r ` R ) |
12 |
|
eqid |
|- ( oppR ` R ) = ( oppR ` R ) |
13 |
|
eqid |
|- ( ||r ` ( oppR ` R ) ) = ( ||r ` ( oppR ` R ) ) |
14 |
3 8 11 12 13
|
isunit |
|- ( A e. ( Unit ` R ) <-> ( A ( ||r ` R ) ( 1r ` R ) /\ A ( ||r ` ( oppR ` R ) ) ( 1r ` R ) ) ) |
15 |
5 14
|
sylib |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> ( A ( ||r ` R ) ( 1r ` R ) /\ A ( ||r ` ( oppR ` R ) ) ( 1r ` R ) ) ) |
16 |
15
|
simpld |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> A ( ||r ` R ) ( 1r ` R ) ) |
17 |
|
eqid |
|- ( ||r ` S ) = ( ||r ` S ) |
18 |
2 11 17
|
rhmdvdsr |
|- ( ( ( F e. ( R RingHom S ) /\ A e. ( Base ` R ) /\ ( 1r ` R ) e. ( Base ` R ) ) /\ A ( ||r ` R ) ( 1r ` R ) ) -> ( F ` A ) ( ||r ` S ) ( F ` ( 1r ` R ) ) ) |
19 |
1 6 10 16 18
|
syl31anc |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> ( F ` A ) ( ||r ` S ) ( F ` ( 1r ` R ) ) ) |
20 |
|
eqid |
|- ( 1r ` S ) = ( 1r ` S ) |
21 |
8 20
|
rhm1 |
|- ( F e. ( R RingHom S ) -> ( F ` ( 1r ` R ) ) = ( 1r ` S ) ) |
22 |
21
|
breq2d |
|- ( F e. ( R RingHom S ) -> ( ( F ` A ) ( ||r ` S ) ( F ` ( 1r ` R ) ) <-> ( F ` A ) ( ||r ` S ) ( 1r ` S ) ) ) |
23 |
22
|
adantr |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> ( ( F ` A ) ( ||r ` S ) ( F ` ( 1r ` R ) ) <-> ( F ` A ) ( ||r ` S ) ( 1r ` S ) ) ) |
24 |
19 23
|
mpbid |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> ( F ` A ) ( ||r ` S ) ( 1r ` S ) ) |
25 |
|
rhmopp |
|- ( F e. ( R RingHom S ) -> F e. ( ( oppR ` R ) RingHom ( oppR ` S ) ) ) |
26 |
25
|
adantr |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> F e. ( ( oppR ` R ) RingHom ( oppR ` S ) ) ) |
27 |
15
|
simprd |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> A ( ||r ` ( oppR ` R ) ) ( 1r ` R ) ) |
28 |
12 2
|
opprbas |
|- ( Base ` R ) = ( Base ` ( oppR ` R ) ) |
29 |
|
eqid |
|- ( ||r ` ( oppR ` S ) ) = ( ||r ` ( oppR ` S ) ) |
30 |
28 13 29
|
rhmdvdsr |
|- ( ( ( F e. ( ( oppR ` R ) RingHom ( oppR ` S ) ) /\ A e. ( Base ` R ) /\ ( 1r ` R ) e. ( Base ` R ) ) /\ A ( ||r ` ( oppR ` R ) ) ( 1r ` R ) ) -> ( F ` A ) ( ||r ` ( oppR ` S ) ) ( F ` ( 1r ` R ) ) ) |
31 |
26 6 10 27 30
|
syl31anc |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> ( F ` A ) ( ||r ` ( oppR ` S ) ) ( F ` ( 1r ` R ) ) ) |
32 |
21
|
breq2d |
|- ( F e. ( R RingHom S ) -> ( ( F ` A ) ( ||r ` ( oppR ` S ) ) ( F ` ( 1r ` R ) ) <-> ( F ` A ) ( ||r ` ( oppR ` S ) ) ( 1r ` S ) ) ) |
33 |
32
|
adantr |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> ( ( F ` A ) ( ||r ` ( oppR ` S ) ) ( F ` ( 1r ` R ) ) <-> ( F ` A ) ( ||r ` ( oppR ` S ) ) ( 1r ` S ) ) ) |
34 |
31 33
|
mpbid |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> ( F ` A ) ( ||r ` ( oppR ` S ) ) ( 1r ` S ) ) |
35 |
|
eqid |
|- ( Unit ` S ) = ( Unit ` S ) |
36 |
|
eqid |
|- ( oppR ` S ) = ( oppR ` S ) |
37 |
35 20 17 36 29
|
isunit |
|- ( ( F ` A ) e. ( Unit ` S ) <-> ( ( F ` A ) ( ||r ` S ) ( 1r ` S ) /\ ( F ` A ) ( ||r ` ( oppR ` S ) ) ( 1r ` S ) ) ) |
38 |
24 34 37
|
sylanbrc |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> ( F ` A ) e. ( Unit ` S ) ) |